Claimed by Maria Rivero
Claimed by Yeon Jae Cho (FALL 2016)

'''Claimed by Stella Chen (Fall 2017)'''

==Ampere-Maxwell Law==

The Ampere-Maxwell Law expands on Ampere's Law relating magnetic field and current on a closed loop. Ampere's Law:

[[File: Amperes.PNG]]

allowed us to explore the relationship between current and magnetic field on a chosen closed loop called an "Amperian path". However, for certain examples in which electric field varies with time, simply using Ampere's Law is not sufficient. The Ampere-Maxwell Law accounts for these situations and establishes that a time dependent electric field is associated with a corresponding magnetic field.

The Ampere-Maxwell Law is also known as the Ampere's Circuital Law, and should not be confused with Ampere's Force Law.

===Mathematical Model===

[[File: Ampere-Maxwell.png]]

Where

*B is the magnetic field,

*dl is the change in path, and

*the sum of I is the sum of the charges inside the path.

==Examples==

===Simple===

Question: A 30-A current is charging a capacitor that has circular plates 5 mm in diameter. The plate separation is 4 mm. What is the magnetic field between the plates 1 mm from the center?

[[File:Example1.png]]

Answer:
:<math>\oint \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \mu_0 ε_0 \frac{d\Phi_E}{dt}</math>

this is simply the Ampere-Maxwell Law

:<math>\oint \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \mu_0 ε_0 \frac{d}{dt}(\frac{q \pi r^2}{ε_0 A})</math>

this step breaks down the electric flux part of the equation

:<math>\mathbf{B} \cdot 2\pi r = \mu_0 ε_0 \frac{dq}{dt}\frac{\pi r^2}{ε_0 A}</math>

Since the path is a circle, replace dl with circumference of the circle

:<math>\mathbf{B} = \frac{\mu_0 I r}{2A} = \frac{\mu_0 I r}{2 \pi R^2}</math>

Continue to simplify and plug in I for dq/dt and plug in values (where I = 30 A, r = 1 mm, R = 2.5 mm)

:<math>\mathbf{B} = 1 mT</math>

===Middling===

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

[[File: AMexample.png]]

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h
Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh
Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh
An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

[[File: AMvelocity.png]]

The example shows that the speed of light relates the a time varying electric and magnetic field.

====What this implies====

In the example above we saw that the speed of light relates a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

[[File: AMwave.png]]

===Difficult===

[[File:Example3.jpg]]

[[File:Example3b.jpg]]

Question 1: First find the line integral of B around a loop of radius R located just outside the left capacitor plate.

Answer:

===Conceptual Questions===
[[File: AMconceptual.png|500px|Image: 500 pixels]]

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

[[File: AMcapacitor.png|500px|Image: 500 pixels]]

Consider a circular capacitor, with an Amperian circular loop (radius r) in the plane midway between the plates. When the capacitor is charging, the line integral of the magnetic field around the circle (in direction shown) is

1) Zero

2) Positive

3) Negative

4) Can’t tell

Answer:

One thing to notice is that there is no enclosed current through the disk. When integrating in the direction shown, the electric flux is positive. Because the plates are charging, the electric flux is increasing. Therefore the line integral is positive.

==Connectedness==

The Ampere-Maxwell Law is extremely useful in many aspects of life. It relates current and time-varying electric fields and allow us to derive a magnetic field from these situations. The Ampere-Maxwell Law can help with understanding and building generators and transformers.

==History==

You may remember Ampere's Law equation from chapter 21 of the textbook ([[Ampere's Law]]). This equation had to be modified to address time-varying electric fields and their associated magnetic fields. If Ampere's Law were applied to a situation with a time-varying electric field, at different points along the chosen Amperian path, different answers for the current enclosed would be obtained. In the same way that Faraday discovered that a time-varying magnetic field is accompanied by an electric field, James Clerk Maxwell resolved this issue when he discovered that an electric field that changed with time was accompanied by a resulting magnetic field. Specifically, when the plates of a capacitor grows as the capacitor is charged, the electric field changes (with time) which produces a magnetic field.

== See also ==

The Ampere-Maxwell Law relates to the other Maxwell equations:

*[[Gauss's Law]]

*[[Magnetic Flux ]]

*[[Faraday's Law ]]

It might also help to read up on [[Maxwell's Electromagnetic Theory ]]

Also for continuity purposes, it can be helpful to look at the incomplete Ampere's Law:

*[[Ampere's Law]]

===Further Reading===

Matter and Interactions: Volume 2 by Ruth Chabay and Bruce Sherwood (4th Edition)

===External Links===

Maxwell's Equations:
http://study.com/academy/lesson/maxwells-equations-definition-application.html

James Maxwell Biography:
http://www.biography.com/people/james-c-maxwell-9403463

==References==

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf
http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/
http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf
http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html

[[Category:Which Category did you place this in?]]

File:Example3.jpg

2017-11-30T04:35:14Z

Schen498:

2017-11-29T14:24:31Z

Schen498:

'''Claimed by Sahil Arora (Fall 2016)'''

'''Claimed by Stella Chen (Fall 2017)'''

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 Energy Principle and 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 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 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>

==Examples==

===Simple===
[[File:SimpleLoopRule.jpg]]

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

Problem:

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]]

Problem:

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]]

Problem:

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> Lastly, we will plug in what we found {I}_{1} 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.

Answers:

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>

==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]]

==Connections==
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.

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

[[Node Rule]]

[[Components]]

[[Resistors and Conductivity]]

[[Current]]

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.

===More Information===

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

[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]

===Video Tutorials===

[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]

===Simulations===

[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.

==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

2017-11-29T14:24:09Z

Schen498:

'''Claimed by Sahil Arora (Fall 2016)'''
'''Claimed by Stella Chen (Fall 2017)'''

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 Energy Principle and 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 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 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>

==Examples==

===Simple===
[[File:SimpleLoopRule.jpg]]

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

Problem:

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]]

Problem:

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]]

Problem:

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> Lastly, we will plug in what we found {I}_{1} 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.

Answers:

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>

==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]]

==Connections==
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.

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

[[Node Rule]]

[[Components]]

[[Resistors and Conductivity]]

[[Current]]

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.

===More Information===

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

[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]

===Video Tutorials===

[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]

===Simulations===

[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.

==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]]