Physics Book - User contributions [en]

Understanding Fundamentals of Current, Voltage, and Resistance

2025-04-13T17:04:20Z

Ielsissi3: I added a summary table. I think this section can be of great help to a student, especially before a test, when they need a quick recap/reminder.

'''Ismail Elsissi (ielsissi3) Spring 2025'''

Circuitry are the building blocks of most electronics in our life. They are what keeps our computers running and our physics equipment functioning. Yet what exactly drives circuitry? The core components of a circuit revolves around three aspects: Current, Voltage, and Resistance.

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow of electrons through the circuit, represented by the symbol I and measured in Amperes. Voltage signifies the driving force behind it, or the pressure behind the source of the current. It is represented by the symbol V and measured in Volts. Resistance encapsulates the opposition encountered in the circuit, slowing and resisting the current. It is represented by the symbol R and measured in Ohms. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Quick Summary==

{| class="wikitable"
! '''Concept''' || '''Equation / Definition''' || '''Notes'''
|-
| Current (I) || <math>I = \frac{dQ}{dt}</math> || Flow of charge per unit time (in amperes)
|-
| Voltage (V) || <math>V = IR</math> || Electric potential difference (Ohm’s Law)
|-
| Resistance (R) || <math>R = \frac{V}{I}</math> || How much a component resists current flow
|-
| Power (P) || <math>P = IV = I^2R = \frac{V^2}{R}</math> || Power dissipated or used by a resistor
|-
| Series Circuit || Same current through all; <math>R_\text{eq} = R_1 + R_2 + \cdots</math> || Voltage divides across resistors
|-
| Parallel Circuit || Same voltage across all; <math>\frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots</math> || Current divides across resistors
|-
| Kirchhoff’s Current Law (KCL) || Sum of currents into a junction = sum of currents out || Conservation of charge
|-
| Kirchhoff’s Voltage Law (KVL) || Sum of voltage changes in a loop = 0 || Conservation of energy
|}

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

Understanding Ohm's law explains how our three components effect a circuit. <math>{I = \frac{V}{R}}</math>, the base component of our formula. The equation states Current is the Voltage divided by the Resistance. That is to say, the flow of electricity is equivalent to the driving force of the current divided by the opposition to the current. As we increase the voltage, or the driving force of the current, the current itself increases. On the other hand, if we increase the opposition to the current, the current decreases. As we examine these interactions we can see how the different components of the circuit affect each other.

[[File:OhmsTriangle.jpg]]
 ''Image Source: Kenneth Jenkins''

==Computational Model==

View model here: https://trinket.io/embed/glowscript/922d8d312818

[[File:Glowscriptmodel-ezgif.com-crop.gif]]

Web VPython 3.2
from vpython import *

# Set up the scene
scene.title = "Visualization of Electric Currents with Graphs"
scene.width = 800
scene.height = 800
scene.background = color.black

# Create the first conductor (voltage increasing)
conductor1 = cylinder(pos=vector(-5, 1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow1 = arrow(pos=conductor1.pos + vector(0, conductor1.radius + 0.1, 0), axis=vector(0, 0, 0),
color=color.red, shaftwidth=0.1)
label1 = label(pos=arrow1.pos + vector(1.5, 0.5, 0), text="", color=color.white, height=12, box=False)

# Create the second conductor (resistance increasing)
conductor2 = cylinder(pos=vector(-5, 0, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow2 = arrow(pos=conductor2.pos + vector(0, -conductor2.radius - 0.1, 0), axis=conductor2.axis,
color=color.green, shaftwidth=0.1)
label2 = label(pos=arrow2.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Create the third conductor (voltage and resistance increase at the same rate)
conductor3 = cylinder(pos=vector(-5, -1.5, 0), axis=vector(10, 0, 0), radius=0.2, color=color.gray(0.5))
arrow3 = arrow(pos=conductor3.pos + vector(0, -conductor3.radius - 0.1, 0), axis=vector(5, 0, 0), # Fixed length (half of cylinder)
color=color.blue, shaftwidth=0.1)
label3 = label(pos=arrow3.pos + vector(1.5, -0.5, 0), text="", color=color.white, height=12, box=False)

# Graph setup for Conductor 1
graph1 = graph(title="Voltage, Current, and Resistance (Conductor 1)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve1 = gcurve(color=color.red, label="Voltage (V)")
current_curve1 = gcurve(color=color.green, label="Current (A)")
resistance_curve1 = gcurve(color=color.blue, label="Resistance (Ohms)")

# Graph setup for Conductor 2
graph2 = graph(title="Voltage, Current, and Resistance (Conductor 2)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve2 = gcurve(color=color.red, label="Voltage (V)", graph=graph2)
current_curve2 = gcurve(color=color.green, label="Current (A)", graph=graph2)
resistance_curve2 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph2)

# Graph setup for Conductor 3
graph3 = graph(title="Voltage, Current, and Resistance (Conductor 3)", width=600, height=300,
xtitle="Time", ytitle="Value (scaled to 100)", foreground=color.white, background=color.black)
voltage_curve3 = gcurve(color=color.red, label="Voltage (V)", graph=graph3)
current_curve3 = gcurve(color=color.green, label="Current (A)", graph=graph3)
resistance_curve3 = gcurve(color=color.blue, label="Resistance (Ohms)", graph=graph3)

# Initialize variables for Conductor 1
voltage1 = 0
max_voltage1 = 10
voltage_rate1 = 0.3 # Increased rate for voltage
resistance1 = 2
current1 = 0

# Initialize variables for Conductor 2
voltage2 = 10 # Constant voltage
resistance2 = 2 # Starts at 2
max_resistance2 = 10
resistance_rate2 = 0.1
current2 = 0

