Physics Book - User contributions [en]

Loop Rule

2026-04-30T19:05:09Z

Ss87:

'''Last edit by Shlesh Sakpal (Spring 2026)'''

The Loop Rule, also known as Kirchhoff's Second Law, is a fundamental principle of electric circuits which states that the sum of potential differences around a closed circuit is equal to zero. More simply, when you travel around an entire circuit loop, you will return to the starting voltage. Note that is only true when the magnetic field is neither fluctuating nor time varying. If a changing magnetic field links the closed loop, then the principle of energy conservation does not apply to the electric field, causing the Loop Rule to be inaccurate in this scenario.
==The Main Idea==
(The Energy Principle and the Loop Rule)

The loop rule simply states that in any round trip path in a circuit,
[[Electric Potential]] equals zero. Keep in mind that this applies through ANY round trip path; there can be
multiple round trip paths through more complex circuits. This principle deals with the conservation of energy within a circuit. Loop Rule and [[Node Rule]] are the two
fundamental principles of electric circuits and are used to determine the behaviors of electric circuits.
This principle is often used to solve for resistance or current passing through of light bulbs and other resistors, as well as the capacitance or charge of capacitors in a circuit.

'''A Visual Model'''

[[File:Visual_Model2.png]]

LOOP 1: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CF} + \Delta {V}_{FA} = 0 </math>

LOOP 2: <math> \Delta {V}_{FC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} = 0 </math>

LOOP 3: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} + \Delta {V}_{FA} = 0 </math>

To figure out the sign of the voltages, act as an observer walking along the path. Start at the negative end of the emf and continue walking along the path. The emf will be positive in the loop rule because you are moving from low to high voltage. Once you reach a resistor or capacitor, this will be negative in the loop rule equation because it is high to low voltage. Continue along the path until you return to the starting position.

[[File:loopexample.png]]

LOOP 1: <math> emf - I_1R_1 = 0 </math>

LOOP 2: <math> -I_1R_1 + I_2R_2 = 0 </math>

LOOP 3: <math> emf - I_2R_2 = 0 </math>

===Sign Conventions===

When applying the Loop Rule, the sign of each voltage term depends on both the
type of component and the direction you traverse it relative to current flow.
A consistent sign convention is essential for setting up correct loop equations.

Choose a direction to traverse the loop (clockwise or counterclockwise — either
works, as long as you stay consistent). Then apply the following rules for each
component you encounter:

{| class="wikitable" style="text-align:center"
! Component !! Traversal Direction !! Voltage Contribution
|-
| Battery/EMF || − terminal → + terminal (with EMF) || <math> +\varepsilon </math>
|-
| Battery/EMF || + terminal → − terminal (against EMF) || <math> -\varepsilon </math>
|-
| Resistor || In direction of assumed current || <math> -IR </math>
|-
| Resistor || Against direction of assumed current || <math> +IR </math>
|-
| Capacitor || In direction of assumed current || <math> -Q/C </math>
|-
| Capacitor || Against direction of assumed current || <math> +Q/C </math>
|}

'''Intuition Behind the Signs'''

* A battery traversed from − to + is like climbing a hill — you gain potential, so the contribution is positive.
* A battery traversed from + to − means you are descending in potential, so the contribution is negative.
* A resistor traversed in the direction of current means you are moving the way charges naturally lose energy, so the voltage drops (negative).
* A resistor traversed against the current means you are moving opposite to the natural energy loss, giving a voltage gain (positive).

'''Common Pitfall'''

If the assumed current direction turns out to be wrong, the solved value for
<math> I </math> will come out negative. This is perfectly valid — it simply means
the current flows in the opposite direction to what was assumed. Do not flip signs
mid-problem; let the algebra resolve the direction for you.

===A Mathematical Model===
A mathematical representation is:

:<math> \sum_{i=1}^n {V}_{i} = 0 </math>

where n is the number of voltages being measured in the loop, as well as

:<math> \Delta {V}_{1} + \Delta {V}_{2} + \space.... = 0 </math>

along any closed path in a circuit.

===A Computational Model===

[http://higheredbcs.wiley.com/legacy/college/cutnell/0470223553/concept_sims/sim34/sim34.html This] is a pretty cool model of how the Loop Rule is applied and calculated. You can change the direction of the current as well as the voltage of the batteries. To turn off the voice, press the Audio Tutorial button. To test your knowledge, click on the Concept Questions and Notes buttons, they have some questions and useful information in them.

[http://www.falstad.com/circuit/ This] is an online circuit simulator. It opens up with an LRC circuit that has current running through it. The graphs on the bottom show the voltage as the current runs through the circuit. You can see that after a full loop the voltage is 0, verifying the loop rule. You can make all sorts of different circuits and loops and see for yourself.

==Examples==

===Simple===
[[File:SimpleLoopRule.jpg|250 px]]

The circuit shown above consists of a single battery and a single resistor. The resistance of the wires is negligible for this problem.

'''Question'''

If the <math> emf </math> is 5 V and the resistance of the resistor is 10 ohms, what is the current passing through the resistor?

'''Solution'''

Although we can solve this using the <math>V = IR</math> equation for the whole loop, let's examine this problem using the loop rule equation.

The loop rule equation would be <math> {V}_{battery} - {V}_{resistor} = 0 </math>

Since we know the <math> emf </math> of the battery we just need to find the potential difference through the resistor. For this we can use the equation of <math> V = IR </math>.

