33.7:
Propagation Speed of Electromagnetic Waves
Electromagnetic waves are consistent with Ampere's law.
Consider a plane wavefront traveling in the +x-direction. Over it, consider a rectangle in the xz-plane, with an area vector in the +y-direction.
The integration is performed counterclockwise around the rectangle.
The magnetic field is either zero or perpendicular to the length elements except for length "RS", where the field is parallel and contributes to the integral.
It implies that the right side of Ampere's law must also be non-zero. Therefore, the electric field must have a y-component that can provide a non-zero time derivative of electric flux.
It also establishes that the electric and magnetic fields must be mutually perpendicular.
The electric flux increases to a positive value in time "dt". The rate of change of electric flux can be substituted in Ampere's law.
Since electromagnetic waves are consistent with all of Maxwell's equations, the obtained expression is compared with the expression derived using Faraday's law, which gives the wave propagation speed in the vacuum.
Substituting the values of permeability and permittivity, the propagation speed is equal to the speed of light.
33.7:
Propagation Speed of Electromagnetic Waves
Electromagnetic waves are consistent with Ampere's law. Assuming there is no conduction current Ampere's law is given as:
Consider a plane wavefront traveling in the positive x-direction as shown in figure. Over it, consider a rectangle in the xz-plane, with an area vector in the positive y-direction. The integration is performed counterclockwise around the rectangle to solve the left-hand side of Ampere's law. The magnetic field is either zero or perpendicular to the length elements except for one length, where the field is parallel. This length contributes to the integral, giving a non-zero value.
To satisfy Ampere's law, the right side of Ampere's law must also be non-zero. Therefore, the electric field must have a y-component that can provide a non-zero time derivative of electric flux. It also establishes that the electric and magnetic fields must be mutually perpendicular. The electric flux increases to a positive value in time, dt. The rate of change of electric flux can be substituted in Ampere's law.
Since electromagnetic waves are consistent with all of Maxwell's equations, the obtained expression is compared with the expression derived using Faraday's law, which gives wave propagation speed in the vacuum.
When the numerical values of permeability and permittivity are substituted, the propagation speed obtained is equal to the speed of light.
It implies that the assumed wave is consistent with all of Maxwell's equations, provided that the wavefront moves with speed given above, which is recognized as the speed of light. Note that the exact value of the speed of light is defined to be 299,792,458 m/s.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. Section 32.2; page 1057.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 16.2; pages 707–709.