Chapter 29, Problem 1FTD

Why is Maxwell’s modification of Ampère’s law essential to the existence of electromagnetic waves?

To determine

The reason why the Maxwell’s modification of Ampere’s law is essential to the existence of electromagnetic waves.

Answer to Problem 1FTD

Maxwell’s modification of Ampere’s law is essential to the existence of electromagnetic waves since the electromagnetic the generation of magnetic field component of the electromagnetic wave is the resultant of a change in electric flux.

Explanation of Solution

According to Faraday’s law, the change in magnetic flux causes electric field. Faraday’s law is given as,

${\displaystyle \oint E\cdot dl}=\frac{d{\varphi }_B}{dt}$

Here,

$E$ is the electric field

$dl$ is the infinitesimal length

$\frac{d{\varphi }_B}{dt}$ is the change in magnetic flux

Ampere’s law is given as,

${\displaystyle \oint B\cdot dl}={\mu }_{0}I$

Here,

$B$ is the electric field

${\mu }_0$ is the permeability of free space

$I$ is the current

Since electricity and magnetism has a near symmetry, the change in electric flux can cause magnetism. Thus, the Ampere’s law should contain an extra term for change in magnetic flux. Amperes law without Maxwell’s modification can only applicable in steady current. Thus, for varying current, there should be a modification of Ampere’s equation without affecting the validity of Ampere’s law in steady case. The Amperes law with Maxwell’s modification is given as,

${\displaystyle \oint B\cdot dl}={\mu }_{0}I+{\mu }_0{\epsilon }_0\frac{d{\varphi }_E}{dt}$

Here,

${\epsilon }_0$ is the permittivity of free space

$\frac{d{\varphi }_E}{dt}$ is the change in electric flux

Since electromagnetic waves propagates by means of propagation of perpendicular electric and magnetic fields, the induction of the fields due to the flux change should be present. Thus, the modification of Ampere’s law is important for the existence of electromagnetic wave.

Conclusion:

Maxwell’s modification of Ampere’s law is essential to the existence of electromagnetic waves since the electromagnetic the generation of magnetic field component of the electromagnetic wave is the resultant of a change in electric flux.