Electric Potential

AUTHOR: Rmohammed7

REVISED BY: HAYOUNG KIM (SPRING 2016)

EDITED BY: ROHITH MACHERLA (SPRING 2017)

CALIMED BY: QIAOHONG WANG (FALL 2018)

This page discusses the Electric Potential and examples of how it is used.

The Main Idea

To understand electric potential, it is important to recognize its unique properties and its difference with other related concepts, such as electric potential energy and electric potential difference.

Electric potential energy is a property of a charged object, defined by the object's location in an electric field. In other words, if there's a charged object in a relative position to other electrically charged objects, electric potential energy exists. Oftentimes we want to find the change in potential energy due to our focus on the interactions of objects. This change is denoted by:

                                             ∆U = qE∆s

This equation states that the change in the potential energy of a charged particle is equivalent to the product of its charge (q), the electric field that the charged particle experiences (E), and the distance that the particle travels (∆s).

In order to normalize change of electric potential energy, we introduce electric potential difference (voltage). By definition, electric potential difference is the change of potential energy experienced by a +1 test charge. It is the external work needed to move a charge from one location to another location in an electric field. It is expressed as:

[math]\displaystyle{ {{U}_{B}}/q - {{U}_{A}}/q }[/math].

Now we can further talk about electric potential. In the pervious equation for electric potential difference, each term represents an electric potential. Electric potential is expressed as:

[math]\displaystyle{ {{U}_{r}}/q }[/math]. which can be further expressed as

[math]\displaystyle{ ∆{V} = -\int_A^B {E}_{x} \, dx }[/math]

It's important to understand that electric potential itself exists at one location as a property of space. Even if there's no charged particle, electric potential still exists.

Models

Now that we have an expression for the potential energy, we can rewrite this expression by introducing a new variable: The potential difference is defined as ∆V = E∆s. Thus we can replace the E∆s part in the equation for potential energy to be defined as:∆U = q∆V. The potential difference is the product of the electric field and the distance a particle travels in that electric field.

The electric potential of a particle at a point is equal to the potential difference of that particle with respect to infinity. Since we know how to calculate the potential difference using the formula, we can see that this implies that the electric potential at infinity is equal to zero. What does this mean? This means that a particle that is extremely far away has no potential energy. This makes sense, because a proton will not be affected by the eletric field of another proton at a distance of infinity apart. This stuble detail aids in solving a case of problems types later on.

Electric potential is a rather difficult concept as it is usually accompanied by other topics. For example, electric potential is not the same as electric potential energy as electric potential is defined as the total potential energy per charge and is basically used to describe the electric field's effect at a certain location. In other words, electric potential is purely dependent on the electric field (whether it is uniform or nonuniform) and the location, whereas the electric potential energy also depends on the amount of charge the object in the system is experiencing. Also, although electric potential is an important topic to learn, most problems encountered will not ask to find just the "electric potential," instead, questions will most likely ask for the "electric potential difference." This is because electric potential is measured using different locations, or more specifically pathways between the different locations, so instead of determining the electric potential of location A and the electric potential of final location B, it would make more sense to determine the "difference in electric potential between locations A and B."

A Mathematical Model

Like mentioned in the Main Idea, instead of electric potential, in most cases, electric potential difference is needed to be found. The general equation for the potential difference is [math]\displaystyle{ ∆{{U}_{electric}} = {q} * ∆{V} }[/math].

[math]\displaystyle{ ∆{{U}_{electric}} }[/math] is the electric potential energy, which is measured in Joules (J). q is the charge of the particle moving through the path of the electric potential difference, which is measured in coulombs (C). ∆V is the electric potential difference, which is measured in Joules per Coulomb (J/C), or just Volts (V).

Aside from the general equation, the electric potential difference can also be found in other ways. The potential difference in an uniform field is [math]\displaystyle{ ∆{V} = -({E}_{x}∆{x} + {E}_{y}∆{y} + {E}_{z}∆{z}) }[/math], which can also be written as [math]\displaystyle{ ∆{V} = -\vec{E}·∆\vec{l} }[/math].

∆V is the electric potential difference, which is measured in Joules per Coulomb (J/C), or just Volts (V). E is the electric field, which is measured in Newtons per Coulomb (N/C), and it is important to note that the different direction components of the electric field are used in the equation. l (or the x, y, z) is the distance between the two described locations, which is measured in meters, and x, y, and z, are the different components of the difference.

