(a) Identify the part of the electromagnetic spectrum used in (i) radar and (ii) eye surgery. Write their frequency range.

(b) Prove that the average energy density of the oscillating electric field is equal to that of the oscillating magnetic field.

(a) (i) Radio waves are used in radar. Frequency range: 500kHz to 1000MHz

(ii) UV rays are used in eye surgery. Frequency range: 3 × 10¹⁶Hz to 8 × 10¹⁴ Hz.

(b)

Concepts/Formulas Used:

Energy stored by electric field:

The energy density of electric field with electric field strength E is given by:

where ϵ₀ is the permittivity of free space.

Energy stored by magnetic field:

The energy density of magnetic field with magnetic field strength B is given by:

where μ₀ is the magnetic permeability of free space.

Electromagnetic waves:

The oscillating electric field and magnetic field of an electromagnetic field wave travelling in the +z direction are in +x and +y directions and are given by:

where k is the wave number given by

where λ is the wavelength of the wave.

Also,

where c is the speed of light in vacuum.

and

Let the electromagnetic field move in +z direction.

Then the electric filed is given by:

Now, the energy density is given by:

Taking the average,

The brackets mean average. We must find the average of the above sin term in brackets. The average is not just over time but also over space. Thus, we will need to integrate twice. However, we can see that both position(z) and time (t) are simply arguments in the sine squared term. We can simply find the average of sine squared term over to .

Now,

The magnetic field is given by:

Now, the energy density is given by:

Taking the average,

Using, and

We can see that