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I am wondering if the Lenz's law is missing in the fourth equation of the Maxwell's equation with regards to the negative sign that exists in the Faraday's law (3rd equation).

$$\begin{align} \nabla\cdot\mathbf{E}&=\frac{\rho}{\epsilon_0} \\ \nabla\cdot\mathbf{B}&=0 \\ \nabla\times\mathbf{E}&=-\frac{\partial\mathbf{B}}{\partial t} \\ \nabla\times\mathbf{B}&=\mu_0\mathbf{J}+\mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t} \end{align}$$

We know that the changing magnetic field will induce an electric field that will generate a current in a manner to oppose the magnetic field itself. Should the fourth equation also not have a negative term to show this effect when the changing electric field generates a magnetic field?

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  • $\begingroup$ The question in your title is different from the one in the body. For the best response you should ask one question per post. $\endgroup$ – my2cts Dec 2 '20 at 23:02
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(Someone seems to have made a couple of mistakes in edits to the question. The curl of E appears twice.)

In a vacuum, and in a system of units where c=1, the two Maxwell's equations you're talking about look like this:

$$\operatorname{curl}\textbf{E} = -\partial\textbf{B}/\partial t$$

$$\operatorname{curl}\textbf{B} = \partial\textbf{E}/\partial t$$

There are really two questions:

(1) Should the two signs be opposite, or should they be the same?

(2) If they're opposite, which should be which?

The answer to 2 is that this is just a matter of convention. Instead of the thing we call the magnetic field $\textbf{B}$, humans could have chosen to define something, call it $\textbf{C}$, where $\textbf{C}=-\textbf{B}$. Under this substitution, the signs in the two Maxwell's equations both flip.

There may be more than one physical way to justify #1. One way is to observe that the combination of these two signs allows us to have solutions to Maxwell's equations that are oscillating waves, whereas if the signs were the same, we would have solutions that were exponentials. That is, the fact that the product of the two signs is $-1$ is what makes it a negative feedback system rather than a positive feedback system.

The version of Lenz's law that you cite is not a great way to think about Maxwell's equations. Induction doesn't really have anything to do with currents or circuits. But note that the formulation of Lenz's law as "resisting the change" is invariant under the change of variables from B to C, since it makes no explicit or implicit reference to any handedness. The "resisting the change" thing only depends on the product of the signs and actually requires the product to be -1.

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  • $\begingroup$ (Someone seems to have made a couple of mistakes in edits to the question...) Yes sorry, my bad. Fixed it now. $\endgroup$ – Thomas Fritsch Dec 3 '20 at 0:12
  • $\begingroup$ My question is straightforward. The changing magnetic field produces a rotating electric field (that's why we denote it with curl right?). And the direction of this rotating electric field is negative to the direction of the changing magnetic field, which is indicated by a negative sign. And that is why I say this is Lenz's law since if the direction of the electric field is opposite the current will be such that will generate a magnetic field that will oppose the changing magnetic field, the very cause of it's own production. $\endgroup$ – GISEnthu Dec 7 '20 at 8:38
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The third equation is the differential form of Faraday's law, which is a quantitative statement of Lenz's law. See the corresponding Wikipedia articles or Jackson's Electrodynamics.

The question in the body about the sign of the displacement current is distinct from this. The answer is "no".

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  • $\begingroup$ Will the changing magnetic field not produce an electric field that will set a current such that the field produced by the current will oppose the changing magnetic field, the root cause of the electric field's and the resultant current's origin? This is why I say the "negative sign" denotes the Lenz's law. $\endgroup$ – GISEnthu Dec 6 '20 at 18:09
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The fourth equation tells the direction of the magnetic field according to the current and E-field. It doesn't say anything about induction. By writing the definition of the curl, we can see that the equation states the hand right rule.

For a demonstration, let a current flows upwards in the z-direction for example. The expression below is always positive if vector B points counter-clockwise around the conductor, when viewed from above: $$(\nabla \times \mathbf B)_z = \frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y}$$

Now, suppose a loop where there is an increasing current counter clockwise, when looking at it from above. The result is an increasing magnetic dipole flowing upwards inside the loop.

According to the third equation, without the minus sign, the same rule (now the thumb in the direction of the magnetic field) would indicate an increasing current also counter clockwise, in violation to Lenz law.

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