# Initialize variables for Conductor 3 (voltage and resistance increase at the same rate)
voltage3 = 1 # Start at 1 to avoid division by zero
resistance3 = 1 # Start at 1 for the same reason
voltage_rate3 = 0.1 # Same rate for voltage and resistance increase
resistance_rate3 = 0.1 # Same rate for resistance increase to maintain constant current
current3 = voltage3 / resistance3 # Initial current is V / R

# Particles for both conductors
particles1 = [sphere(pos=vector(4 - i, conductor1.pos.y + conductor1.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles2 = [sphere(pos=vector(4 - i, conductor2.pos.y + conductor2.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]
particles3 = [sphere(pos=vector(4 - i, conductor3.pos.y + conductor3.radius, 0),
radius=0.05, color=color.blue, make_trail=True) for i in range(10)]

# Animate the currents and graphs
dt = 0.05
time = 0
while True:
rate(30)
time += dt

# Update Conductor 1 (Voltage Increasing)
if voltage1 < max_voltage1:
voltage1 += voltage_rate1 * dt
current1 = voltage1 / resistance1
arrow1.axis = vector(voltage1 * 0.4, 0, 0) # Arrow grows twice as fast
label1.text = f"Voltage: {voltage1:.1f} V\nCurrent: {current1:.2f} A"
voltage_curve1.plot(time, voltage1 * 10) # Scale voltage to graph's max height
current_curve1.plot(time, current1 * 10) # Scale current to graph's max height
resistance_curve1.plot(time, resistance1 * 10) # Scale resistance to graph's max height

# Update Conductor 2 (Resistance Increasing)
if resistance2 < max_resistance2:
resistance2 += resistance_rate2 * dt
current2 = voltage2 / resistance2 if resistance2 != 0 else 0
arrow2.axis = vector(conductor2.axis.x * (1 - resistance2 / max_resistance2), 0, 0) # Arrow shrinks
label2.text = f"Resistance: {resistance2:.1f} Ohms\nCurrent: {current2:.2f} A"
voltage_curve2.plot(time, voltage2 * 10) # Scale voltage to graph's max height
current_curve2.plot(time, current2 * 10) # Scale current to graph's max height
resistance_curve2.plot(time, resistance2 * 10) # Scale resistance to graph's max height

# Update Conductor 3 (Voltage and Resistance Increasing at the Same Rate)
voltage3 += voltage_rate3 * dt
resistance3 += resistance_rate3 * dt
current3 = voltage3 / resistance3 # Current stays constant as V and R increase at the same rate
arrow3.axis = vector(5, 0, 0) # Fixed length (half of cylinder length)
label3.text = f"Voltage: {voltage3:.1f} V\nResistance: {resistance3:.2f} Ohms\nCurrent: {current3:.2f} A"
voltage_curve3.plot(time, voltage3 * 10) # Scale voltage to graph's max height
current_curve3.plot(time, current3 * 10) # Scale current to graph's max height
resistance_curve3.plot(time, resistance3 * 10) # Scale resistance to graph's max height

# Animate particles for Conductor 1 (speed depends on current1)
speed1 = current1 * 0.5 # Adjust speed scaling factor if needed
for electron in particles1:
electron.pos.x -= speed1 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 2 (speed depends on current2)
speed2 = current2 * 0.5 # Adjust speed scaling factor if needed
for electron in particles2:
electron.pos.x -= speed2 * dt
if electron.pos.x < -5:
electron.pos.x = 5

# Animate particles for Conductor 3 (speed depends on current3)
speed3 = current3 * 0.5 # Adjust speed scaling factor if needed
for electron in particles3:
electron.pos.x -= speed3 * dt
if electron.pos.x < -5:
electron.pos.x = 5

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.
By convention, current is measured in the opposite direction of which electrons flow. This is likely because when current was first discovered, electrons had not yet been established in the scientific world. Hence, it could seem like positive charges were moving in a certain direction, rather than negative ones in the opposite as we know today. This convention still stands to this day, and in fact most likely makes calculations simpler, due to not having to account for the negative sign (-), which could cause for mistakes in calculations. However, it is certainly possible to solve a circuit problem with the opposite direction current, just with everything negated.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It can be thought of as a change in electric potential, meaning that if there is no change, the voltage must be 0. Voltage does NOT measure anything at a given point, which is why a voltmeter must be connected at two different points in order to display a reading. These points should have some sort of change in current between them to be non-zero, such as the existence of a resistor or a node.
It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. This is because, when given the choice (such as a node splitting into two), current will try to flow through the wire with less resistance. This is why voltmeters are made with such high resistance - to avoid affecting the current flow (by having current flow through the voltmeter instead of the circuit), such an example why high resistance might be beneficial. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Examples==

===Simple===

'''Question 1'''

If a wire has 2 ohms of resistance and 3 volts of voltage, what is the current?

'''Answer'''

<math>{I = \frac{V}{R}}</math>

<math>{I = \frac{3}{2}}</math>

<math>{I = 1.5 \text{ Amps}}</math>

----
'''Question 2'''

If a wire has 6 ohms of resistance and 2 amperes of current, what is the voltage?

'''Answer'''

<math>{V = IR}</math>

<math>{V = 6*2}</math>

<math>{V = 12 \text{ Volts}}</math>

----

'''Question 3'''

If a wire has 18 ohms of resistance and 10 volts of voltage, what is the current?

