Lecture Supplement from Walter Lewin - confusion about derivation of Kirchoff's law in the presence of resistance-free solenoid

Question

In this lecture supplement (lecture supplement of the MIT class $8.02$ by Walter Lewin), there is an interesting discussion about self-inductance - Kirchoff's law and Faraday's law. From what I have understood, the induced field produced by a changing magnetic field is always uniform along a loop (the magnetic field is normal to the loop), that is, it has a constant value (along a closed circular path). That happens independently of the material that we choose to put (or not to put) in the space where there's the induced field. Now, the lecture supplement raises the question: "what if there a low resistive material on the left half of the loop and a more resistive material on the right half of the loop?". The author says "Nature charges up" the top frontier between the two halves (of the material) with, say, positive charges (or negative charges, depending on whether $\vec{E}$ is pointing clockwise or counterclockwise) and the bottom frontier between the two halves with negative charges (or positive charges, see previous parenthesis), so that an extra field between the charges comes and reduces the induced field on the left half, and boosts the induced field on the right half, so that $\vec{E^{total}}$ on the left could be almost zero (recall $\vec{E} = \rho \vec{j}$, and $\rho_{left} \simeq 0$) and so that $\vec{E^{total}}$ on the right could be pretty strong.

The author concludes the lecture supplement by saying that this phenomena is what happens in a resistance-free solenoid, that is, that $\oint_C \vec{E}^{total}. d\vec{s}=0$ where $C$ is a closed loop between the two endpoints of the solenoid, provided $\vec{E}^{total}=\vec{0}$ along $C$.

I agree with this last affirmation but, there is from my point of view a conceptual mistake in the lecture supplement. The mistake lies in the way the author derives Kirchoff's law.

What he does is he applies Faraday's law in a closed loop circling all the circuit (which consists in a resistor $R$, a resistance-free solenoid and a generator whose tension is $\epsilon$). When it comes to integrating $\int \vec{E}. d\vec{s}$ between the two endpoints of the resistance-free solenoid he says this integral is $0$. But there is, in my opinion, a confusion from the author between $\vec{E}^{total}$ (whose integral, I agree, is zero) and $\vec{E}_{induced}$ (whose integral in the solenoid is not zero, because there is a changing magnetic field in the solenoid). What the author seems to have forgotten is that, in Faraday's law, the electric field involved is the induced electric field.

What do you think ? Is there a mistake in the way he derives Kirchoff's law or am I wrong ?

Answer 1

From what I have understood, the induced field produced by a changing magnetic field is always uniform along a loop

This is true if the loop is circular and the magnetic field is centered within the loop. That is, if our experimental setup is symmetrical. If that is the case, the induced electric field is uniform around the loop.

The author says "Nature charges up" the top frontier between the two halves (of the material) with, say, positive charges (or negative charges, depending on whether $\vec{E}$ is pointing clockwise or counterclockwise) and the bottom frontier between the two halves with negative charges (or positive charges, see previous parenthesis), so that an extra field between the charges comes and reduces the induced field on the left half,

This is part of the truth, but not the whole story. Yes, charges build up on the interfaces where the resistivity of the loop changes. But charges also build up on the surface of the loop wherever they might be needed to ensure 1) that the current throughout the loop is uniform, and 2) that the current follows the contours of the wire loop, instead of going straight. Since the loop is not made of uniform material, the induced $\vec{E}$ field will not ensure this alone, but must be supplemented by surface charges.

But there is, in my opinion, a confusion from the author between $\vec{E}_{total}$ (whose integral, I agree, is zero)

The integral would only be exactly $0$ if the ring were a perfect conductor. Otherwise, there must be a field corresponding to the microscopic version of Ohm's Law

$\vec{E}_{total} = \frac{\vec{J}}{\sigma}$

where $\vec{J}$ is the current density, and $\sigma$ is the resistivity of the material.

Is there a mistake in the way he derives Kirchoff's law or am I wrong?

I am going to quote from Lewin's Lecture notes

Kirchhoff's Second Law was originally based on the fact that the integral of $\vec{E}$ around a closed loop was zero.

