Missing Lenz's Law in Maxwell's Equation?

Question

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?

Answer 1

(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.

Answer 2

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".

Answer 3

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.

Answer 4

I had the same doubt.

The reason it may seem odd and violate conservation of energy is that we thought about the exponential increase of kinetic energy on ELECTRIC charges.

look up the maxwell equations with magnetic monopoles. If you're comfortable with the curls orientation, and it's relation to the line integral. You can work through the steps to see if the sign causes an exponential increase in kinetic energy of a magnetic monopole.

If done correctly, you will see that the positive sign actually has the same effect with magnetic charge, as the negative does with electric charge. The positive sign acts to oppose the magnetic current.

Without magnetic monopoles there isn't any violation with energy since it is the electric field that does work on charges, not the magnetic field.

( but as commenter above stated, the convention doesn't matter, but they have to be opposite)

Answer 5

Assuming you mean this by Lenz's law: $$\mathcal{E} = -\frac{\partial \Phi_B}{\partial t}$$ from https://en.wikipedia.org/wiki/Lenz%27s_law, this is easy to show.

Taking the emf around a closed loop $C$ which is the edge of surface $S$, we get

\begin{align} \mathcal{E} &= \oint_C \mathbf{E} \cdot d\mathbf{l} \tag{1}\\ &= \iint_S \nabla \times \mathbf{E} \cdot d\mathbf{A} \tag{2}\\ &= \iint_S -\frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A} \tag{3}\\ &= -\frac{\partial}{\partial t} \iint_S \mathbf{B}\cdot d\mathbf{A} \tag{4}\\ &= -\frac{\partial \Phi_B}{\partial t}\tag{5}. \end{align}

(1) is by definition of emf, (2) is Stokes' law, (3) is the third Maxwell equation, (4) can be done if the surface $S$ is not changing in time and (5) is by definition.

So since lenz's law can be derived from the Maxwell eqs, it is not missing from them. I hope from this you see why the minus sign is there.