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 the magnetic fields at the centers of two circular coils with different radii but equal number of turns, we can follow these steps:

### Step 1: Understand the formula for the magnetic field at the center of a circular coil
The magnetic field \( B \) at the center of a circular coil can be calculated using the formula:
\[ B = \frac{\mu_0 N i}{4R} \]
where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the number of turns in the coil,
- \( i \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

### Step 2: Identify the parameters for both coils
Let:
- For Coil 1 (radius \( R_1 = 10 \) cm):
- \( N_1 = N \) (number of turns)
- \( R_1 = 0.1 \) m (convert cm to m for standard SI units)

- For Coil 2 (radius \( R_2 = 40 \) cm):
- \( N_2 = N \) (number of turns)
- \( R_2 = 0.4 \) m (convert cm to m for standard SI units)

### Step 3: Write the magnetic field equations for both coils
Using the formula for magnetic field:
- For Coil 1:
\[ B_1 = \frac{\mu_0 N i}{4 R_1} = \frac{\mu_0 N i}{4 \times 0.1} \]
- For Coil 2:
\[ B_2 = \frac{\mu_0 N i}{4 R_2} = \frac{\mu_0 N i}{4 \times 0.4} \]

### Step 4: Find the ratio of the magnetic fields
To find the ratio \( \frac{B_1}{B_2} \):
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0 N i}{4 \times 0.1}}{\frac{\mu_0 N i}{4 \times 0.4}} \]
The \( \mu_0 \), \( N \), and \( i \) cancel out:
\[ \frac{B_1}{B_2} = \frac{0.4}{0.1} = 4 \]

### Step 5: Conclusion
Thus, the ratio of the magnetic fields at their centers is:
\[ \frac{B_1}{B_2} = 4:1 \]