'''Answer'''

<math>{R = \frac{I}{V}}</math>

<math>{R = \frac{18}{10}}</math>

<math>{R = 1.8 \text{ Ohms}}</math>

----

===Middling===

'''Question 4'''

A circuit has 2 3 volt batteries powering it. The circuit has 4 light bulbs, each providing 2 ohms of resistance. Assuming all components add up, what is the magnitude of the flow of the circuit?

'''Answer'''

<math>{V = 2*3 = 6}</math>

<math>{R = 4*2 = 8}</math>

<math>{I = \frac{6}{8} = 0.75 \text{ Amps}}</math>

----

===Difficult===

'''Question 5'''

A circuit has a constant current according to the total resistance, but the voltage drop is variable across each resistor according to its resistance and Ohm's law. A Circuit has a 10 volt battery at location 1, a 2 ohm resistor at location 2, a 3 ohm resistor at location 3, a 2 volt battery at location 4, a 5 ohm resistor at location 5, and a 6 ohm resistor at location 6 which connects back to the battery at location 1. What is the total current and the voltage drop at each location.

'''Answer'''

Current:
<math>{V_{total} = 10+2 = 12}</math>
<math>{R_{total} = 2+3+5+6 = 16}</math>
<math>{I_{total} = \frac{12}{16} = 0.75 Amps}</math>
Voltage Drops:

Resistor 1 (2 ohm): <math>{V = IR = 0.75 * 2 = 1.5 \text{ Volts}}</math>

Resistor 2 (3 ohm): <math>{V = IR = 0.75 * 3 = 2.25 \text{ Volts}}</math>

Resistor 3 (5 ohm): <math>{V = IR = 0.75 * 5 = 3.75 \text{ Volts}}</math>

Resistor 4 (6 ohm): <math>{V = IR = 0.75 * 6 = 4.5 \text{ Volts}}</math>

Checking answer (total voltage drop should be equivalent to total voltage provided, which as stated previously should be 12):

<math>{V = 1.5+2.25+3.75+4.5 = 12 \text{ Volts}}</math>

Correct! It is 12!

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==Initial vs Steady State==
A critical difference in steady state vs. initial state circuits is how capacitors behave. Initially they are uncharged, and so charge will flow to them to be stored. At steady state, the capacitor is fully charged and the current there is zero. Therefore, if there is a loop with a capacitor, it can be treated as "open" (essentially as if the wire were not connected there, and loop rules as such would apply.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]

Gauss's Law

2025-04-13T16:59:56Z

Ielsissi3: I added a summary table. I think this section can be of great help to a student, especially before a test, when they need a quick recap/reminder.

'''Ismail Elsissi (ielsissi3) Spring 2025'''


Gauss's Law is a very powerful law that spans a diverse array of fields, with applications in physics, mathematics, chemistry, and engineering, among others. Along with [[James Maxwell]]'s other three equations, Gauss's Law forms the foundation of classical electrodynamics.

==Quick Summary==

{| class="wikitable"
! Symmetry !! Gaussian Surface !! Electric Field Result
|-
| Point Charge (spherical) || Sphere || <math>E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{Q}{r^2}</math>
|-
| Infinite Line of Charge || Cylinder || <math>E = \frac{\lambda}{2\pi\varepsilon_0 r}</math>
|-
| Infinite Plane || Box || <math>E = \frac{\sigma}{2\varepsilon_0}</math>
|}

==The Main Idea==

[[File:Gaussian Box Ex.jpg|thumb|Three-dimensional Gaussian surface]]
Picture a three-dimensional solid object in space that does not have any charges around it. Suppose that you discover an electric field is directed radially outward from all faces of the object. You then measure the magnitude of the electric field at each face. Without being able to open the object and determine its charge distribution, how might you determine the charge inside the object?

This simple situation illustrates one basic application of Gauss's Law: Using the measured electric field flowing through the faces of a closed surface to determine the charge that lies inside.

===A Mathematical Model===

Gauss's Law elegantly relates the net charge enclosed within a ''Gaussian surface'' to the patterns of electric field that flow over its faces ([[Electric Flux]]). A Gaussian surface is any surface belonging to a closed three-dimensional object. 
In words, the law states that the net electric flux outside a surface is equal to the ratio of the total enclosed charge inside the surface to the permittivity of free space. Symbolically, this is written as the area integral

<math>\oint \vec E \cdot \hat{n} ~dA = \frac{\sum Q_{inside}}{\varepsilon_{0}}</math>

where <math> \vec E </math> is the net electric field acting through the surface, <math> \hat{n}</math> is the unit normal vector perpendicular to the face of the surface, and <math> dA </math> is the surface area of the face being examined.



From the definition of [[Electric Flux]], Gauss's Law may also be written equivalently as

<math> \Phi_{el} = \frac{Q_{enclosed}}{\varepsilon_{0}}</math>

====Relation to Coulomb's Law====

Additionally, Coulomb's Law and Gauss's Law are innately connected. Coulomb's Law relates charge to electric field. Gauss's Law relates charge to electric flux. Both laws, relate charge to an electrical property. For clarification, the electric flux is a quantity that is equal to the product of the perpendicular component of E-field and the area of the closed surface. Not only do these two laws look similar at the surface, but using one law it is quite easy to derive the other law.

====Relationship Between Charge and Flux====

We know from the previous introduction of flux that there is a relationship between the electric flux on a closed surface and the charges inside the surface. However, we would like to figure out what that particular factor is.

We start by considering a point charge of <math>+Q</math> enclosed by an imaginary spherical shell.

[[File:gaussv2212.jpg]]

Everywhere on the imaginary shell, the electric field produced is parallel to the normal unit vector. Cos(0) = 1, so the dot product is simply the magnitude of the electric field. Recall that the surface area of the imaginary sphere is <math>4 \pi r^{2}</math>. With Coulomb's Law and the surface area of a sphere, we get that electric flux is equal to <math>\frac{+Q}{\varepsilon_{0}}</math>. This implies that the factor is <math>\frac{1}{\varepsilon_{0}}</math>.