That is just historically inaccurate. Kirchhoff own statement of his 2nd law is something like this:

The sum of the IR products for all of the segments around a loop is equal to the sum of the emfs for all the segments around that loop

i.e.

$$\sum_i I_iR_i = \sum \mathscr{E}_i$$

However, many English speaking authors, who may have never read Kirchhoff's paper, believe, as Lewin does, that Kirchhoff stated something along the lines of

$$\oint_C \vec{E}\cdot d\vec{s} = 0 $$

which is true enough when time-varying magnetic fields do not induce any $\vec{E}$ fields in your circuit, but becomes incorrect when such magnetically induced $\vec{E}$ fields are present.

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For the mathematically inclined:

We can define the emf $\mathscr{E}_{\partial\vec{B}/\partial t}$ induced in a conductor by time-varying magnetic fields as the path integral of the component of the electric field induced by the time-varying magnetic field, $\vec{E}_{\partial\vec{B}/\partial t}$.

$$\vec{E}_{\partial\vec{B}/\partial t} =\nabla\times\frac{1}{4\pi}\int_{R^3}\frac{\partial\vec{B}(r')}{\partial t}\frac{1}{|r-r'|}d^3r'$$

$$\mathscr{E}_{\partial\vec{B}/\partial t} =\int_a^b \vec{E}_{\partial\vec{B}/\partial t} \cdot d\vec{s}$$

Since $\vec{E}_{\partial\vec{B}/\partial t}$ is the curl of a vector field, its divergence is $0$. That is, it is solenoidal.

$\nabla\cdot\vec{E}_{\partial\vec{B}/\partial t}=0$

Although the presentation of Kirchhoff's Voltage Law as

$$\oint_C\vec{E}_{total}\cdot d\vec{s} = 0$$

does NOT always hold, the following formulation DOES always hold

$$\oint_C\vec{E}_{total}-\vec{E}_{\partial\vec{B}/\partial t}\cdot d\vec{s} =\oint_C\vec{E}_{\rho}\cdot d\vec{s} = 0$$

That is, $\vec{E}_{\rho}$ is conservative, even in regions in which a time varying $\vec{B}$ field is present.

$$\nabla\times\vec{E}_{\rho}=0$$

Answer 2

Lewin and Belcher are correct.

The essence of your issue is really down to two different misunderstandings. First of all, superconductors just cause a lot of confusion. That is, in $$\tag1\vec E=\rho\,\vec\jmath\to\vec0\qquad\text{as}\qquad\rho\to0$$ it does not matter if you were talking about $\vec E_\text{total}$ or $\vec E_\text{induced}$ when this is happening. Both are zeroed.

That is also why the rest of the lecture supplement talked about what happens when the resistance isn't zero. The situation is much more interesting when the superconducting wires approximation is relaxed.

What the author seems to have forgotten is that, in Faraday's law, the electric field involved is the induced electric field.

That is your misunderstanding, the 2nd misunderstanding. After taking a lot of effort to build up to Maxwell's equations, you are supposed to stop everything that you are doing, and then restart your learning of electrodynamics by taking Maxwell's equations as true, and derive everything, including Coulomb's law, from Maxwell's equations. Faraday's law is one of them, and it always means the total electric field. But of course, you can estimate and show that certain contributions are negligible, or that others are dominant. Here, the self-inductance of a single loop is negligible compared to a huge external magnetic field imposed upon it, which is made ever more the case as you increase the size of the loop.

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Note that in the above, no part of invocation of Kirchoff's Voltage Law was used. That is, you had been so confused that you misattributed where your confusion was.

However, I'd like to draw an analogous parallel that Lewin did not emphasise. The modification from Equation (1) to Equation (2) in the lecture supplement, i.e. the hazardous step to salvage KVL, is the same kind of step that moves the centripetal acceleration term over to the other side of the equal sign to produce the centrifugal acceleration term. The mathematical manipulation is clearly allowed, but then one cannot claim to still be in an inertial frame of reference. The ensuing confusion cast upon students is about equally pervasive.

The fact of the matter is that, for all electrodynamic phænomena, Maxwell's equations is the thing we consult to dispel arguments. You have just one theory to handle all of these issues. If, instead, you prefer a patchwork of hatch jobs to handle things, you will inevitably run into some edge case that breaks down. Faraday's Law clearly asserts that a time-varying magnetic field causes this behaviour, and derives it unambiguously, and so you should not have any other law disagreeing with what it says.