Question

Poynting vector problems?

A radio station transmits a 10-kW signal at frequency of 100 MHz. For simplicity, assume that it radiates at a point source. At a distance of 1 km from the (point source) antenna,

(a) the amplitudes of the electric and magnetic field strengths are___；

(b) the energy incident normally on a square plate of side 10 cm in 5 min is__.

Answer 1

(a) We know that the intensity `I` of a wave is power per unit area and follows inverse square law.
Also that if a source of power `P_s` emits isotropically then intensity at a distance `R` is found from the expression

`I=P_s/(4piR^2)`

Inserting given values we get

`I=(10xx10^3)/(4pi(1xx10^3)^2)`
`=>I=1/(4pi)xx10^-2\ Wm^-2`

Let `vecS` be the Poynting vector. Modulus `|vecS|` gives us the intensity of wave and direction of `vecS` is the direction of flow of energy.
In an electromagnetic wave, the magnitude of Poynting vector
fluctuates rapidly with time. Therefore, a practical and useful quantity, the average (over one cycle) intensity, is used. We thus have

`S_"ave"=E_m^2/(2mu_0c)`
where `E_m` is maximum amplitude of electric field, `mu_0` is permeability of free space and `c` is velocity of light.

Inserting various values and equating intensity with calculated value of `I` we get

`S_"ave"=E_m^2/(2(4pixx10^-7)(3xx10^8))=1/(4pi)xx10^-2`
`=>E_m^2=10^-2xx60`
`=>E_m=sqrt0.60`
`=>E_m=color (blue)(0.775\ Vm^-1)` ......(1)

Since `E=cB`, we get

`B_m=E_m/c`
`=>B_m=0.775/(3xx10^8)`
`=>B_m=color (blue)(2.58xx10^-9\ T)` ......(2)

(b) Energy incident normally on a square plate `DeltaU` is

`DeltaU=IAt`
where `A` is the area and `t` is the time.

Inserting given values in SI units we get

`DeltaU=1/(4pi)xx10^-2xx(10/100)^2xx(5xx60)`
`DeltaU= color (blue)(0.0024\ J)` .......(3)