33.6:
Plane Electromagnetic Waves II
The relation between electric and magnetic fields is obtained by applying Faraday's law to a plane wavefront traveling in the +x-direction with a constant speed.
Consider a rectangle of height "a" having an area vector in the +z-direction.
While integrating around the rectangle counterclockwise, the electric field is found to be zero along length "PQ", or perpendicular to the elements except for length "RS". It only contributes to the integral giving a non-zero value.
To satisfy Faraday's law, there must be a magnetic field component in the +z-direction, which can provide a non-zero magnetic flux through the rectangle and hence a non-zero time derivative of magnetic flux.
In time "dt", the wave moves a distance "c dt". While moving, it sweeps an area equal to "ac dt".
During this time interval, the magnetic flux through the rectangle increases, giving the rate of change of magnetic flux.
This value is substituted in Faraday's law, resulting in a relationship between E and B in terms of the speed of wave propagation in a vacuum.
33.6:
Plane Electromagnetic Waves II
Consider a plane wavefront traveling in position x-direction with a constant speed. This wavefront can be utilized to obtain the relationship between electric and magnetic fields with the help of Faraday's law.
To apply Faraday's law, consider a rectangle of width a, as shown in the figure.1, whose area vector is in the positive z-direction. To solve the left-hand side integral in Faraday's law, integrate counterclockwise along the rectangle. The electric field is zero along one of the lengths lying outside the wavefront, while the electric field is perpendicular to the other two length elements. These three sides give no contribution to the integral. Only along one side, the electric field has a non-zero value, and it is parallel to the length. This side contributes to the integral, giving a non-zero value.
Since the left side of Faraday's law is non-zero, to satisfy the equation there must be a magnetic field component in the positive z-direction, which can provide a non-zero magnetic flux through the rectangle and hence a non-zero time derivative of magnetic flux. To obtain this value, consider that in time dt, the wave moves a distance c dt. While moving, it sweeps an area equal to ac dt. During this time interval, the magnetic flux through the rectangle increases, giving the rate of change of magnetic flux. This value is substituted in Faraday's law, resulting in a relationship between E and B in terms of the speed of wave propagation in a vacuum.
This expression shows that the wave is consistent with Faraday's law only if the wave speed and the magnitudes of the perpendicular vectors are related, as in the above equation.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. Section 32.2; page 1056.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from https://openstax.org/details/books/university-physics-volume-2; section 16.2; pages 710–711.