The magnetic field of a beam emerging from a filter facing a floodlight is given by `B=12xx10^-8sin(1.20xx10^7z-3.60xx10^14t)T`.
What is the average intensity of the beam?

Options

A. `1.71 "W m"^(-2)`
B. `2.1"W m"^(-2)`
C. `3.2"W m"^(-2)`
D. `2.9"W m"^(-2)`

Explanation

To find the average intensity of the beam given the magnetic field, we can follow these steps:

### Step 1: Identify the amplitude of the magnetic field (B0)
The magnetic field is given by the equation:
\[ B = 12 \times 10^{-8} \sin(1.20 \times 10^{7} z - 3.60 \times 10^{14} t) \, T \]
From this equation, we can see that the amplitude \( B_0 \) is:
\[ B_0 = 12 \times 10^{-8} \, T \]

### Step 2: Use the formula for average intensity (I)
The average intensity \( I_{\text{average}} \) of an electromagnetic wave can be calculated using the formula:
\[ I_{\text{average}} = \frac{1}{2} \frac{B_0^2 c}{\mu_0} \]
where:
- \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, m/s \)
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, T \cdot m/A \)

### Step 3: Substitute the values into the formula
Now, substituting the values we have:
\[ I_{\text{average}} = \frac{1}{2} \times (12 \times 10^{-8})^2 \times (3 \times 10^8) \div (4\pi \times 10^{-7}) \]

### Step 4: Calculate \( B_0^2 \)
First, calculate \( B_0^2 \):
\[ B_0^2 = (12 \times 10^{-8})^2 = 144 \times 10^{-16} = 1.44 \times 10^{-14} \, T^2 \]

### Step 5: Substitute \( B_0^2 \) into the intensity formula
Now substituting \( B_0^2 \) back into the intensity formula:
\[ I_{\text{average}} = \frac{1}{2} \times (1.44 \times 10^{-14}) \times (3 \times 10^8) \div (4\pi \times 10^{-7}) \]

### Step 6: Calculate the numerator
Calculating the numerator:
\[ \text{Numerator} = \frac{1}{2} \times 1.44 \times 10^{-14} \times 3 \times 10^8 = 2.16 \times 10^{-6} \]

### Step 7: Calculate the denominator
Calculating the denominator:
\[ \text{Denominator} = 4\pi \times 10^{-7} \approx 1.2566 \times 10^{-6} \]

### Step 8: Final calculation of average intensity
Now, divide the numerator by the denominator:
\[ I_{\text{average}} = \frac{2.16 \times 10^{-6}}{1.2566 \times 10^{-6}} \approx 1.72 \, W/m^2 \]

### Final Answer
Thus, the average intensity of the beam is approximately:
\[ I_{\text{average}} \approx 1.71 \, W/m^2 \]

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