Question

A laser lamp is of $9mW$ and diameter$= 2mm$. Then what is the amplitude of the magnetic field associated with it?
A. $49\mu T$
B. $98\mu T$
C. $9.8\mu T$
D. $4.9\mu T$

Answer 1

In the question, they’ve given the power and diameter of the laser. From this we can find the intensity of the laser. Once we find the intensity, we write it in the terms of the electric field associated. We can use the relation between the electric field and magnetic field and find the magnetic field associated with the laser beam.

Formula used:
$I = \dfrac{1}{2}{\varepsilon _0}B_0^2{c^3}$

Complete step-by-step answer:
In the question, they’ve given a laser of power, $P = 9mW$ and diameter, $d = 2mm$. From this, we have the intensity of the laser, $I$ as
\eqalign{ & I = \dfrac{P}{A} \cr & \Rightarrow I = \dfrac{{9mW}}{{\pi {r^2}}} \cr & \Rightarrow I = \dfrac{{9mW}}{{\pi {{\left( {\dfrac{d}{2}} \right)}^2}}} \cr & \Rightarrow I = \dfrac{{9 \times {{10}^{ - 3}} \times 4}}{{\pi \times {{\left( {2 \times {{10}^{ - 3}}} \right)}^2}}} \cr & \Rightarrow I = 2.86 \times {10^3}W/{m^2} \cr}
The intensity can be written in the terms of the electric field associated for any electro-magnetic radiation, as with the laser as
$I = \dfrac{1}{2}{\varepsilon _0}E_0^2c$
Where,
$I$ is the intensity of the laser
${\varepsilon _0}$ is the permittivity of free space
${E_0}$ is the electric field associated with the laser beam
$c$ is the speed of light
But we have the relationship between the Electric field and Magnetic field as
${E_0} = c{B_0}$
Where,
${E_0}$ is the electric field
$c$ is the speed of the light
${B_0}$ is the magnetic field
So, we have the intensity of the magnetic field associated with the laser beam as
\eqalign{ & I = \dfrac{1}{2}{\varepsilon _0}E_0^2c = \dfrac{1}{2}{\varepsilon _0}{\left( {c{B_0}} \right)^2}c \cr & \Rightarrow I = \dfrac{1}{2}{\varepsilon _0}B_0^2{c^3} \cr}
Substituting the values, we have
\eqalign{ & I = \dfrac{1}{2}{\varepsilon _0}B_0^2{c^3} \cr & \Rightarrow {B_0} = \sqrt {\dfrac{{2I}}{{{\varepsilon _0}{c^3}}}} \cr & \Rightarrow {B_0} = \sqrt {\dfrac{{2 \times 2.86 \times {{10}^3}}}{{8.85 \times {{10}^{ - 12}} \times {{\left( {3 \times {{10}^8}} \right)}^3}}}} = 4.9 \times {10^{ - 6}}T \cr & \therefore {B_0} = 4.9\mu T \cr}
Therefore, the magnetic field associated with the laser beam is $4.9\mu T$.

So, the correct answer is “Option A”.

Note: Normally, energy is directly proportional to amplitude squared. But, for electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic field. Hence, the intensity is proportional to the square of the electric field or magnetic field.