Note that if the point charge was negative, the electric field would still be parallel, just opposite direction. Cos(180) = -1 so the dot product would be -1 times the magnitude of the electric field.

To understand this law more fully, one can look at the diagram below. As can be seen on the left side of this diagram, change in flux equals electric field multiplied by change in area. This is the basis for the equation ф=∮E da. When you integrate the change of A, you get the flux.

Image Taken from Hyperphysics

[[File:Gaulaw.gif]]

To more clearly state it, in integral form, the formula for this Law is the electric flux equals the total charge contained by a closed surface, divided by the permittivity (<math>\varepsilon_{0} = 8.854187817...×10^{-12} ~F*m^{-1}</math>).

[[File:Adc2dff3156800a39ef0a9df76a7d868.png]]

Additionally, Gauss's law can be written in differential form.

[[File:Screen Shot 2018-04-18 at 5.31.54 PM.png]]

In this equation, ρ is electric charge density, ∇*E is divergence of electric field, and <math>\varepsilon_{0} = 8.854187817...×10^{-12} ~F*m^{-1}</math>.

Below is a brief summary of what has been covered.

[[File:Gauss_14.jpg]]

==A Computational Model==

An extremely rudimentary explanation of Gauss's Law is: https://www.youtube.com/watch?v=f2Cccp6XBUY

To visualization of Gauss's law with computational model: https://www.youtube.com/watch?v=i5N36mWsdGo

Experimental demonstration of Gauss's law: https://www.youtube.com/watch?v=r3w4FAUOLuM

Visual explaination of Gauss Divergence Theorem(Khan Academy): https://www.youtube.com/watch?v=XyiQ2dwJHXE

To view the applications of Gauss's law in a coding setting (Python GLowScript): https://trinket.io/glowscript/f618920f61

==Examples==

===Easy Examples===
====Easy Example 1====

Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).

1. Recall Gauss's Formula:

[[File:Gauss for.jpg|400px]]

2. Now look carefully at the diagram below and go through each of the surfaces one by one. Ask yourself, in what direction does the normal vector point?

[[File:Gauss ex.jpg|400px]]

3. Therefore, in the sides where there is 90 degree angle made with E, the flux will just be zero!

[[File:Gauss fl.jpg|400px]]

4. In the only case where the flux is not zero is in the sides A and B of the box (which are the same in magnitude). We compute the flux as follows:

[[File:Gauss ans.jpg|600px]]

Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.

====Easy Example 2====
Consider a point charge <math> Q = +2 \, \text{μC} </math> enclosed by a spherical Gaussian surface with a radius of <math> r = 10 \, \text{cm} </math>. Calculate the electric flux through the surface.

**Solution:**
Using Gauss's Law, the electric flux <math>( \Phi_{el} )</math> through a closed surface is given by <math> \Phi_{el} = \frac{Q_{enclosed}}{\varepsilon_{0}} </math>. For a point charge enclosed by a spherical surface, the electric flux is simply <math> \frac{Q}{\varepsilon_{0}} </math>. Substituting the given values:

<math> \Phi_{el} = \frac{2 \times 10^{-6} \, \text{C}}{8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2} </math>

===Intermediate Examples===

====Example 1====
The example below shows how to determine the net charge located inside a box and is fundamental to understanding Gauss's Law. It demonstrates the idea that to calculate the net charge inside a box, we have to start off by calculating the fluxes acting on all surfaces on the box by using the Gauss's Law.

[[File:Wiki resource.jpg|600px]]

====Example 2====
The example below shows how to calculate the net charge enclosed by a box.

[[File:Wiki resource2.jpg|600px]]

====Example 3====
In order to apply Gauss's Law, it is important to be certain you are working with a closed surface, then set electric flux equal to the internal field divided by the permittivity (<math>\varepsilon_{0} = 8.854187817...×10^{-12} ~F*m^{-1}</math>). An example of this Law being applied can be found below.

[[File:Gauss_law3.png|600px]]

====Example 4====
In this example, the Qenc is equal to zero. This was determined by adding up all of the fluxes through every surface. This can also be determined by seeing that the same fluxes travel through the same areas of the tent. The equation Ф=∮B da=0 helps explain this when considering that Ф is the same in both equations.

[[File:Gauss's_Law1234.jpeg|600px]]

===Hard Examples===

====Example 1====

Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.

=====Solution:=====
To find the electric field inside the sphere, create a Gaussian surface and take advantage of the spherical symmetry. The charge enclosed within a Gaussian sphere of radius <math>r</math> is <math>\frac{4}{3}\pi \rho_0 r^3 </math>. Gauss's Law then gives <math>E = \frac{\rho_0 r}{3\varepsilon_0} </math>.

====Example 2====

A long, straight wire carries a non-uniform linear charge density <math> \lambda(z) = \lambda_0 e^{-az} </math>, where <math>\lambda_0</math> and a are constants, and z is the distance along the wire. Determine the electric field at a distance r from the wire using Gauss's Law.

=====Solution:=====
Set up a Gaussian cylindrical surface around the wire to take advantage of the cylindrical symmetry. The total charge enclosed within the Gaussian surface is found by integrating the linear charge density along the length of the cylinder. Gauss's Law then gives <math>E = \frac{\lambda_0}{2\pi\varepsilon_0 r} \int_{-\infty}^{\infty} e^{-az} \, dz </math>. Solving this integral yields the expression for the electric field as a function of r.

