The earth's magnetic field at a given point is `0.5xx10^(-5) Wb-m^(-2)`. This field is to be annulled by magnetic indcution at the centre of a circular conducting loop of radius `5.0 cm`. The current required to be flown in the loop is nearly

Options

A. `0.2A`
B. `0.4A`
C. `4A`
D. `40 A`

Explanation

To solve the problem of finding the current required to annul the Earth's magnetic field at the center of a circular conducting loop, we can follow these steps:

### Step 1: Understand the given data
- The Earth's magnetic field \( B_E = 0.5 \times 10^{-5} \, \text{Wb/m}^2 \).
- The radius of the circular loop \( r = 5.0 \, \text{cm} = 0.05 \, \text{m} \).

### Step 2: Use the formula for the magnetic field at the center of a circular loop
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 3: Set the magnetic field equal to the Earth's magnetic field
To annul the Earth's magnetic field, we need the magnetic field produced by the loop to be equal in magnitude but opposite in direction to the Earth's magnetic field:
\[ \frac{\mu_0 I}{2r} = 0.5 \times 10^{-5} \]

### Step 4: Substitute the values into the equation
Substituting \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) and \( r = 0.05 \, \text{m} \):
\[ \frac{4\pi \times 10^{-7} I}{2 \times 0.05} = 0.5 \times 10^{-5} \]

### Step 5: Simplify the equation
This simplifies to:
\[ \frac{4\pi \times 10^{-7} I}{0.1} = 0.5 \times 10^{-5} \]
\[ 4\pi \times 10^{-6} I = 0.5 \times 10^{-5} \]

### Step 6: Solve for \( I \)
Now, isolate \( I \):
\[ I = \frac{0.5 \times 10^{-5}}{4\pi \times 10^{-6}} \]
\[ I = \frac{0.5}{4\pi} \, \text{A} \]

### Step 7: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ I \approx \frac{0.5}{4 \times 3.14} \approx \frac{0.5}{12.56} \approx 0.0398 \, \text{A} \approx 0.04 \, \text{A} \]

### Conclusion
The current required to annul the Earth's magnetic field at the center of the circular conducting loop is approximately \( 0.04 \, \text{A} \) or \( 40 \, \text{mA} \).

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