A laser beam of diameter `2mm` is of `9 mW`. What is the amplitude of magnetic field (in `mu T` ) associated with it? (Answer should be in positive integer form)

Answer

To find the amplitude of the magnetic field associated with a laser beam of diameter 2 mm and power 9 mW, we can follow these steps:

### Step 1: Calculate the Area of the Laser Beam
The area \( A \) of the laser beam can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
where \( r \) is the radius of the beam. The diameter is given as 2 mm, so the radius \( r \) is:
\[ r = \frac{2 \text{ mm}}{2} = 1 \text{ mm} = 1 \times 10^{-3} \text{ m} \]
Now, substituting the radius into the area formula:
\[ A = \pi (1 \times 10^{-3})^2 = \pi \times 10^{-6} \text{ m}^2 \]

### Step 2: Convert Power to Watts
The power \( P \) of the laser beam is given as 9 mW. We need to convert this to watts:
\[ P = 9 \text{ mW} = 9 \times 10^{-3} \text{ W} \]

### Step 3: Calculate the Intensity of the Laser Beam
The intensity \( I \) of the laser beam can be calculated using the formula:
\[ I = \frac{P}{A} \]
Substituting the values of \( P \) and \( A \):
\[ I = \frac{9 \times 10^{-3}}{\pi \times 10^{-6}} = \frac{9 \times 10^{-3}}{3.14 \times 10^{-6}} \approx 2865.9 \text{ W/m}^2 \]

### Step 4: Relate Intensity to the Amplitude of the Magnetic Field
The intensity \( I \) of an electromagnetic wave can also be expressed in terms of the amplitude of the electric field \( E_0 \) and the amplitude of the magnetic field \( B_0 \):
\[ I = \frac{1}{2} \epsilon_0 c E_0^2 = \frac{1}{2} \frac{B_0^2}{\mu_0} c \]
where:
- \( \epsilon_0 \) is the permittivity of free space \( \approx 8.85 \times 10^{-12} \text{ F/m} \)
- \( \mu_0 \) is the permeability of free space \( \approx 4\pi \times 10^{-7} \text{ H/m} \)
- \( c \) is the speed of light \( \approx 3 \times 10^8 \text{ m/s} \)

Using the relationship \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \), we can express the intensity in terms of \( B_0 \):
\[ I = \frac{B_0^2}{2 \mu_0 c} \]

### Step 5: Solve for \( B_0 \)
Rearranging the equation to solve for \( B_0 \):
\[ B_0^2 = 2 I \mu_0 c \]
Substituting the values:
\[ B_0^2 = 2 \times 2865.9 \times (4\pi \times 10^{-7}) \times (3 \times 10^8) \]
Calculating \( B_0^2 \):
\[ B_0^2 = 2 \times 2865.9 \times (4\pi \times 10^{-7}) \times (3 \times 10^8) \approx 24 \times 10^{-12} \]

### Step 6: Calculate \( B_0 \)
Taking the square root:
\[ B_0 \approx \sqrt{24 \times 10^{-12}} \approx 4.9 \times 10^{-6} \text{ T} \]
Converting to microtesla:
\[ B_0 \approx 4.9 \mu T \]

### Final Answer
The amplitude of the magnetic field associated with the laser beam is approximately:
\[ \boxed{5} \mu T \]