A condenser has two conducting plates of radius `10cm` separated by a distance of `5 mm`. It is charged with a constant current of `0.15A`. The magnetic field at a point `2cm` from the axis in the gap is

Options

A. `1.5xx10^(6)T`
B. `3xx10^(-8)T`
C. `6xx10^(-8)T`
D. `3xx10^(-6)T`

Explanation

To find the magnetic field at a point 2 cm from the axis of a charged parallel plate capacitor, we can use the formula for the magnetic field around a current-carrying conductor. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the plates, \( R = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Distance between the plates, \( d = 5 \, \text{mm} = 0.005 \, \text{m} \)
- Current, \( I = 0.15 \, \text{A} \)
- Distance from the axis where we want to find the magnetic field, \( r = 2 \, \text{cm} = 0.02 \, \text{m} \)

2. **Determine the Relationship Between r and R:**
- Since \( r = 0.02 \, \text{m} < R = 0.1 \, \text{m} \), we will use the formula for the magnetic field inside the region between the plates.

3. **Use the Formula for the Magnetic Field:**
- The formula for the magnetic field \( B \) at a distance \( r \) from the axis of a long straight conductor is given by:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
- Here, \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

4. **Substitute the Values into the Formula:**
- Plugging in the values:
\[ B = \frac{4\pi \times 10^{-7} \times 0.15}{2 \pi \times 0.02} \]

5. **Simplify the Expression:**
- The \( \pi \) cancels out:
\[ B = \frac{4 \times 10^{-7} \times 0.15}{2 \times 0.02} \]
- Calculate the denominator:
\[ 2 \times 0.02 = 0.04 \]
- Now, substituting this back:
\[ B = \frac{4 \times 10^{-7} \times 0.15}{0.04} \]

6. **Calculate the Magnetic Field:**
- Now calculate the numerator:
\[ 4 \times 0.15 = 0.6 \]
- Therefore:
\[ B = \frac{0.6 \times 10^{-7}}{0.04} = 0.6 \times 10^{-7} \times 25 = 15 \times 10^{-6} \, \text{T} = 60 \times 10^{-9} \, \text{T} \]

7. **Final Answer:**
- The magnetic field at a point 2 cm from the axis in the gap is:
\[ B = 60 \, \text{nT} \, (\text{nanotesla}) \]