Question 35: EM waves

Question

A laser beam has intensity of $4.0 	imes 10^{14} \mathrm{~W} / \mathrm{m}^2$. The amplitude of magnetic field associated with beam is $\_\_\_\_$ T. (Take $\epsilon_{\mathrm{o}}=8.85 	imes 10^{-12} \mathrm{C}^2 / \mathrm{Nm}^2$ and $\mathrm{c}=3 	imes 10^8 \mathrm{~m} / \mathrm{s}$ )

Accepted Answer

The intensity I of an electromagnetic wave is related to the maximum electric field ( $\mathrm{E}_0$ ) and the maximum magnetic field $\left(\mathrm{B}_0ight)$. The formula for the intensity to the magnetic field amplitude is :
$$ I=\frac{c B_0^2}{2 \mu_0} $$
Where :
I is the intensity.
c is the speed of light in a vacuum.
$\mathrm{B}_0$ is the amplitude of the magnetic field.
$\mu_0$ is the permeability of free space.
The relationship between the speed of light (c), permeability ( $\mu_0$ ), and permittivity ( $\epsilon_0$ ) of free space is,
$$ c=\frac{1}{\sqrt{\mu_0 \epsilon_0}} \Rightarrow \mu_0=\frac{1}{\epsilon_0 c^2} $$
By substituting $\mu_0$ into the intensity equation, we get a formula directly relating the given variables :
$$ I=\frac{c B_0^2}{2\left(\frac{1}{\epsilon_0 c^2}ight)}=\frac{1}{2} c^3 \epsilon_0 B_0^2 $$
The intensity of the laser beam is, $\mathrm{I}=4.0 	imes 10^{14} \mathrm{~W} / \mathrm{m}^2$
The permittivity of free space, $\epsilon_0=8.85 	imes 10^{-12} \mathrm{C}^2 / \mathrm{N} \cdot \mathrm{m}^2$
The speed of light, $\mathrm{c}=3 	imes 10^8 \mathrm{~m} / \mathrm{s}$
From the intensity formula, $B_0=\sqrt{\frac{2 I}{c^3 \epsilon_0}}$
Putting the values,
$$ B_0=\sqrt{\frac{2 	imes 4.0 	imes 10^{14}}{\left(3 	imes 10^8ight)^3 	imes 8.85 	imes 10^{-12}}}=\sqrt{\frac{8.0 	imes 10^{14}}{27 	imes 10^{24} 	imes 8.85 	imes 10^{-12}}} $$
$$ \mathrm{B}_0=\sqrt{\frac{8.0}{238.95} 	imes 10^2} \approx 1.83 \mathrm{~T} $$
Therefore, the amplitude of the magnetic field associated with beam is approximately 1.83 T.
Hence, the correct option is (A).

Answer

1.83

Answer

2.0

Answer

5.5

Answer

18.3

EM waves

A laser beam has intensity of 4.0 \times 10^{14} \mathrm{~W} / \mathrm{m}^2. The amplitude of magnetic field associated with beam is \_\_\_\_ T. (Take \epsilon_{\mathrm{o}}=8.85 \times 10^{-12} \mathrm{C}^2 / \mathrm{Nm}^2 and \mathrm{c}=3 \times 10^8 \mathrm{~m} / \mathrm{s} )

Choose the correct answer: