Question

The mean intensity of radiation on the surface of the Sun is about \[{10^8}W/{m^2}\]. The rms value of the corresponding magnetic field is closest to:
(A) ${10^2}T$
(B) ${10^{ - 4}}T$
(C) $1T$
(D) ${10^{ - 2}}T$

Answer 1

Hint: Intensity of electromagnetic waves is given by:
${I_{mean}} = \dfrac{{B_{rms}^2}}{{{\mu _0}}}c$ …… (1)
Where,
${I_{mean}}$ is the mean intensity of an electromagnetic wave.
\[{B_{rms}}\] is the rms magnetic field.
\[{\mu _0}\] is the permeability of free space.
$c$ is the speed of light in vacuum.

Formula Used:
1. $B_{rms}^2 = \dfrac{{{I_{mean}}{\mu _0}}}{c}$

Complete step by step answer:
Given: Mean intensity of radiation (${I_{mean}}$) = \[{10^8}W/{m^2}\].
To find: rms value of magnetic field (\[{B_{rms}}\]).

Step 1 of 3:
Rearranging eq (1):
$B_{rms}^2 = \dfrac{{{I_{mean}}{\mu _0}}}{c}$ …… (2)
${B_{rms}} = \sqrt {\dfrac{{{I_{mean}}{\mu _0}}}{c}} $ …… (3)

Step 2 of 3:
We know that, \[{\mu _0}\]= $4\pi {10^{ - 7}}H/m$
and $c = 3 \times {10^8}m/s$

Step 3 of 3:
Putting the values of \[{\mu _0}\],$c$ and ${I_{mean}}$ in eq(3):
${B_{rms}} = \sqrt {\dfrac{{{{10}^8} \times 4\pi \times {{10}^{ - 7}}}}{{3 \times {{10}^8}}}} \\
  {B_{rms}} \approx {10^{ - 4}}T \\ $

Final Answer:
(B) ${10^{ - 4}}T$

Additional Information:
If a signal is a periodic signal. Then, the RMS value of that particular signal is said to be the DC equivalent of a given signal. Here, Radiation from the sun is a periodic time varying function having both $\overrightarrow E $ and $\overrightarrow B $ as sinusoidal functions. Time average of any sinusoidal function over it’s time period T is 0. But we still extract energy from the signal that is why for simplicity we calculate RMS value which has always net positive magnitude.

Note: The relation, $\overrightarrow B = \overrightarrow {\dfrac{E}{c}} $, is the equivalent magnetic field required for given electric field value in all cases. Here, in our problem, we used the RMS value relation which is the same as the above formula by replacing $\overrightarrow B $ by ${\overrightarrow B _{rms}}$, similarly we use corresponding electric field equivalence of same kind.