==Connectedness==
Gauss's Law can be found in many areas of science, technology, engineering, and mathematics. In fact, Gauss's Law and other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study. Gauss's Law is also connected to the Gaussian/Normal distribution, which is a continuous probability distribution that is characterized by a bell-shaped curve. This distribution is found throughout statistics and probability and is used everyday by people in STEM fields.

As Industrial and Systems Engineering majors, we deal with many statistical distributions. A very important fundamental distribution is the Gaussian distribution or the Normal Distribution. With inference, the Gaussian Distribution comes up in confidence intervals for single statistics. With comparison inference, like finding the pairwise difference between statistics, generally we use other distributions such as the T-Distribution. However, the T-Distribution approximates the Gaussian distribution with degrees of freedom greater than 29.

As civil and environmental engineering majors, we also deal with the Gaussian/Normal Distribution in our fields. As an example, the speed data of traffic on a highway is said to follow the normal distribution.

Additional to many engineering majors. Gauss's Law has applications in many different STEM job fields. Engineering majors learn about the different Gaussian laws in college and apply them in there jobs. Millions of people are applying his laws to the world and will continue to do so. Gauss has had a huge impact on the world and will continue to have a big impact on the future. Gauss has done great work with his work.

==History==

[[File:220px-Carl_Friedrich_Gauss_(C._A._Jensen).jpg||Portrait of Gauss]]

Gauss's Law was first 'discovered' in 1773, by the great mathematician Joseph-Louis Lagrange. It was further developed by Carl Friedrich Gauss in 1813. The law was inspired by Coulomb's law.

Gauss was a prolific German mathematician and physicist. He made significant contributions to a variety of fields in mathematics and physics. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians".

Gauss is also known for his advancements in the astronomy field. While the dwarf planet Ceres was discovered by Giuseppe Piazza (Italian astronomer), Gauss was able to predict the position for its rediscovery. Due to the methodology behind this rediscovery, Gauss's Method, there was also significant advancements in the theory of the motion of planetoids, which introduced the Gaussian gravitation constant.

The foundation for Gauss's law was laid by Isaac Newton, who formulated the inverse-square law of gravitation in 1687. This idea of forces diminishing with the square of distance inspired analogous formulations for electrostatics. In 1785, Charles-Augustin de Coulomb experimentally confirmed that electric forces between charges follow an inverse-square law. The concept of a field—an invisible influence extending through space—began to take shape in the late 18th and early 19th centuries.
Mathematicians and physicists, including Joseph Priestley and others, laid groundwork for understanding how charges influence surrounding regions. Gauss formulated the law in his personal notes in 1835. These notes contained his ideas on how electric fields could be mathematically related to charges through surface integrals. His reluctance to publish might have been due to the lack of experimental confirmation at the time or his tendency to prioritize mathematical rigor over immediate dissemination. The law became widely known as part of James Clerk Maxwell’s unification of electromagnetic theory in the 1860s. Maxwell included Gauss’s law as one of his four foundational equations in Maxwell’s Equations. Although Maxwell credited earlier physicists, including Gauss, Faraday, and Ampère, the incorporation of Gauss’s law into a cohesive electromagnetic theory marked its formal introduction into mainstream physics. By the late 19th century, Gauss’s law was being taught in academic settings as a fundamental part of electrostatics and electromagnetism. It gained recognition in its integral and differential forms, largely through Maxwell’s equations and the broader development of vector calculus.

== See also ==
===Further Reading===

Gauss's Law is tied in closely with the other of Maxwell's equations that can be found here in the Physics Book.

http://physicsbook.gatech.edu/Gauss%27s_Flux_Theorem

http://physicsbook.gatech.edu/Faraday%27s_Law

http://physicsbook.gatech.edu/Magnetic_Flux

http://physicsbook.gatech.edu/Ampere%27s_Law

===External links===

http://physics.info/law-gauss/

http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter24/Chapter24.html

https://en.wikipedia.org/wiki/Gauss%27s_law

https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://physicscatalyst.com/elec/guass_0.php

==References==

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

http://www.pas.rochester.edu/~stte/phy114S09/lectures/lect04.pdf

https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/Gauss_Law.pdf

http://www.sciencedirect.com/science/article/pii/S2090447911000165

spiff.rit.edu

study.com

[[Category: Electric Field]]
[[Category: Fields]]
[[Category: Gauss's Law]]
[[Category: Maxwell's Equations]]

Potential Energy

2025-04-13T16:49:19Z

Ielsissi3: I added a summary table. I think this section can be of great help to a student, especially before a test, when they need a quick recap/reminder.

'''Ismail Elsissi (ielsissi3) Spring 2025'''

Potential energy (referred as <math>U</math>) is the stored energy of position possessed by an object and is that some body possesses due to their position relative to other bodies, configuration or stresses within itself, electric charges, and other factors. These factors can include a variety of many things, the main one being the pull of the earth. However, other common sources of potential energy include springs and other planets.
Potential Energy can be pictured as a stored energy which has the potential to do some work but hasn't done it yet.
Changing the parameters that give a system a potential energy (e.g. distance between objects), <math>U</math> changes as well. The change in <math>U</math> can result in a change in <math>K</math> (Kinetic Energy) For a lot of beginner-level physics the total energy in a system is generally created by the addition of <math>U</math> and <math>K</math>.

''Examples of Potential Energy''

''Example 1: A rock sitting at the edge of a cliff. At the moment it will have potential energy since it is being stored on the height H from the surface of the Earth. If the rock falls, the potential energy will be converted to kinetic energy because now the object is in motion.''

