A long straight non-conducting string carriers a charge density of `40 muC//m`. It is pulled along its length at a speed of `300 m//sec`. What is the magnetic field at a normal distance of `5 mm` from the moving string?

Options

A. `B=4.8 xx 10^(-7)T`
B. `B=3.2 xx 10^(-7)T`
C. `B=2.5 xx 10^(-7)T`
D. `B=5 xx 10^(-7)T`

Explanation

To solve the problem step by step, we will follow the outlined method in the video transcript.

### Step 1: Identify the given values
- Charge density (\( \lambda \)) = \( 40 \, \mu C/m = 40 \times 10^{-6} \, C/m \)
- Speed (\( v \)) = \( 300 \, m/s \)
- Distance from the string (\( r \)) = \( 5 \, mm = 5 \times 10^{-3} \, m \)

### Step 2: Calculate the current (\( I \))
The current \( I \) can be calculated using the formula:
\[ I = \lambda \cdot v \]
Substituting the values:
\[ I = (40 \times 10^{-6} \, C/m) \cdot (300 \, m/s) \]
Calculating this gives:
\[ I = 12 \times 10^{-3} \, A = 0.012 \, A \]

### Step 3: Use the formula for the magnetic field (\( B \))
The magnetic field \( B \) at a distance \( r \) from a long straight current-carrying conductor is given by:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
Where \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, T \cdot m/A \).

### Step 4: Substitute the values into the magnetic field formula
Substituting the values we have:
\[ B = \frac{(4\pi \times 10^{-7} \, T \cdot m/A) \cdot (0.012 \, A)}{2 \pi (5 \times 10^{-3} \, m)} \]

### Step 5: Simplify the expression
The \( \pi \) terms cancel out:
\[ B = \frac{4 \times 10^{-7} \cdot 0.012}{2 \cdot 5 \times 10^{-3}} \]
Calculating the denominator:
\[ 2 \cdot 5 \times 10^{-3} = 10 \times 10^{-3} = 0.01 \]
Now substituting back:
\[ B = \frac{4 \times 10^{-7} \cdot 0.012}{0.01} \]
Calculating the numerator:
\[ 4 \times 10^{-7} \cdot 0.012 = 4.8 \times 10^{-9} \]
Now substituting this into the equation:
\[ B = \frac{4.8 \times 10^{-9}}{0.01} = 4.8 \times 10^{-7} \, T \]

### Final Answer
The magnetic field at a normal distance of \( 5 \, mm \) from the moving string is:
\[ B = 4.8 \times 10^{-7} \, T \]