The electric potential difference in a nonuniform field is [math]\displaystyle{ ∆{V} = -∑ \vec{E}·∆\vec{l} }[/math]. The different parts in this particular equation resembles the equation for the potential difference in an uniform field, except that with the nonuniform field, the potential difference in the different fields are summed up. This situation can be quite easy, but when the system gets difficult, first, choose a path and divide it into smaller pieces of [math]\displaystyle{ ∆\vec{l} }[/math]; second, write an expression for [math]\displaystyle{ ∆{V} = -\vec{E}·∆\vec{l} }[/math] of one piece; third, add up the contributions of all the pieces; last, check the result to make sure the magnitude, direction, and units make sense.

Aside from just calculating the value of the electric potential difference, determining the sign is also quite crucial to be successful. If the path being considered is in the same direction as the electric field, then the sign with be negative (-), or the potential is decreasing. If the path being considered is in the opposite direction as the electric field, then the sign will be positive (+), or the potential is increasing. If the path being considered is perpendicular to the electric field, then the potential difference will just be zero and have no direction. With these simple tips, the direction of the potential difference can be rechecked with the answer calculated using vectors.

Also, when working with different situations, it is nice to keep in mind that in a conductor, the electric field is zero. Therefore, the potential difference is zero as well. In an insulator, the electric field is [math]\displaystyle{ {E}_{applied} / K }[/math] where K is the dielectric constant. Also, the round trip potential difference is always zero, or in other words, if you start from a certain point and end at the same point, then, the potential difference will be zero.

A Computational Model

Click on the link to see Electric Potential through VPython!

Make sure to press "Run" to see the principle in action!

Teach hand-on with GlowScript

Watch this video for a more visual approach!

Electric Potential: Visualizing Voltage with 3D animations

Examples

Simple

In a capacitor, the negative charges are located on the left plate, and the positive charges are located on the right plate. Location A is at the left end of the capacitor, and Location B is at the right end of the capacitor, or in other words, Location A and B are only different in terms of their x component location. The path moves from A to B. What is the direction of the electric field? Is the potential difference positive or negative?

Answer: The electric field is to the left. The potential difference is increasing, or is positive.

Explanation: The electric field always moves away from the positive charge and towards the negative charge, which means the electric field in this example is to the left. Because the direction and the electric field and the direction of the path are opposite, the potential difference is increasing, or is positive.

Middling

Calculate the difference in electric potential between two locations A, which is at <-0.4, 0,0>m, and B, which is at <0.2,0,0>m. The electric field in the location is <500,0,0> N/C.

Answer: -300V

Explanation:

[math]\displaystyle{ ∆\vec{l} }[/math] = <0.2,0,0>m - <-0.4,0,0>m = <0.6,0,0>m

[math]\displaystyle{ ∆{V} = -({E}_{x}∆{x} + {E}_{y}∆{y} + {E}_{z}∆{z}) }[/math]

[math]\displaystyle{ ∆{V} = -(500 N/C * 0.6 m + 0*0 + 0*0) }[/math]

[math]\displaystyle{ ∆{V} = -300 V }[/math]

Difficult

Calculate the potential difference from x = 1e-10 to x=2e-8 of a point charge with Q = 1.6e-19 Columbs. Then use your answer to calculate the work done on this particle.

Answer: -14.3 V ; 2.3e-18 J

Explanation:

[math]\displaystyle{ ∆{V} = -\int_C^D {E}_{x} \, dx }[/math]

[math]\displaystyle{ ∆{V} = -\int_{1e-10}^{2e-8} \tfrac{1}{4π{ε}_{0}}*\tfrac{Q}{{x}^{2}} \, dx }[/math]

[math]\displaystyle{ ∆{V} = \tfrac{1}{4π{ε}_{0}}*{1.6e-19 C}*({\tfrac{1}{2e-8 m} - \tfrac{1}{1e-10 m}}) }[/math]

[math]\displaystyle{ ∆{V} }[/math] = -14.3 V

∆K + ∆U = [math]\displaystyle{ {W}_{ext} }[/math]

[math]\displaystyle{ {W}_{ext} }[/math] = 0 + (-e)(∆V)

[math]\displaystyle{ {W}_{ext} }[/math] = (-1.6e-19 C)(-14.3 V)

[math]\displaystyle{ {W}_{ext} }[/math] = 2.3e-18 J

Connectedness

How is this topic connected to something that you are interested in?

[Author] I am interested in robotic systems and building circuit boards and electrical systems for manufacturing robots. While studying this section in the book, I was able to connect back many of the concepts and calculations back to robotics and the electrical component of automated systems.

[Revisionist] Since high school, I never really understood how to work with the voltmeter and what it measured, and I have always wanted to know, but although this particular wiki page did not go into the details and other branches of electric potential, it led me to find the answers to something I was interested in since high school, the concept of electric potential.