Thus we now have an loop rule equation of <math> emf - IR = 0 </math>
From here it is a relatively simple process to find the current. We can rewrite the loop rule equation as <math> emf = IR </math> and then plug in 5 for the emf and
10 for the resistance, leaving us with I = .5 amperes.

===Middling===

[[File:ExampleLoopRule2.png|250 px]]

'''Question'''

The circuit shown above consists of a single battery, whose <math> emf </math> is 1.3 V, and three wires made of the same material, but having different cross-sectional areas.
Let the length of the thin wires be <math> {L}_{thick} </math> and the length of the thin wire be <math> {L}_{thin} </math>
Find a loop rule equation that starts at the negative end of the battery and goes counterclockwise through the circuit.

'''Solution'''

When beginning this problem, you must notice that the difference in cross-sectional areas affects the electric field in each wire. Because of this we will denote the electric field at D. as <math> {E}_{D} </math> and the electric field everywhere else as <math> {E}_{A} </math>.
To begin we will go around the circuit clockwise and add up each component. First, we know that the <math> emf </math> of the battery is 1.3 V. Then, we will add up the potential
voltage of each of the wires.

Remember that the electric potential of a wire is equal to the product of the electric field and the length of the wire. From this we can now find the potential difference
of each section of the wires. The electric potential of location A - C is <math> {E}_{A} * {L}_{thick} </math>. This is the same for the electric potential of locations E - G of the wire.
For the thin section of the wire, the electric potential is <math> {E}_{D} * {L}_{thin} </math>. From here we just go around the circuit counterclockwise and add each potential difference to the loop rule equation.

Thus we can find that a loop rule equation is: <math> emf - 2 ({E}_{A} * {L}_{thick}) - {E}_{D} * {L}_{thin} = 0 </math>

This can also be rewritten as: <math> emf = 2 ({E}_{A} * {L}_{thick}) + {E}_{D} * {L}_{thin} = 0 </math>

===Difficult===

[[File:HardLoopRule.jpg|250 px]]

'''Question'''

For the circuit above, imagine a situation where the switch has been closed for a long time.
Calculate the current at a,b,c,d,e and charge Q of the capacitor. Answer these using <math> emf, {R}_{1}, {R}_{2},
and \space C </math>

'''Solution'''

First, write loop rule equations for each of the possible loops in the circuit. There are 3 loop equations that are possible.

<math> emf - {I}_{1}{R}_{1} - {I}_{2}{R}_{2} = 0 </math>

<math> emf - {I}_{1}{R}_{1} - Q/C = 0 </math>

<math> {I}_{2}{R}_{2} - Q/C = 0 </math>

From here, we can then solve for the current passing through a,b,d and e. We also know that the current passing through
these points must be the same so <math> {I}_{1} = {I}_{2} </math>

<math> emf - {I}_{1}{R}_{1} - {I}_{1}{R}_{2} = 0 </math>

<math> emf = {I}_{1}({R}_{1} + {R}_{2}) = 0 </math>

<math> emf/({R}_{1} + {R}_{2}) = {I}_{1} </math>

So the current at a,b,d,e = <math> emf/({R}_{1} + {R}_{2}) </math>

You must also know that once a capacitor is charging for a long time, current no longer flows through the capacitor. We can then
easily solve for c because since current is no longer flowing through the capacitor, the current at c = 0.

Lastly, we will use the loop rule equation of <math> {I}_{2}{R}_{2} - Q/C = 0 </math> to solve for Q.

<math> {I}_{2}{R}_{2} = Q/C </math>

<math> C*({I}_{2}{R}_{2}) = Q </math>

Since <math> {I}_{1} = {I}_{2} </math> , <math> C*({I}_{1}{R}_{2}) = Q </math>

We can plug in what we found <math> {I}_{1} </math> equals from before.

<math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}) = Q </math>

Lastly, Q = <math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math> and we have now solved the problem.

Current at a,b,d,e = <math> emf/({R}_{1} + {R}_{2}) </math>

Current at c = 0

Q = <math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math>

==Connectedness==

The Loop Rule is simply an extension of the conservation of energy applied to circuits. Circuits are ubiquitous as they are featured in almost every technology today. Our understanding of these technologies is rooted in the empirical discoveries made in the mid 19th century, and further boosted by the the advancement of theoretical knowledge due to the likes of Faraday and Maxwell.

One interesting note about Loop Rule is that is does not apply universally to all circuits. In particular, AC (alternating current) circuits at high frequencies, have a fluctuating
electric charge that changes direction. This causes the electric potential of a round trip path around the circuit to no longer be zero. However, DC (direct current) circuits, and low frequency circuits in general, still follow the loop rule.

==History==

The Loop Rule is formally known as the [https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s_voltage_law_.28KVL.29 Kirchhoff Voltage Laws], named after [[Gustav Kirchhoff]], the scientist who discovered and defined this fundamental concept of electric circuits. He discovered this during his time as a student at Albertus University of Königsberg in 1845. Kirchoff went on to explore the topics of spectroscopy and black body radiation after his graduation from Albertus. Nowadays, it is used very often in electrical engineering.

[[File:KirchoffLoopRule.jpg|thumb|right||Gustav Kirchhoff]]

==Practical Applications==
The Loop Rule is not just a theoretical concept - it's actively used by engineers and technicians in the real world as well, something we tend to forget from our physics textbook.

