Question

An $EM$ wave radiates outwards from a dipole antenna,with $E_o$ as the amplitude of its electric field vector. The electric field $E_o$ which transports significant energy from the source falls off as:

• A. Remains constant
• B. $\frac{1}{r^3}$
• C. $\frac{1}{r}$
• D. $\frac{1}{r^2}$

Answer 1

The correct option is C $\frac{1}{r}$

Given:
Amplitude of electric field vector $=E_{\circ }$
As we know that,
Intensity of radiation is given,

$I=\frac{W}{At}=\frac{E}{At}...\left(i\right)$

where,
W is the energy emitted, t is the time, A is the area on which the radiation is incident.
Therefore, assuming the energy and time to be constant from equation (i) we have,

$I\propto \frac{1}{A}...\left(ii\right)$

Consider an antenna which radiates energy is in a spherical shape around it, thus

$A=4\pi r^2$

By putting the value of A in equation (ii) we get

$I\propto \frac{1}{r^2}...\left(iii\right)$

As we know that the electric field and average intensity are related as

$I=U_{av}=\frac{1}{2}{ϵ}_0{E}_0^2...\left(iv\right)$

From equation (iv)

$I\propto E_0^2$

Now, the equation becomes,

$E_0^2\propto \frac{1}{r^2}$

Hence,

$E_0\propto \frac{1}{r}$

Final Answer: Option (c)