Two circular coils of radii 10 cm and 40 cm and equal number of turns are connected in series to a battery. The ratio of magnetic fields at their centres is

Options

A. 1 : 4
B. 4 : 1
C. 2 : 1
D. 1 : 1

Answer

To solve the problem of finding the ratio of magnetic fields at the centers of two circular coils with different radii, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Formula for Magnetic Field**:
The magnetic field \( B \) at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0 n I}{2R} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns per unit length,
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

2. **Define the Parameters for Each Coil**:
- For the first coil (radius \( R_1 = 10 \) cm):
\[ B_1 = \frac{\mu_0 n I}{2 \times 0.1} = \frac{\mu_0 n I}{0.2} \]
- For the second coil (radius \( R_2 = 40 \) cm):
\[ B_2 = \frac{\mu_0 n I}{2 \times 0.4} = \frac{\mu_0 n I}{0.8} \]

3. **Calculate the Magnetic Fields**:
- For the first coil:
\[ B_1 = \frac{\mu_0 n I}{0.2} \]
- For the second coil:
\[ B_2 = \frac{\mu_0 n I}{0.8} \]

4. **Find the Ratio of the Magnetic Fields**:
To find the ratio \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0 n I}{0.2}}{\frac{\mu_0 n I}{0.8}} = \frac{0.8}{0.2} = 4 \]

5. **Express the Ratio**:
The ratio of the magnetic fields at the centers of the two coils is:
\[ \frac{B_1}{B_2} = 4:1 \]

### Final Answer:
The ratio of the magnetic fields at the centers of the two coils is \( 4:1 \).

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