===Circuit Design and Analysis===
Electrical engineers use it when designing circuits for electronic devices like smartphones, computers, and medical equipment and a vast plethora of other stuff that we see in our daily life. Before building a prototype, they apply the Loop Rule to calculate voltage drops across components and ensure each part receives the correct voltage. This prevents component damage as well as ensures the device functions properly.

===Troubleshooting and Repair===
When a circuit isn't working perfectly, technicians as well as electricians use the infamous Loop Rule to diagnose problems. By measuring voltages at different points in a circuit and comparing them to expected values from Kirchof's Loop Rule calculations, they can identify faulty components like burned-out resistors as well as dead batteries. For example, if a loop should sum to zero and it doesn't, there's likely a short circuit somewhere or broken connection.

===Power Distribution Systems===
Utility companies use principles based on the Loop Rule to manage gigantic electrical grids. When power flows through transmission lines from power plants to homes and businesses, engineers must account for voltage drops across long distances and utilizing the Loop Rule helps them calculate these drops and determine where to place voltage boosters to maintain proper power delivery.

===Battery Management Systems===
Modern electric vehicles and portable electronics use sophisticated battery management systems. These systems apply the Loop Rule to monitor individual cells in battery packs, ensuring each cell charges and discharges properly. This extends battery life and prevents dangerous conditions like overcharging. This is seen a lot in EV car racing teams, including Georgia Tech's own Hytech team which undergoes a long discussion on energy and current flow for both low and high voltage electronics.

===Sensor Networks===
Many sensor systems, including home thermostats and industrial monitoring equipment, use circuits with multiple branches. The Loop Rule helps engineers design these networks to ensure accurate readings and reliable operation even when multiple sensors share power sources.

== See also ==
===Further Reading===
Other Circuit Concepts you can check out :

[[Node Rule]]

[[Components]]

[[Resistors and Conductivity]]

[[Current]]

[https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws Kirchoff's Circuit Laws - Wikipedia]

===External Links===

If you want to test your knowledge, Khan Academy's DC Circuit Analysis under the Electrical Engineering Topic is an excellent resource. There are questions, videos, and written explanations to help you understand not just the loop rule, but the node rule as well. Click [https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic#ee-dc-circuit-analysis here] to access it.

[https://www.boundless.com/physics/textbooks/boundless-physics-textbook/circuits-and-direct-currents-20/kirchhoff-s-rules-152/the-loop-rule-540-5636/ Loop Rule - Boundless.com Textbook]

[https://www.youtube.com/watch?v=IlyUtYRqMLs Doc Physics Video Lecture]

[https://www.youtube.com/watch?v=TdCuu-4wm44 Doc Physics Worked Example]

[https://www.youtube.com/watch?v=paDs-Hnmklo Bozeman Science Lecture]

==References==

* Dr. Nicholas Darnton's lecture notes
* https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s_voltage_law
* https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-dc-circuit-analysis
* Chabay, Ruth W. Matter and Interactions: Electric and Magnetic Interactions. John Wiley, 2015. Print.
* http://www.ux1.eiu.edu/~cfadd/1360/28DC/Loop.html
* http://higheredbcs.wiley.com/legacy/college/cutnell/0470223553/concept_sims/sim34/sim34.html
* http://www.falstad.com/circuit/
* Schuster, D. (Producer). (2013). Kirchhoff's Loop and Junction Rules Theory [Motion picture]. United States of America: YouTube.
* Schuster, D. (Producer). (2013). Kirchhoff's Rules (Laws) Worked Example [Motion picture]. United States of America: YouTube.
* Anderson, P. (Producer). (2015). Kirchoff's Loop Rule [Motion picture]. United States of America: YouTube.

[[Category: Electric Fields and energy in circuits]]

Loop Rule

2026-04-30T19:04:41Z

Ss87: added section on sign conventions

'''Claimed by Aazam Alam (Fall 2025)'''

The Loop Rule, also known as Kirchhoff's Second Law, is a fundamental principle of electric circuits which states that the sum of potential differences around a closed circuit is equal to zero. More simply, when you travel around an entire circuit loop, you will return to the starting voltage. Note that is only true when the magnetic field is neither fluctuating nor time varying. If a changing magnetic field links the closed loop, then the principle of energy conservation does not apply to the electric field, causing the Loop Rule to be inaccurate in this scenario.
==The Main Idea==
(The Energy Principle and the Loop Rule)

The loop rule simply states that in any round trip path in a circuit,
[[Electric Potential]] equals zero. Keep in mind that this applies through ANY round trip path; there can be
multiple round trip paths through more complex circuits. This principle deals with the conservation of energy within a circuit. Loop Rule and [[Node Rule]] are the two
fundamental principles of electric circuits and are used to determine the behaviors of electric circuits.
This principle is often used to solve for resistance or current passing through of light bulbs and other resistors, as well as the capacitance or charge of capacitors in a circuit.

'''A Visual Model'''

[[File:Visual_Model2.png]]

LOOP 1: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CF} + \Delta {V}_{FA} = 0 </math>

LOOP 2: <math> \Delta {V}_{FC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} = 0 </math>

LOOP 3: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} + \Delta {V}_{FA} = 0 </math>

To figure out the sign of the voltages, act as an observer walking along the path. Start at the negative end of the emf and continue walking along the path. The emf will be positive in the loop rule because you are moving from low to high voltage. Once you reach a resistor or capacitor, this will be negative in the loop rule equation because it is high to low voltage. Continue along the path until you return to the starting position.

