At the centre of a current - carrying circular coil of radius 5 cm, magnetic field due to earth is `0.5xx10^(-5)"Wb m"^(-2)`. The current flowing through the coil, so that it equals the Earth's magnetic field, is

Options

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

Answer

To find the current flowing through a circular coil such that the magnetic field at its center equals the Earth's magnetic field, we can use the formula for the magnetic field at the center of a circular coil:

\[ B = \frac{\mu_0 I}{2R} \]

Where:
- \( B \) is the magnetic field at the center of the coil,
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{Wb/A·m} \),
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

### Step 1: Identify the given values
- The radius of the coil, \( R = 5 \, \text{cm} = 0.05 \, \text{m} \)
- The magnetic field due to the Earth, \( B = 0.5 \times 10^{-5} \, \text{Wb/m}^2 \)

### Step 2: Substitute the values into the formula
We need to set the magnetic field \( B \) equal to the Earth's magnetic field:

\[ 0.5 \times 10^{-5} = \frac{4\pi \times 10^{-7} \cdot I}{2 \cdot 0.05} \]

### Step 3: Simplify the equation
First, simplify the denominator:

\[ 2 \cdot 0.05 = 0.1 \]

Now the equation becomes:

\[ 0.5 \times 10^{-5} = \frac{4\pi \times 10^{-7} \cdot I}{0.1} \]

### Step 4: Rearranging to solve for \( I \)
Multiply both sides by \( 0.1 \):

\[ 0.1 \times 0.5 \times 10^{-5} = 4\pi \times 10^{-7} \cdot I \]

Calculating the left side:

\[ 0.05 \times 10^{-5} = 5 \times 10^{-7} \]

So we have:

\[ 5 \times 10^{-7} = 4\pi \times 10^{-7} \cdot I \]

### Step 5: Isolate \( I \)
Now, divide both sides by \( 4\pi \times 10^{-7} \):

\[ I = \frac{5 \times 10^{-7}}{4\pi \times 10^{-7}} \]

### Step 6: Simplify further
The \( 10^{-7} \) cancels out:

\[ I = \frac{5}{4\pi} \]

### Step 7: Calculate the value of \( I \)
Using \( \pi \approx 3.14 \):

\[ I \approx \frac{5}{4 \times 3.14} \approx \frac{5}{12.56} \approx 0.398 \, \text{A} \]

### Step 8: Round to appropriate significant figures
Thus, rounding to two significant figures, we get:

\[ I \approx 0.4 \, \text{A} \]

### Final Answer
The current flowing through the coil is approximately \( 0.4 \, \text{A} \).

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