Question

Write Maxwell's equation in integral form and reduce speed of plane electromagnetic wave in vaccum.

Answer 1

Answer:

Maxwell’s four equations describe the electric and magnetic fields arising from distributions of electric charges and currents, and how those fields change in time. They were the mathematical distillation of decades of experimental observations of the electric and magnetic effects of charges and currents, plus the profound intuition of Michael Faraday. Maxwell’s own contribution to these equations is just the last term of the last equation—but the addition of that term had dramatic consequences. It made evident for the first time that varying electric and magnetic fields could feed off each other—these fields could propagate indefinitely through space, far from the varying charges and currents where they originated. Previously these fields had been envisioned as tethered to the charges and currents giving rise to them. Maxwell’s new term (called the displacement current) freed them to move through space in a self-sustaining fashion, and even predicted their velocity—it was the velocity of light!

Here are the equations:

1. Gauss’ Law for electric fields:

∫E→⋅dA→=q/ε0.

(The integral of the outgoing electric field over an area enclosing a volume equals the total charge inside, in appropriate units.)

2. The corresponding formula for magnetic fields:

∫B→⋅dA→=0.

(No magnetic charge exists: no “monopoles”.)

3. Faraday’s Law of Magnetic Induction:

∮E→⋅dℓ→=−d/dt(∫B→⋅dA→).

The first term is integrated round a closed line, usually a wire, and gives the total voltage change around the circuit, which is generated by a varying magnetic field threading through the circuit.

4. Ampere’s Law plus Maxwell’s displacement current:

∮B→⋅dℓ→=μ0(I+ddt(ε0∫E→⋅dA→)).

This gives the total magnetic force around a circuit in terms of the current through the circuit, plus any varying electric field through the circuit (that’s the “displacement current”).

The purpose of this lecture is to review the first three equations and the original Ampere’s law fairly briefly, as they were covered earlier in the course, then to demonstrate why the displacement current term must be added for consistency, and finally to show, without using differential equations, how measured values of static electrical and magnetic attraction are sufficient to determine the speed of light.

Explanation:

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