[[File:loopexample.png]]

LOOP 1: <math> emf - I_1R_1 = 0 </math>

LOOP 2: <math> -I_1R_1 + I_2R_2 = 0 </math>

LOOP 3: <math> emf - I_2R_2 = 0 </math>

===Sign Conventions===

When applying the Loop Rule, the sign of each voltage term depends on both the
type of component and the direction you traverse it relative to current flow.
A consistent sign convention is essential for setting up correct loop equations.

Choose a direction to traverse the loop (clockwise or counterclockwise — either
works, as long as you stay consistent). Then apply the following rules for each
component you encounter:

{| class="wikitable" style="text-align:center"
! Component !! Traversal Direction !! Voltage Contribution
|-
| Battery/EMF || − terminal → + terminal (with EMF) || <math> +\varepsilon </math>
|-
| Battery/EMF || + terminal → − terminal (against EMF) || <math> -\varepsilon </math>
|-
| Resistor || In direction of assumed current || <math> -IR </math>
|-
| Resistor || Against direction of assumed current || <math> +IR </math>
|-
| Capacitor || In direction of assumed current || <math> -Q/C </math>
|-
| Capacitor || Against direction of assumed current || <math> +Q/C </math>
|}

'''Intuition Behind the Signs'''

* A battery traversed from − to + is like climbing a hill — you gain potential, so the contribution is positive.
* A battery traversed from + to − means you are descending in potential, so the contribution is negative.
* A resistor traversed in the direction of current means you are moving the way charges naturally lose energy, so the voltage drops (negative).
* A resistor traversed against the current means you are moving opposite to the natural energy loss, giving a voltage gain (positive).

'''Common Pitfall'''

If the assumed current direction turns out to be wrong, the solved value for
<math> I </math> will come out negative. This is perfectly valid — it simply means
the current flows in the opposite direction to what was assumed. Do not flip signs
mid-problem; let the algebra resolve the direction for you.

===A Mathematical Model===
A mathematical representation is:

:<math> \sum_{i=1}^n {V}_{i} = 0 </math>

where n is the number of voltages being measured in the loop, as well as

:<math> \Delta {V}_{1} + \Delta {V}_{2} + \space.... = 0 </math>

along any closed path in a circuit.

===A Computational Model===

[http://higheredbcs.wiley.com/legacy/college/cutnell/0470223553/concept_sims/sim34/sim34.html This] is a pretty cool model of how the Loop Rule is applied and calculated. You can change the direction of the current as well as the voltage of the batteries. To turn off the voice, press the Audio Tutorial button. To test your knowledge, click on the Concept Questions and Notes buttons, they have some questions and useful information in them.

[http://www.falstad.com/circuit/ This] is an online circuit simulator. It opens up with an LRC circuit that has current running through it. The graphs on the bottom show the voltage as the current runs through the circuit. You can see that after a full loop the voltage is 0, verifying the loop rule. You can make all sorts of different circuits and loops and see for yourself.

==Examples==

===Simple===
[[File:SimpleLoopRule.jpg|250 px]]

The circuit shown above consists of a single battery and a single resistor. The resistance of the wires is negligible for this problem.

'''Question'''

If the <math> emf </math> is 5 V and the resistance of the resistor is 10 ohms, what is the current passing through the resistor?

'''Solution'''

Although we can solve this using the <math>V = IR</math> equation for the whole loop, let's examine this problem using the loop rule equation.

The loop rule equation would be <math> {V}_{battery} - {V}_{resistor} = 0 </math>

Since we know the <math> emf </math> of the battery we just need to find the potential difference through the resistor. For this we can use the equation of <math> V = IR </math>.

Thus we now have an loop rule equation of <math> emf - IR = 0 </math>
From here it is a relatively simple process to find the current. We can rewrite the loop rule equation as <math> emf = IR </math> and then plug in 5 for the emf and
10 for the resistance, leaving us with I = .5 amperes.

===Middling===

[[File:ExampleLoopRule2.png|250 px]]

'''Question'''

The circuit shown above consists of a single battery, whose <math> emf </math> is 1.3 V, and three wires made of the same material, but having different cross-sectional areas.
Let the length of the thin wires be <math> {L}_{thick} </math> and the length of the thin wire be <math> {L}_{thin} </math>
Find a loop rule equation that starts at the negative end of the battery and goes counterclockwise through the circuit.

'''Solution'''

When beginning this problem, you must notice that the difference in cross-sectional areas affects the electric field in each wire. Because of this we will denote the electric field at D. as <math> {E}_{D} </math> and the electric field everywhere else as <math> {E}_{A} </math>.
To begin we will go around the circuit clockwise and add up each component. First, we know that the <math> emf </math> of the battery is 1.3 V. Then, we will add up the potential
voltage of each of the wires.

Remember that the electric potential of a wire is equal to the product of the electric field and the length of the wire. From this we can now find the potential difference
of each section of the wires. The electric potential of location A - C is <math> {E}_{A} * {L}_{thick} </math>. This is the same for the electric potential of locations E - G of the wire.
For the thin section of the wire, the electric potential is <math> {E}_{D} * {L}_{thin} </math>. From here we just go around the circuit counterclockwise and add each potential difference to the loop rule equation.

Thus we can find that a loop rule equation is: <math> emf - 2 ({E}_{A} * {L}_{thick}) - {E}_{D} * {L}_{thin} = 0 </math>

This can also be rewritten as: <math> emf = 2 ({E}_{A} * {L}_{thick}) + {E}_{D} * {L}_{thin} = 0 </math>

===Difficult===

[[File:HardLoopRule.jpg|250 px]]

'''Question'''

For the circuit above, imagine a situation where the switch has been closed for a long time.
Calculate the current at a,b,c,d,e and charge Q of the capacitor. Answer these using <math> emf, {R}_{1}, {R}_{2},
and \space C </math>

'''Solution'''

First, write loop rule equations for each of the possible loops in the circuit. There are 3 loop equations that are possible.

<math> emf - {I}_{1}{R}_{1} - {I}_{2}{R}_{2} = 0 </math>

<math> emf - {I}_{1}{R}_{1} - Q/C = 0 </math>

<math> {I}_{2}{R}_{2} - Q/C = 0 </math>

From here, we can then solve for the current passing through a,b,d and e. We also know that the current passing through
these points must be the same so <math> {I}_{1} = {I}_{2} </math>

<math> emf - {I}_{1}{R}_{1} - {I}_{1}{R}_{2} = 0 </math>

<math> emf = {I}_{1}({R}_{1} + {R}_{2}) = 0 </math>

<math> emf/({R}_{1} + {R}_{2}) = {I}_{1} </math>

So the current at a,b,d,e = <math> emf/({R}_{1} + {R}_{2}) </math>

You must also know that once a capacitor is charging for a long time, current no longer flows through the capacitor. We can then
easily solve for c because since current is no longer flowing through the capacitor, the current at c = 0.

Lastly, we will use the loop rule equation of <math> {I}_{2}{R}_{2} - Q/C = 0 </math> to solve for Q.

<math> {I}_{2}{R}_{2} = Q/C </math>

<math> C*({I}_{2}{R}_{2}) = Q </math>

Since <math> {I}_{1} = {I}_{2} </math> , <math> C*({I}_{1}{R}_{2}) = Q </math>

We can plug in what we found <math> {I}_{1} </math> equals from before.

<math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}) = Q </math>

Lastly, Q = <math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math> and we have now solved the problem.

Current at a,b,d,e = <math> emf/({R}_{1} + {R}_{2}) </math>

Current at c = 0

Q = <math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math>

==Connectedness==

The Loop Rule is simply an extension of the conservation of energy applied to circuits. Circuits are ubiquitous as they are featured in almost every technology today. Our understanding of these technologies is rooted in the empirical discoveries made in the mid 19th century, and further boosted by the the advancement of theoretical knowledge due to the likes of Faraday and Maxwell.

One interesting note about Loop Rule is that is does not apply universally to all circuits. In particular, AC (alternating current) circuits at high frequencies, have a fluctuating
electric charge that changes direction. This causes the electric potential of a round trip path around the circuit to no longer be zero. However, DC (direct current) circuits, and low frequency circuits in general, still follow the loop rule.

==History==

The Loop Rule is formally known as the [https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s_voltage_law_.28KVL.29 Kirchhoff Voltage Laws], named after [[Gustav Kirchhoff]], the scientist who discovered and defined this fundamental concept of electric circuits. He discovered this during his time as a student at Albertus University of Königsberg in 1845. Kirchoff went on to explore the topics of spectroscopy and black body radiation after his graduation from Albertus. Nowadays, it is used very often in electrical engineering.

[[File:KirchoffLoopRule.jpg|thumb|right||Gustav Kirchhoff]]

==Practical Applications==
The Loop Rule is not just a theoretical concept - it's actively used by engineers and technicians in the real world as well, something we tend to forget from our physics textbook.

===Circuit Design and Analysis===
Electrical engineers use it when designing circuits for electronic devices like smartphones, computers, and medical equipment and a vast plethora of other stuff that we see in our daily life. Before building a prototype, they apply the Loop Rule to calculate voltage drops across components and ensure each part receives the correct voltage. This prevents component damage as well as ensures the device functions properly.

===Troubleshooting and Repair===
When a circuit isn't working perfectly, technicians as well as electricians use the infamous Loop Rule to diagnose problems. By measuring voltages at different points in a circuit and comparing them to expected values from Kirchof's Loop Rule calculations, they can identify faulty components like burned-out resistors as well as dead batteries. For example, if a loop should sum to zero and it doesn't, there's likely a short circuit somewhere or broken connection.

===Power Distribution Systems===
Utility companies use principles based on the Loop Rule to manage gigantic electrical grids. When power flows through transmission lines from power plants to homes and businesses, engineers must account for voltage drops across long distances and utilizing the Loop Rule helps them calculate these drops and determine where to place voltage boosters to maintain proper power delivery.

===Battery Management Systems===
Modern electric vehicles and portable electronics use sophisticated battery management systems. These systems apply the Loop Rule to monitor individual cells in battery packs, ensuring each cell charges and discharges properly. This extends battery life and prevents dangerous conditions like overcharging. This is seen a lot in EV car racing teams, including Georgia Tech's own Hytech team which undergoes a long discussion on energy and current flow for both low and high voltage electronics.