[[File:Yo-yo player Antikensammlung Berlin F2549.jpg|thumb|''Fig.1'' The energy stored in person's hand with holding the yo-yo and not moving it. [[Attica|Attic]] [[kylix (drinking cup)|kylix]], c. 440 BC, [[Antikensammlung Berlin]] (F 2549)]]

''Example 2: Even the simplest example would be even holding a yo-yo in ones hand, for example, the yo-yo has the potential to do some work once you release it from the string it's tied on.''

''Example 3: When throwing a ball up into the air the speed is constantly changing. However, the total energy in the ball-earth system is still the same. This is because as the kinetic energy increases/decreases the potential energy does the opposite. The ball has the most speed (and thus the most kinetic energy) at the points where the ball is nearest to the ground. However, at the apex of the flight the speed of the ball is 0. This is where the potential energy is at its maximum because the kinetic energy of the ball is 0 (because of the speed being equal to 0.)''

When referring to <math>U</math> being at a minimum or maximum, it is important to note that these can be in absolutes. If a system has all potential energy, then all of its energy is stored and there is absolutely no kinetic energy. An example to help think of this is a stationary ball on top of a cliff. It is not moving whatsoever, but if it were to be pushed off of the cliff, the potential energy would begin to be transformed into kinetic energy as it falls. There is also a moment, just before the ball hits the ground and while its velocity is at its theoretical maximum, that virtually zero potential energy is present. All of the energy would be kinetic energy in this case.

Energy is conserved, so the total amount of energy in a system never changes, but it can change forms, such as from potential to kinetic energy, or the other way around. In non-ideal systems, so most realistic earthly systems, energy can be transformed into other forms such as thermal energy. An example of this would be a toy racecar rolling along a flat surface. It eventually slows down and stops due to friction, which results in a decrease in kinetic energy since that energy is transformed into thermal energy on the tires.

In terms of potential energy, its capacity for doing [[work]] is a result of its position in a gravitational field (gravitational potential energy), an [[Electric Field|electric field]] (electric potential energy), or a [[magnetic field]] (magnetic potential energy). It may have elastic potential energy due to a stretched spring or other elastic deformation.

It is interesting to note, that the universe naturally prefers lower potential configuration of systems. This peculiarity is partly why balancing a pencil on one finger is so hard; the pencil contains potential energy that can be released easily. As taught in many engineering classes, this can be explained by one of the thermodynamic laws, where the universe always increase randomness, or entropy.

==Quick Summary==

{| class="wikitable"
| '''Type''' || '''Equation''' || '''Depends On''' || '''Example'''
|-
| Gravitational Potential Energy || <math>U = mgh</math> || Mass <math>(m)</math>, Gravity <math>(g)</math>, Height <math>(h)</math> || A book on a shelf
|-
| Elastic Potential Energy || <math>U = \frac{1}{2}kx^2</math> || Spring constant <math>(k)</math>, Compression/stretch <math>(x)</math> || A compressed spring
|-
| Electric Potential Energy || <math>U = \frac{kq_1q_2}{r}</math> || Charge magnitudes <math>(q_1, q_2)</math>, Distance <math>(r)</math> || Two charged particles
|}

==The Main Idea==

When we want to discuss Energy, the first step is always defining your system.
In fact, the Energy Principle states: <math> \vartriangle \!\, Esys = Wsurr </math>.
Up to this point, energy was always studied in mono-particular systems.
However, most situations in life are more complex and involve multi-particular systems.

''An example would be any object that is displaced, but still has stuff interacting and working in it. We can imagine the earth, on which work is done by the Sun and other planets. Above the surface of the earth, there is a ball which also interacts with it. Surrounding planets also do work on the ball. So how can we analyze the behavior of both the ball and the Earth when they interact with each other and the Sun?''

This analysis is infinite and can be very difficult to handle. That's why sometimes you need to consider multi-particular systems for your analysis. We might want to fix earth+ball as a point to analyze the work done by the Sun on this system.
But the problem is, we'll be neglecting the interaction happening within the system.

''If the ball is the system, the Earth represents the surroundings (image 1). The Kinetic energy of the system (ball only) increases gradually due to the work done by the Earth. Imagine the ball is 1 kg and is released from rest 10 m above the surface. When the ball has fallen to 5 m, what is <math>K</math>?''

''
<math> \vartriangle \!\, Eball = WEarth </math>

<math> \vartriangle \!\, Kball = WEarth </math>

<math> \vartriangle \!\, Kball = Fy\vartriangle \!\,y = -mg\vartriangle \!\,y </math>

<math> Kf = -(1 kg)(9.8 N/Kg)(-5 m) = 5 J </math>''

''What if we decide the choose the ball and the Earth as the system?

In this case, we consider there is nothing significant in the surroundings. As a result

<math> \vartriangle \!\, Eball = 0 </math>

<math> \vartriangle \!\, Kball + \vartriangle \!\, Kearth = 0 </math>''

However, we know that the kinetic energy of the ball increased, and the kinetic energy of the Earth also increased. We also know the surroundings did zero work. But the change in kinetic energy of the system is bigger than zero and so different from the work done by the surroundings.
This is happening since we are overlooking a kind of energy in the system related to the interaction between both bodies: it is potential energy.

In any systems containing two or more interacting bodies (stars interacting in a galaxy), there is energy associated with the interactions between pairs of particles inside the system.

So, since the ball+earth system contains interacting objects, its energy is:

<math>Esys = EEarth + Eball + Uball/earth </math>

Since the interaction force is gravity (dependent of distance), a change in potential energy should be associated with a change of the separation between objects. This can also mean a change of shape of the multi-particle system, such as a spring stretching or compressing.

