How do Electric and magnetic fields generate each other (mathematically)?

Question

Regarding electromagnetism, a changing magnetic flux$(\phi_B)$ produces emf by-$$EMF= -\frac{d \phi_B}{dt}\tag1$$ This emf creates a current which again creates a magnetic field given by-(bio-savart law) $$dB= \frac{kI×dl}{r^2}\space\space\space\space\space\space (2)$$ assume appropriate constants and vectors.

But if this emf is such that, it is also changing with time then it creates a time varying current I(t) which will then produce a time varying magnetic field by (2) (maybe) and the process is again back to (1) i.e we have a changing magnetic flux. This is obviously wrong and shouldn't happen as it may become endless loop creating $\infty$ E and B fields.

I am highly unsure about everything I have written, as I am not confident in this topic but please explain my misunderstanding here. Also I am fully aware of the four Maxwell equations and their basic meaning. I know this topic can get too complicated as it uses tensors and such so keep math (if any) moderate graduation level

Answer 1

Linked differential equations may reinforce one another leading to unbounded exponential growth - but they do not have to.

For example, the linked equations

$\displaystyle y = \frac {dx}{dt} \\ \displaystyle x = \frac {dy}{dt}$

with initial conditions $x(0)=y(0)=1$ have solution

$x = e^t \\ y = e^t$

which is unbounded as $t \rightarrow \infty$. But if we introduce a negative sign as follows:

$\displaystyle y = -\frac {dx}{dt} \\ \displaystyle x = \frac {dy}{dt}$

with the same initial conditions $x(0)=y(0)=1$ then the solution is

$x = \cos(t) - \sin(t) \\y = \cos(t) + \sin(t)$

which is bounded for all $t$.

The interaction between electric and magnetic fields is like the latter case (although more complicated since we have spatial dimensions to consider as well).

Answer 2

It is commonly believed that a changing magnetic field generates a rotation of an electric field. This is not correct.

A changing magnetic field and the rotation of an electric field are equivalent expressions of the same underlying effect, namely a time dependent vector potential as in* $${\bf \nabla} \times {\bf E} \equiv {\bf \nabla} \times {\bf A}_t \equiv ({\bf \nabla} \times {\bf A})_t \equiv {\bf B}_t \,.$$

The origin of $${\bf \nabla} \times {\bf B} \equiv {\bf E}_t$$ is the wave equation, in its homogeneous form, $${\bf A}_{tt} = \Delta {\bf A} \,,$$ which in this form holds in the Lorenz gauge. Here indeed one could say that ${\bf A}_{tt}$ causes $\Delta {\bf A}$ and that brings the question 'How?'. All attempts to explain this have failed, among which the famous Aether theory.

You posed an excellent question!

*All uncompensated charges assumed stationary, that is $V_t=0$.

Answer 3

From a spatial point of view, it very much looks like a changing electric field generates a changing magnetic field. And vice-versa. But this raises the problem of causality and locality.

Its general taken in physics the cause and effect are local. Hence, if we wave an electron, the changing electric field will generate a changing magnetic field. But then by locality, you can equally say, after the initial cause, that a changing magnetic field generates the changing electric field. Thus what causes which becomes ambiguous. Of course one could posit instead Liebniz pre-established harmony. (Actually, this view, which sounds incredible begins to make sense when there is no time, for then there can be no temporal cause amd effect).

This puzzle is resolved once we move into a spacetime view. Then the electric and magnetic spatial wave are combined into a single spacetime EM field strength. It is this that varies. And its only when we choose a split of spacetime into space and time that we obtain the electric and magnetic fields.

Answer 4

Towards the path of universal conservation of energy in an energetically isolated system, Lenz (1834) formulated his law as a principle avoiding self-amplification of magnetic systems: The fields of the induced currents reduce the inducing field in the same sense as the field of the induced charge density on a conducting surface reduces the inducing electric field. Last line: The squares of both electromagnetic fields represent an energy density, its integral value is conserved together with all other forms of energy in an isolated system.