===Sensor Networks===
Many sensor systems, including home thermostats and industrial monitoring equipment, use circuits with multiple branches. The Loop Rule helps engineers design these networks to ensure accurate readings and reliable operation even when multiple sensors share power sources.

== See also ==
===Further Reading===
Other Circuit Concepts you can check out :

[[Node Rule]]

[[Components]]

[[Resistors and Conductivity]]

[[Current]]

[https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws Kirchoff's Circuit Laws - Wikipedia]

===External Links===

If you want to test your knowledge, Khan Academy's DC Circuit Analysis under the Electrical Engineering Topic is an excellent resource. There are questions, videos, and written explanations to help you understand not just the loop rule, but the node rule as well. Click [https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic#ee-dc-circuit-analysis here] to access it.

[https://www.boundless.com/physics/textbooks/boundless-physics-textbook/circuits-and-direct-currents-20/kirchhoff-s-rules-152/the-loop-rule-540-5636/ Loop Rule - Boundless.com Textbook]

[https://www.youtube.com/watch?v=IlyUtYRqMLs Doc Physics Video Lecture]

[https://www.youtube.com/watch?v=TdCuu-4wm44 Doc Physics Worked Example]

[https://www.youtube.com/watch?v=paDs-Hnmklo Bozeman Science Lecture]

==References==

* Dr. Nicholas Darnton's lecture notes
* https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s_voltage_law
* https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-dc-circuit-analysis
* Chabay, Ruth W. Matter and Interactions: Electric and Magnetic Interactions. John Wiley, 2015. Print.
* http://www.ux1.eiu.edu/~cfadd/1360/28DC/Loop.html
* http://higheredbcs.wiley.com/legacy/college/cutnell/0470223553/concept_sims/sim34/sim34.html
* http://www.falstad.com/circuit/
* Schuster, D. (Producer). (2013). Kirchhoff's Loop and Junction Rules Theory [Motion picture]. United States of America: YouTube.
* Schuster, D. (Producer). (2013). Kirchhoff's Rules (Laws) Worked Example [Motion picture]. United States of America: YouTube.
* Anderson, P. (Producer). (2015). Kirchoff's Loop Rule [Motion picture]. United States of America: YouTube.

[[Category: Electric Fields and energy in circuits]]

Spring Potential Energy

2025-11-19T10:27:10Z

Ss87: Added tables to the main idea sections that include common mistakes made when conceptualizing springs, and added a few more examples to the connectedness section

'''Shlesh Sakpal (Ss87) Fall 2025 11/19/2025'''

This topic covers Spring Potential Energy.

==The Main Idea==

[[File:spring2456.png|thumb|500 px| Spring Potential Energy]]

Spring potential energy, also known as elastic potential energy,is the stored energy in a spring, that can potentially be converted into kinetic energy. The energy stored in the spring is due to the deformation of the spring, often from stretching and compressing. The force excreted to stretch or compress a spring is known as Hooke's law, '''Fs = -ksx''', where '''Fs''' is force, '''x''' is the displacement, and '''-ks''' is the spring constant. The spring constant being unique for every spring depends on factors such as material and thickness of coiled wire. If a spring is not stretched or compressed, then it is at equilibrium. At equilibrium a spring has no potential energy, assuming there is no force being applied to the spring.

===A Mathematical Model===
The formula for Force of a Spring:

'''Fs = -ksx

The formula for Spring Potential Energy:

'''Us = 1&frasl;2ksx2'''

where:

'''ks''' = spring constant

'''x2''' = stretch measured from the equilibrium point;

===A Computational Model===
'''An oscillating spring on the floor:'''

GlowScript 2.9 VPython
display(width=600,height=600,center=vector(6,0,0),background=color.black)
mbox=2
L0 = vector(9,0,0)
ks = 1
deltat = .01
t = 0
wall=box(pos=vector(0,1,0),size=vector(0.2,3,2),color=color.cyan)
floor=box(pos=vector(7.2,-0.6,0),size=vector(14,0.2,4),color=color.cyan)
box=box(pos=vector(12,0,0),size=vector(1,1,1),color=color.red)
pivot=vector(0,0,0)
spring=helix(pos=pivot,axis=box.pos-pivot,radius=0.4,constant=1,thickness=0.1,coils=20,color=color.orange)
box.p = vector(0,0,0)
while (t<50):
rate(100)
s = wall.pos - box.pos
Fspring=(L0-box.pos)*(ks)
box.p= box.p +Fspring*deltat
box.pos = box.pos + box.p*deltat
spring.axis = box.pos - spring.pos
t = t+ deltat

[[File:spring2457.png|500 px| Oscillating Spring on Floor]]

Link to Simulation:
https://trinket.io/glowscript/3146b836dc

'''A hanging oscillating spring:'''

from __future__ import division
from visual import *
from visual.graph import *
scene.width=600
scene.height = 760
g = 9.8
mball = .2
Lo = 0.3
ks = 12
deltat = 1e-3
t = 0
ceiling = box(pos=(0,0,0), size = (0.5, 0.01, 0.2))
ball = sphere(pos=(0,-0.3,0), radius=0.025, color=color.yellow)
spring = helix(pos=ceiling.pos, color=color.green, thickness=.005, coils=10, radius=0.01)
spring.axis = ball.pos - ceiling.pos
vball = vector(0.02,0,0)
ball.p = mball*vball
scene.autoscale = 0
scene.center = vector(0,-Lo,0)
while t < 10:
rate(1000)
L_vector = (mag(ball.pos) - Lo)* ball.pos.norm()
Fspring = -ks * L_vector
Fgrav = vector(0,-mball * g,0)
Fnet = Fspring + Fgrav
ball.p = ball.p + Fnet * deltat
ball.pos = ball.pos + (ball.p/mball) * deltat
spring.axis = ball.pos-ceiling.pos
t = t + deltat