Getting back to our example above, as the ball and Earth get closer, the kinetic energy of the system increases but it is compensated by a decrease in potential energy (interaction energy). In fact, since there are no surroundings:

<math> \vartriangle \!\, Eball = 0 </math>

<math> \vartriangle \!\, Kball + \vartriangle \!\, Kearth + \vartriangle \!\, Uball/earth = 0 </math>

===A Mathematical Model===

Taking ball+earth as the system makes the force exerted on the ball interior to the system: it is an internal force which makes internal work and so changes <math>U</math>.

<math>\vartriangle \!\, Kball = Wearth </math>

<math>\vartriangle \!\, Kball + (- Wearth) = 0 </math>

'''Definition:''' for a multi-particle system we define the change of potential energy <math>\vartriangle \!\, U </math> to be the negative of the internal work:

<math>\vartriangle \!\, U = -Wearth </math>

For the system of the ball + Earth:

<math>\vartriangle \!\, U = -(Fy \vartriangle \!\, y) = -(-mg)\vartriangle \!\, y = \vartriangle \!\,(mgy) </math>

To conclude, the difference between a single-particle and a multi-particle one is that the second has pairwise potential energy on top of the particular energy.

'''Revised energy principle for a multiparticle system'''

<math> \vartriangle \!\, (E1 + E2 + E3 + ...) + \vartriangle \!\, (U12 + U13 + U23 + ...) = W </math>

with <math> E1 = E1rest + K1 </math>

'''An algebraic process can help us derive equations for all kinds of Potential energies with respect to the parameters that affect it.'''

A force is considered conservative if it is acting on an object as a function of position only.

We can relate work to potential energy using the equation

<math>U = -\int \vec{F}\cdot\vec{dr} = -W</math>

This says that the potential energy U is equal to the[[work]] you must do to move an object from an arbitrary reference point <math>U=0</math> to the position <math>r</math>.

''The potential energy of a rock on the top of a cliff is equal to the work you've done to bring the rock up to this point.''

If we take the derivative of both sides of this equation and obtain:

<math>\frac{-dU}{dx} = F(x)</math>

Which means that the force on an object is the negative of the derivative of the potential energy function U. Therefore, the force on an object is the negative of the slope of the potential energy curve. Plots of potential functions are valuable aids to visualizing the change of the force in a given region of space.

Let's apply this relationship. If the potential energy function U is known, the force at any point can be obtained by taking the derivative of the potential. Let's consider gravitational potential and elastic potential.

The potential energy function U of gravitational potential is <math>mgh</math>, where <math>m</math> is mass, <math>g</math> is the gravitational constant, and <math>h</math> is some distance away from the reference point at which U = 0. Then the force is

<math>F = \frac{-d}{dh}mgh = -mg</math>

We can go the other way as well. We know the force of gravity is <math>-mg</math>, and integrating with respect to h we obtain <math>U = mgh</math>.

This process can be done with elastic potential as well, where the force <math>F = -kx</math> and the potential energy function is <math>U = \frac{1}{2}k^{2}</math>

Here are the potential energy functions for all forms:

{| class="wikitable"
| '''Type''' || '''Equation''' || '''Variables'''
|-
| Gravitational Potential || <math>U = \frac{GMm}{r}</math> <math>U = mgh</math> close to Earth's surface || <math>G</math> is the gravitational constant, <math>M</math> and <math>m</math>, and <math>r</math> is distance
|-
| Elastic Potential || <math>U = \frac{1}{2}kx^{2}</math> || <math>k</math> is the spring constant, <math>x</math> is how much the spring is stretched.
|-
| Electric Potential || <math>U = k\frac{Qq}{r}</math> || <math>k</math> is Coulomb's constant, <math>Q</math> and <math>q</math> are point charges, <math>r</math> is distance
|-
| Magnetic Potential || <math>U = -μ \cdot B</math> || <math>μ</math> is the dipole moment and <math>μ = IA</math> in a current loop and <math>I</math> is the current and <math>A</math> is the area
|-
|}

===A Computational Model===

A spring is like a rubber band, as you stretch it, it stores potential energy. Once released, this stored energy is converted to kinetic energy. The simulation at this link shows it https://trinket.io/embed/glowscript/fa7fbeaf3b[https://trinket.io/embed/glowscript/fa7fbeaf3b]

''It is interesting to see how potential(yellow) and kinetic(cyan) energy alter. When U is at a maximum, K is at a minimum. When K is at a maximum, U is at a minimum. This suggests an energy transfer: U is transformed into kinetic energy which causes the ball to move. Also total Energy is constant because there are no surroundings.''

"See "Demo 3" at: https://drive.google.com/drive/folders/0B5m-O1TAXl8GWkxYNGhwQm1xMkU?usp=sharing"

==Examples==

This is a link to a video displaying an exercise resolved step by step by Dr. Greco [https://vimeo.com/38926717]

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===

'''Question'''

An object of mass 3 kg is pressed by a spring with stiffness = 8 and is compressed a length of .02 meters. How much potential energy does this object have?

'''Solution'''

Using the equation <math>U = \frac{1}{2}kx^{2}</math> we can say <math>U = 1/2*8*(.02)^2 = .0016</math> J.

'''Question'''

An object of mass 5 kg is held 10 meters above the Earth's surface. Relative to the surface, how much potential energy does this object have?

'''Solution'''

Using the equation <math>U = mgh</math> we can say <math>U = 5*9.8*10 = 490</math> J.

===Middling===

'''Question'''

If it takes 4J of work to stretch a Hooke's law spring 10 cm from its unstretched length, determine the extra work required to stretch it an additional 10 cm.

