An `AC rms` voltage of `2V` having a frequency of `50KHz` is applied to a condenser of capacity of `10muF`. The maximum value of the magnetic field between the plates of the condenser if the radius of plate is `10cm` is

Options

A. `0.4 p mu`
B. `4pi mu T`
C. `2 mu T`
D. `40pi mu T`

Explanation

To find the maximum value of the magnetic field between the plates of a capacitor when an AC rms voltage is applied, we can follow these steps:

### Step 1: Convert Given Values
- Given:
- AC rms voltage, \( V_{rms} = 2 \, V \)
- Frequency, \( f = 50 \, kHz = 50 \times 10^3 \, Hz \)
- Capacitance, \( C = 10 \, \mu F = 10 \times 10^{-6} \, F \)
- Radius of the plates, \( r = 10 \, cm = 0.1 \, m \)

### Step 2: Calculate Maximum Voltage
The maximum voltage \( V_{max} \) can be calculated from the rms voltage using the formula:
\[ V_{max} = V_{rms} \times \sqrt{2} \]
Substituting the values:
\[ V_{max} = 2 \times \sqrt{2} \approx 2.828 \, V \]

### Step 3: Calculate Current
The current \( I \) through the capacitor can be calculated using the formula:
\[ I = C \frac{dV}{dt} \]
For an AC voltage, \( dV/dt \) can be expressed as:
\[ \frac{dV}{dt} = 2 \pi f V_{max} \]
Substituting the values:
\[ \frac{dV}{dt} = 2 \pi (50 \times 10^3) (2.828) \approx 628318.53 \, V/s \]
Now, substituting back to find \( I \):
\[ I = C \cdot \frac{dV}{dt} = (10 \times 10^{-6}) \cdot (628318.53) \approx 6.283 \, A \]

### Step 4: Calculate Magnetic Field
The magnetic field \( B \) between the plates of the capacitor can be calculated using the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
Where \( \mu_0 = 4 \pi \times 10^{-7} \, T \cdot m/A \). Substituting the values:
\[ B = \frac{(4 \pi \times 10^{-7}) (6.283)}{2 \pi (0.1)} \]
This simplifies to:
\[ B = \frac{(4 \times 10^{-7}) (6.283)}{0.2} = \frac{2.513 \times 10^{-6}}{0.2} \approx 1.2565 \times 10^{-5} \, T \]

### Step 5: Convert to Microtesla
To convert the magnetic field from Tesla to microtesla:
\[ B \approx 12.565 \, \mu T \]

### Final Answer
The maximum value of the magnetic field between the plates of the capacitor is approximately \( 12.57 \, \mu T \).

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