[[File:Animated-mass-spring-faster.gif|200 px]]

'''Code to visualize a spring's motion given x and y coordinate input:'''

Web VPython 3.2
scene.background = color.white
ball = sphere(radius=0.03, color=color.blue)
trail = curve(color=ball.color)
origin = sphere(pos=vector(0,0,0), color=color.yellow, radius=0.015)
spring = helix(color=color.cyan, thickness=0.006, coils=40, radius=0.015)
spring.pos = origin.pos
xplot = graph(title="x-position vs time", xtitle="time (s)", ytitle="x-position (m)")
xposcurve = gcurve(color=color.blue, width=4, label="model")
xpos2curve = gcurve(color=color.red, width=4, label="experiment")
yplot = graph(title="y-position vs time", xtitle="time (s)", ytitle="y-position (m)")
yposcurve = gcurve(color=color.blue, width=4, label="model")
ypos2curve = gcurve(color=color.red, width=4, label="experiment")
eplot = graph(title="Change in Energy vs Time", xtitle="Time (s)", ytitle="Change in Energy (J)")
dKcurve = gcurve(color=color.blue, width=4, label="deltaK")
dUgcurve = gcurve(color=color.red, width=4, label="deltaUgrav")
dUscurve = gcurve(color=color.green, width=4, label="deltaUspring")
dEcurve = gcurve(color=color.orange, width=4, label="deltaE")
ball2 = sphere(radius=0.025, color=color.red)
ball.m = 0.402
ball.pos = vector(0.55,-0.0039,0)
ball.vel = vector(0,0,0)
X = []
Y = []
obs = read_local_file(scene.title_anchor).text;
for line in obs.split('\n'):
if line != '':
line = line.split(',')
X.append(float(line[0]))
Y.append(float(line[1]))
idx = 0 #variable used to select data from list.
cnt = 0 #variable to keep track of predictions made between each measurement
t = 0
deltat = (5.5/len(X))/20 #choose this small AND an integer multiple of the time interval between frames of experiment video
g = 9.8
k_s = 8.87
L0 = 0.123
L = spring.pos-ball.pos
Lhat = L/mag(L)
s = mag(L)-L0
K = (1/2)*ball.m*mag(ball.vel)**2 # kinetic energy
Ug = ball.m * g * ball.pos.y # gravitational potential energy
Us = 1/2*k_s*s**2 # spring potential energy
E = K + Ug + Us # total energy
while t < 5.5:
K_i = K
Ug_i = Ug
Us_i = Us
E_i = E
Fgrav = vector(0,-ball.m*g,0)
Fspring = -k_s*s*Lhat
Fnet = Fspring + Fgrav
ball.vel = ball.vel+(Fnet/ball.m)*deltat
ball.pos = ball.pos+ball.vel*deltat
L = ball.pos-spring.pos
Lhat = L/mag(L)
s = mag(L)-L0
spring.axis = L
trail.append(pos=ball.pos)
K = (1/2)*ball.m*mag(ball.vel)**2
deltaK = K-K_i
Ug = ball.m*g*ball.pos.y
deltaUg = Ug-Ug_i
Us = 1/2*k_s*s**2
deltaUs = Us-Us_i
E = K + Ug + Us
deltaE = deltaK+deltaUg+deltaUs
dKcurve.plot(t,deltaK) # blue
dUgcurve.plot(t,deltaUg) # red
dUscurve.plot(t,deltaUs) # green
dEcurve.plot(t,deltaE) # orange
xposcurve.plot(t,ball.pos.x)
yposcurve.plot(t,ball.pos.y)
# Update time
t = t + deltat
rate(1000)

===Common Mistakes and Misconceptions with Spring Problems===

Students usually struggle with signs and displacement when working with Hooke's Law. Here are some common misconceptions that are made when working with springs.

{| class = "wikitable"
! Misconception
! Why It's Wrong
! Correct Idea
|-
| The sign in Hooke's Law doesn't matter
| The negative sign in Hooke's Law is often dropped students
| The negative sign shows that the force always points toward equilibrium
|-
| x is the total length of the spring
| Students often plug in L instead of <math>(L - L_0)</math>
| x must be the displacement from equilibrium, not the total length
|-
| Springs store energy only when stretched
| Students often forget that compression also stores energy
| Spring potential energy depends on **any** displacement from equilibrium
|-
| Spring potential energy is negative when the spring is compressed
| Students confuse spring energy with gravitational potential energy
| Spring potential energy is always positive because it depends on x²
|}

===Spring Quantities at a Glance===

{| class = "wikitable"
! Quantity
! Symbol
! Formula
! Units
! Meaning
|-
| Spring force
| <math>F_s</math>
| <math>F_s = -kx</math>
| N
| Restoring force toward equilibrium.
|-
| Spring potential energy
| <math>U_s</math>
| <math>U_s = \frac{1}{2}kx^2</math>
| J
| Stored elastic energy.
|-
| Stretch/compression
| <math>x</math>
| <math>x = L - L_0</math>
| m
| Displacement from equilibrium.
|-
| Spring constant
| <math>k</math>
| —
| N/m
| Measures stiffness of the spring.
|}

==Examples==

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

===Simple===

'''Question'''

If a spring's spring constant is 200 N/m and it is stretched 1.5 meters from rest, what is the potential spring energy?