[Editor] I think electively is really interesting. When I was younger, I participated in this demo where a group of people hold hands and someone touches this special ball full of charge. We all could feel the tingling sensation of the current passing through us. It’s cool to learn the theory behind the supposed magic that occurs.

How is it connected to your major?

[Author] I am a Mechanical Engineering major, so I will be dealing with the electrical components of machines when I work. Therefore, I have to know these certain concepts such as electric potential in order to fully understand how they work and interact.

[Revistionist] As a biochemistry major, electric potential and electric potential difference is not particularly related to my major, but in chemistry classes, we use electrostatic potential maps (electrostatic potential energy maps) that shows the charge distributions throughout a molecule. Although the main use in electric potential is different in physics and biochemistry (where physicists use it identify the effect of the electric field at a location), I still found it interesting as the concept of electric potential (buildup) was being used in quite a different way.

[Editor] I am a computer science major. Although I deal mostly with software, the hardware aspect is still important. The algorithms that I design run differently on different machines. The time complexity of an algorithm is sometimes useless when worrying about constant factors that are determined by a system’s hardware. Quicksort, for instance, is usually faster than many other sorts that have lower time complexities. The hardware of computers heavily relies on electricity and current (which is induced by a potential difference) to switch transistors on and off and thereby process information.

Is there an interesting industrial application?

[Author] Electrical potential is used to find the voltage across a path. This is useful when working with circuit components and attempting to manipulate the power output or current throughout a component.

[Revisionist] Electric potential sensors are being used to detect a variety of electrical signals made by the human body, thus contributing to the field of electrophysiology.

[Editor] The study of electric potential has lead scientists to generate very safe wires that will not overheat and cause fires. Connecting circuits to ground is important and the third prong in an electrical outlet is this ground connection.

History

The idea of electric potential, in a way, started with Ben Franklin and his experiments in the 1740s. He began to understand the flow of electricity, which eventually paved the path towards explaining electric potential and potential difference. Scientists finally began to understand how electric fields were actually affecting the charges and the surrounding environment. Benjamin Franklin first shocked himself in 1746, while conducting experiments on electricity with found objects from around his house. Six years later, or 261 years ago for us, the founding father flew a kite attached to a key and a silk ribbon in a thunderstorm and effectively trapped lightning in a jar. The experiment is now seen as a watershed moment in mankind's venture to channel a force of nature that was viewed quite abstractly.

By the time Franklin started experimenting with electricity, he'd already found fame and fortune as the author of Poor Richard's Almanack. Electricity wasn't a very well understood phenomenon at that point, so Franklin's research proved to be fairly foundational. The early experiments, experts believe, were inspired by other scientists' work and the shortcomings therein.

source: http://www.benjamin-franklin-history.org/kite-experiment/

That early brush with the dangers of electricity left an impression on Franklin. He described the sensation as "a universal blow throughout my whole body from head to foot, which seemed within as well as without; after which the first thing I took notice of was a violent quick shaking of my body." However, it didn't scare him away. In the handful of years before his famous kite experiment, Franklin contributed everything from designing the first battery designs to establishing some common nomenclature in the study of electricity. Although Franklin is often coined the father of electricity, after he set the foundations of electricity, many other scientists contributed his or her research in the advancement of electricity and eventually led to the discovery of electric potential and potential difference.

References

[1] "Benjamin Franklin and Electricity." Benjamin Franklin and Electricity. N.p., n.d. Web. 17 Apr. 2016. <http://www.americaslibrary.gov/aa/franklinb/aa_franklinb_electric_1.html>.

[2] Bottyan, Thomas. "Electrostatic Potential Maps." Chemwiki. N.p., 02 Oct. 2013. Web. 17 Apr. 2016. <http://chemwiki.ucdavis.edu/Core/Theoretical_Chemistry/Chemical_Bonding/General_Principles_of_Chemical_Bonding/Electrostatic_Potential_maps>.

[3] "Electric Potential Difference." Electric Potential Difference. The Physics Classroom, n.d. Web. 14 Apr. 2016. <http://www.physicsclassroom.com/class/circuits/Lesson-1/Electric-Potential-Difference>.

[4] Harland, C. J., T. D. Clark, and R. J. Prance. "Applications of Electric Potential (Displacement Current) Sensors in Human Body Electrophysiology." International Society for Industrial Process Tomography, n.d. Web. 16 Apr. 2016. <http://www.isipt.org/world-congress/3/269.html>.

[5] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.