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

Answer

To solve the problem, we need to find the current required to annul the Earth's magnetic field at the center of a circular conducting loop. Here are the steps to find the solution:

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

### 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 Up the Equation
Since we want to annul the Earth's magnetic field with the magnetic field produced by the loop, we set:
\[ B = B_E \]
Thus, we have:
\[ \frac{\mu_0 I}{2r} = 0.5 \times 10^{-5} \]

### Step 4: Substitute the Values into the Equation
Substituting \( \mu_0 \) and \( r \) into the equation:
\[ \frac{(4\pi \times 10^{-7}) I}{2 \times 0.05} = 0.5 \times 10^{-5} \]

### Step 5: Simplify the Equation
First, simplify the left side:
\[ \frac{4\pi \times 10^{-7} I}{0.1} = 0.5 \times 10^{-5} \]
This simplifies to:
\[ 40\pi \times 10^{-7} I = 0.5 \times 10^{-5} \]

### Step 6: Solve for Current \( I \)
Now, isolate \( I \):
\[ I = \frac{0.5 \times 10^{-5}}{40\pi \times 10^{-7}} \]
Calculating the right side:
\[ I = \frac{0.5 \times 10^{-5}}{40 \times 3.14 \times 10^{-7}} = \frac{0.5 \times 10^{-5}}{1.256 \times 10^{-5}} \approx 0.398 \, \text{A} \]

### Step 7: Round the Answer
Rounding off gives approximately:
\[ I \approx 0.4 \, \text{A} \]

### Final Answer
The current required to annul the Earth's magnetic field at the center of the loop is nearly \( 0.4 \, \text{A} \).

---