Kirchoff's loop law

Can any one provide me with mathematical proof of kirchoff's loop law? I am not able to understand from where to start.

Physicist137 took a crack at showing why this follows from the structure of the electric field in the absence of time-dependent magnetic fields. That doesn't hold once you have induction in the system, however.

So let's look at a simpler approach.

Anything that you want to treat as a potential (whether it is a real potential or not) has to have a simple property: it has to be a function of position. That is it has to have only one value associated with every point in space.

If it has only one value at a point in space than the of the changes in that function around a path have to be equal to zero.

• I'm sorry. I couldn't understand what you said. How so induction in the system? Why doesn't hold? You mean AC circuits? Kirchhoff's laws are invalid on AC circuits (at high frequency). But holds as good approximation for low frequency. – Physicist137 Nov 27 '14 at 17:25
• If there is an externally imposed, changing magnetic field then (a) you can't formally define a "potential" anymore and (b) If you're going to fake it (which you can and generally should) you have to include a EMF from the induction or KLL doesn't work. See WetSavanna... answer on the linked question. – dmckee Nov 27 '14 at 17:29
• Strange. I agree completely with (a) and (b). Maybe you saw my answer before the edit 45min ago? =D. Maybe I was not clear enough? I think it's clear. – Physicist137 Nov 27 '14 at 17:33
• Do you think I object to your answer? I don't and in fact voted for it, but I thought there was room for another approach to the problem. – dmckee Nov 27 '14 at 17:36
• Ahhh! I thought my answer were wrong, and hence my understanding of electromagnetism wrong as well: A mistake which should be fixed. Well, since this is not the case, =). – Physicist137 Nov 27 '14 at 17:42

The rotational of the electric field: $$\nabla\times\mathbf E = -\frac{\partial\mathbf B}{\partial t}$$

Using Stokes' Theorem on this equation, we get the integral form of this equation: $$\varepsilon = \oint_{\gamma}\mathbf E\cdot\mathbf{dl} = -\frac{d}{dt}\iint_S\mathbf B\cdot\mathbf{dS} = -\frac{d\Phi}{dt}$$

Which means, the electric field in a loop line $\gamma$ depends on the variation of the magnetic flux $\Phi$. If there's no external magnetic fields acting in the circuit, there will be no magnetic flux, and then we have: $$\oint_{\gamma}\mathbf E\cdot\mathbf{dl} = 0 \quad\Longrightarrow\quad \varepsilon = 0$$

Which means, the emf of a loop, assuming no external magnetic fields, and no voltage sources, is zero. But notice: This loop line is the circuit, not including its passive/active elements. Once you include this, you will get Kirchhoff's voltage law. For the current law, a similar consideration is made with the density current vector $\mathbf J$.

The best-known name of this principle is Kirchhoff's Voltage Law.

It is pure consequence of the energy conservation in broader sense.

Imagine positives charges that follows the normal convention.

Each electric potential (voltage) is potential energy divided by charge (coulombs), so when a charge logically closes a loop, it returns to the same point, like a car in a roller coaster with no friction.

The difference here is that energy loss is included in the total energy conservation account and it's predictable because physics properties of the circuit are stable.

Let's suppose, for instance, a battery-powered circuit. Inside the battery, chemical energy is converted in electric potential (so the voltage increases).

I's like the charge was raised to the top of a waterfall.

During the "fall" of the waterfall, there is potential energy loss due to resistance in the wire or in resistors. The energy loss is directy represented by voltage drop and generally converted to heat.

A electric circuit is like a dynamic balance. If you suddenly increase the battery voltage or change resistance, there is a very small lapse of time that this balance is broken, but very, very soon the balance is restored.