33.10:
Intensity Of Electromagnetic Waves
When a capacitor is connected to an alternating current, the direction of the electric field between the plates changes with time, which induces a varying magnetic field.
The energy flow given by the Poynting vector is parallel to the plate. It varies rapidly with time.
Taking the time average of the Poynting vector gives the wave's intensity.
Consider a plane electromagnetic wave with the electric field oscillating along the y-axis and the magnetic field along the z-axis. Then, the Poynting vector points in the positive x-direction.
By replacing the fields with wave functions, the expression for the Poynting vector can be rewritten.
As the fields are mutually perpendicular, the Poynting vector magnitude equals the product of the peak amplitudes of the fields and a square of cosine term. By integrating the Poynting vector magnitude over a wave period, the expression for the intensity can be obtained.
Using the relationship between fields and the speed of light, the equivalent expressions for intensity can be defined in terms of the magnitudes of the electric and magnetic fields alone.
33.10:
Intensity Of Electromagnetic Waves
The energy transport per unit area per unit time, or the Poynting vector, gives the energy flux of an electromagnetic wave at any specific time. For a plane electromagnetic wave with E0 and B0 as the peak electric and magnetic fields and traveling along the x-axis, the time-varying energy flux can be given by the following equation:
As the frequency of the electromagnetic wave is very high, for example, the frequency of visible light is in the order of 1014 Hz, the energy flux rapidly varies with time. The energy flux for visible light through any area is an extremely rapidly varying quantity. Most measuring devices, including our eyes, detect only an average over many cycles. The time average of the energy flux is the intensity of the electromagnetic wave, which is the power per unit area. It can be expressed by averaging the cosine function in the expression of over one complete cycle, which is the same as time-averaging over many cycles (here, T is one period). Hence, the average of
the Poynting vector, or the intensity, can be given as,
The average of cos2θ or sin2θ gives 1/2. Hence, the intensity of light moving at speed c in a vacuum is then found to be
The equivalent expressions for intensity are,
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from 16.1 Maxwell’s Equations and Electromagnetic Waves – University Physics Volume 2 | OpenStax