'''Solution'''

The work done in stretching or compressing a spring is proportional to the square of the displacement. If we double the displacement, we do 4 times as much work. It takes 16 J to stretch the spring 20 cm from its unstretched length, so it takes 12 J to stretch it from 10 cm to 20 cm.

Formally:
<math>W = \frac{1}{2}kx^{2}.</math> Given W and x we find k. 
<math>4 J = \frac{1}{2}k(0.1)^{2}</math> 
<math>k =\frac{8}{0.1^{2}} = 800</math> N/m. 
Using <math>x = 0.2</math> m, <math>W = \frac{1}{2}(800)(0.2)^{2} = 16</math> J 
Extra work: 16 J - 4 J = 12 J.

===Difficult===

'''Question'''

We have a point charge A of charge +Q at the origin. Let's say we want to move another charge B of +q, located 10m away from particle A, to a location 5m away from particle B. How much work does it require to move the particle B 5m closer to particle A?

'''Solution'''

We have a nonuniform electric field, so we need to integrate the potential energy function to find the amount of work needed.
<math>W = \int_{10}^{5} \frac{-kQq}{r^2}dr= -kQq\int_{10}^{5}\frac{1}{r^2}dr = \frac{kQq}{10} </math>

==Connectedness==
Potential energy is the driving force behind voltage, or the electric potential difference, expressed in volts. In fact, the circuit has to have a potential in order to light stuff and do some electrical work. The energy use for this work is subtracted from this potential which explains the difference of potential known as V. As a computer engineering major, I recognize the importance of this concept, as without potential difference we would not have transistors or circuits to power our machines.

Nuclear potential energy also exists, and is the potential energy of the particles inside an atomic nucleus. Most nuclear elements like Pd and Ur have unstable nucleus due to a disproportion of nuclear particles like protons and neutrons. Unstable nucleus means a nucleus which is susceptible to change to another configuration: they have a potential. Elements used in nuclear reactions thus have kernels with a high potential energy. As we previously discussed, nature always try to go to the most stable state of energy, the state where it has the least potential energy. That's why most radioactive elements naturally decay and change composition to be in a more stable stage. The difference of energy between previous and next stage is the energy liberated in nuclear reactions as rays.

==History==

The term "potential energy" was coined by William Rankine, a nineteenth-century Scottish engineer and physicist. It corresponds to energy that is stored within a system. It exists when there is a force that tends to pull an object back towards some original position when the object is displaced.

William Rankine contributed to Civil engineering and was a founding contributor, with Rudolf Clausius and William Thomson (Lord Kelvin), to the science of thermodynamics.

He unfortunately passed away on December 24 1872 in Glasglow, Scotland at the age 52. Cause of death remains unknown as of now

[[File:William Rankine 1870s.jpg|thumb|Rankine in the 1870s]]

== Gravitational Potential Energy ==
Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released when the objects fall towards each other.

For two pairwise interacting point particles, the gravitational potential energy <math>U</math> is given by
:<math>U = -\frac{GMm}{R},</math>

Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to
:<math>U = mgh</math>
where <math>m</math> is the object's mass, <math>g = GM_{\oplus}/R_{\oplus}^2</math> is the [[gravity of Earth]], and <math>h</math> is the height of the object's [[center of mass]] above a chosen reference level.<ref name=gpe/>

''Examples of Gravitational Energy''

*A raised weight
*Water that is behind a dam
*A car that is parked at the top of a hill
*A yoyo before it is released
*River water at the top of a waterfall
*A book on a table before it falls
*A child at the top of a slide
*Ripe fruit before it falls

''m represents the mass of the object, h represents the height of the object and g represents the gravity of the earth surface which is (9.8 N/kg on Earth).''

== Elastic Potential Energy ==
Elastic energy, or elastic potential energy, is the energy that an object has when being compressed by a spring. Found by the equation :
<math>U = \frac{1}{2}kx^{2}</math>
Elastic potential energy deals with springs and conservation of energy. When a mass in put next to a compressed spring, the spring has the potential to extend and thus push the mass. The value of how much energy the spring can push with is seen with the elastic potential energy.
== See also ==

Electric Potential Energy

When an electric field accelerates a free positive charge Q, kinetic energy is associated. According to LibreTexts physics, "the electrostatic or Coulomb force is conservative, which means that the work done on q is independent of the path taken, as we will demonstrate later. This is exactly analogous to the gravitational force." (LibreTexts).

According to Khan Academy, electric potential energy is the amount of energy required to change the state of or move a charge in a field. If two charges are positively charged, they require less energy to be pulled away from each other while if the charges are oppositely charged, they would require more energy to be pulled away from each other since they are attracted to each other. The closer to the plate the charges are, the more energy we need to put in resulting in more electric potential energy. Voltage or electric potential energy is the difference between the electric potential energy of two charges according to Khan Academy.

[[Kinetic Energy]] 
[[Potential Energy for a Magnetic Dipole]] 
[[Potential Energy of a Multiparticle System]] 
[[Work]]

===External links===
[https://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy]
[https://www.youtube.com/watch?v=KoZ61FujkRk]
[https://www.youtube.com/watch?v=DyaVgHGssos]
[https://www.youtube.com/watch?v=g7u6pIfUVy4]

==References==

“Potential Energy.” The Physics Classroom, https://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy.<be>

https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/07%3A_Electric_Potential/7.02%3A_Electric_Potential_Energy

“What Is Gravitational Potential Energy? (Article).” Khan Academy, Khan Academy, https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-gravitational-potential-energy.<be>

https://www.khanacademy.org/test-prep/mcat/physical-processes/electrostatics-1/a/electric-potential#:~:text=Electric%20potential%20energy%20is%20the,through%20a%20stronger%20electric%20field.

[[Category:Energy]]