A laser beam has intensity `2.5 xx 10^14W m^-2`. Find the amplitudes of electric and magnetic fields in the beam.

Explanation

To find the amplitudes of the electric and magnetic fields in a laser beam with a given intensity, we can follow these steps:

### Step 1: Understand the relationship between intensity and electric field amplitude
The intensity \( I \) of an electromagnetic wave can be expressed in terms of the electric field amplitude \( E_0 \) and the speed of light \( c \) using the formula:

\[ I = \frac{1}{2} \epsilon_0 c E_0^2 \]

Where:
- \( I \) is the intensity of the wave,
- \( \epsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \, \text{F/m} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( E_0 \) is the amplitude of the electric field.

### Step 2: Rearrange the formula to solve for \( E_0 \)
To find the electric field amplitude \( E_0 \), we can rearrange the formula:

\[ E_0 = \sqrt{\frac{2I}{\epsilon_0 c}} \]

### Step 3: Substitute the known values
Given:
- \( I = 2.5 \times 10^{14} \, \text{W/m}^2 \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \)
- \( c = 3 \times 10^8 \, \text{m/s} \)

Now substitute these values into the equation:

\[ E_0 = \sqrt{\frac{2 \times (2.5 \times 10^{14})}{(8.85 \times 10^{-12}) \times (3 \times 10^8)}} \]

### Step 4: Calculate the value inside the square root
First, calculate the denominator:

\[ \epsilon_0 c = (8.85 \times 10^{-12}) \times (3 \times 10^8) = 2.655 \times 10^{-3} \]

Now calculate the numerator:

\[ 2 \times (2.5 \times 10^{14}) = 5.0 \times 10^{14} \]

Now, substitute these values back into the equation:

\[ E_0 = \sqrt{\frac{5.0 \times 10^{14}}{2.655 \times 10^{-3}}} \]

### Step 5: Perform the division and take the square root
Calculating the division:

\[ \frac{5.0 \times 10^{14}}{2.655 \times 10^{-3}} \approx 1.886 \times 10^{17} \]

Now take the square root:

\[ E_0 \approx \sqrt{1.886 \times 10^{17}} \approx 4.34 \times 10^8 \, \text{N/C} \]

### Step 6: Find the amplitude of the magnetic field \( B_0 \)
The amplitude of the magnetic field \( B_0 \) can be calculated using the relationship:

\[ B_0 = \frac{E_0}{c} \]

Substituting the values:

\[ B_0 = \frac{4.34 \times 10^8}{3 \times 10^8} \approx 1.45 \times 10^0 \, \text{T} = 1.45 \, \text{T} \]

### Final Answers
- The amplitude of the electric field \( E_0 \) is approximately \( 4.34 \times 10^8 \, \text{N/C} \).
- The amplitude of the magnetic field \( B_0 \) is approximately \( 1.45 \, \text{T} \).