'''Solution'''

'''ks'''= 200 N/m

'''s'''= 1.5 m

'''Us'''=(0.5)kss2

'''Us'''= (0.5)(200 N/m)(1.5 m)2

'''Us'''= 225 J

===Middling===

'''Question'''

A horizontal spring with stiffness 0.6 N/m has a relaxed length of 10 cm. A mass of 25 g is attached and you stretch the spring to a length of 20 cm. The mass is released and moves with little friction. What is the speed of the mass at the moment when the spring returns to its relaxed length of 10cm?

'''Solution'''

'''ks'''= 0.6 N/m

'''s'''= 0.1 m

'''Us'''=(.5)kss2 = (.5)(0.6 N/m)(0.1 m)2

'''Us''' = 0.003 J

Potential Energy is Converted into Kinetic Energy (K):

'''Us''' = K

'''Us''' =(0.5)mv2

0.003 J = (0.5)(0.025 kg)v2

'''v2''' = (0.003 J)&frasl;((0.5)(0.025 kg))

'''v2''' = 0.24 J/kg*s

'''v''' = 0.49 m/s

===Difficult===

'''Question'''

A package of mass 9 kg sits on an airless asteroid with mass 8.0x1020 kg and radius 8.7x105 m. Your goal is to launch the package so that it will never come back and when it is very far away it will have a speed of 226 m/s. You have a spring whose stiffness is 2.8x105 N/m. How much must you compress the spring?

'''Solution'''

The initial condition for escape from the asteroid is:

Ki+Ui=1&frasl;2mvesc2 + (-G*Mm&frasl;R)=0

Potential energy of the spring equals the total energy in the system.

'''1&frasl;2kss2'''=1&frasl;2mvesc2 +(-G*Mm&frasl;R)

'''s2'''= m&frasl;ks(2GM&frasl;R +v2)=s2 = 9&frasl;2.8x105(2G8.0x1020&frasl;8.7x105+2262)

'''s2'''=5.58 m

'''s'''=2.36 m

===Graphing===
'''Energy graph'''

The energy graph of a oscillating spring continuously switches between kinetic and potential energy because springs are continuously returning to equilibrium. The following is an example of a computational model of a hanging spring with an initial force in the positive x direction. As seen below, the energies the spring contains will always equal out to zero despite each fluctuating individually.

[[File:Spring energy.jpg]]

==Connectedness==

Because springs are all around us, from Slinkies to parts in automobiles, spring potential energy is useful in everyday life. One example of this is a trampoline. Without potential spring energy to allow for bounce, a trampoline would simply be a boring stretch of fabric. Spring potential is also used to absorb shock in vehicles. This allows for a smoother ride while traveling over bumps in the road. An extreme example of spring energy we commonly come across is the mechanism for garage doors. These doors contain large springs that allow them to open and close. These springs can lift an average of 400 pounds, giving them a huge amount of potential energy.

===Some Real World Applications of Spring Potential Energy===

Examples of springs being used can be found all over our daily lives, even in places you wouldn't expect. For example, they are found in:
* Car Suspension Systems
* Mechanical Watches
* Trampolines
* Archery
* Exercise Equipment

{| class="wikitable"
! Device
! Type of “Spring”
! What the Energy Becomes
|-
| Car suspension
| Metal coil + damper
| Mostly heat due to damping.
|-
| Trampoline
| Metal springs and elastic fabric
| Kinetic and gravitational potential energy when being used.
|-
| Archery
| Elastic bow (not a metal spring)
| Kinetic energy of the arrow.
|-
| Mechanical watches
| The mainspring winds up and slowly releases spring potential energy to turn the watch
| Mechanical motion of gears.
|}

[[File:trampoline.jpg|thumb|Trampoline Potential Energy]]

==History==

Elastic Potential Energy stemmed from the ideas of Robert Hooke, a 17th century British physicist who studied the relationship between forces applied to springs and elasticity. Hooke’s Law, which is a principle that states that the that the force needed to extend or compress a spring by a distance is proportional to that distance.

== See also ==

Spring potential energy is related to [[Hooke's Law]] and [[Potential Energy]].

===Further reading===

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

https://www.khanacademy.org/science/ap-physics-1/ap-work-and-energy/spring-potential-energy-and-hookes-law-ap/a/spring-force-and-energy-ap1

https://openstax.org/books/university-physics-volume-1/pages/8-1-potential-energy-of-a-system

===External links===

[http://www.physicsclassroom.com/class/energy/Lesson-1/Potential-Energy]

==References==
#http://www.universetoday.com/55027/hookes-law/
#http://www.scienceclarified.com/everyday/Real-Life-Physics-Vol-2/Oscillation-Real-life-applications.html
#http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html
#Chabay and Bruce A. Sherwood. Matter & Interactions. 4th ed.

[[Category:Energy]]