A condenser of capacity `50pF` is connected to an `AC` supply of `220 V 50Hz`. The `rms` value of magnetic field at a distance of `5cm` from the axis is

Options

A. `22pixx10^(-1)T`
B. `22pixx10^(-12)T`
C. `44pixx10^(-13)T`
D. `(11)/(5)pixx10^(-12)T`

Explanation

To solve the problem, we need to find the RMS value of the magnetic field at a distance of 5 cm from the axis of a condenser connected to an AC supply. Here is a step-by-step solution:

### Step 1: Identify the given values
- Capacitance \( C = 50 \, \text{pF} = 50 \times 10^{-12} \, \text{F} \)
- Voltage \( E = 220 \, \text{V} \)
- Frequency \( f = 50 \, \text{Hz} \)

### Step 2: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 50 = 100\pi \, \text{rad/s} \]

### Step 3: Calculate the capacitive reactance \( X_C \)
The capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{\omega C} \]
Substituting the values of \( \omega \) and \( C \):
\[ X_C = \frac{1}{100\pi \times 50 \times 10^{-12}} = \frac{1}{5 \times 10^{-10} \pi} = \frac{2 \times 10^{9}}{\pi} \, \Omega \]

### Step 4: Calculate the RMS current \( I_{RMS} \)
The RMS current \( I_{RMS} \) can be calculated using:
\[ I_{RMS} = \frac{E}{X_C} \]
Substituting the values of \( E \) and \( X_C \):
\[ I_{RMS} = \frac{220}{\frac{2 \times 10^{9}}{\pi}} = \frac{220 \pi}{2 \times 10^{9}} = \frac{110 \pi}{10^{9}} \, \text{A} \]

### Step 5: Calculate the RMS magnetic field \( B_{RMS} \)
The RMS magnetic field \( B_{RMS} \) at a distance \( R = 5 \, \text{cm} = 0.05 \, \text{m} \) is given by:
\[ B_{RMS} = \frac{\mu_0 I_{RMS}}{2\pi R} \]
Where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \). Substituting the values:
\[ B_{RMS} = \frac{4\pi \times 10^{-7} \times \frac{110 \pi}{10^{9}}}{2\pi \times 0.05} \]
Simplifying this:
\[ B_{RMS} = \frac{4 \times 110 \times 10^{-7}}{2 \times 0.05 \times 10^{9}} = \frac{440 \times 10^{-7}}{0.1 \times 10^{9}} = \frac{440 \times 10^{-7}}{10^{-1} \times 10^{9}} = \frac{440 \times 10^{-7}}{10^{8}} = 4.4 \times 10^{-6} \, \text{T} = 8.8 \times 10^{-4} \, \text{T} \]

### Final Answer
The RMS value of the magnetic field at a distance of 5 cm from the axis is:
\[ B_{RMS} = 8.8 \times 10^{-4} \, \text{T} \]