The radii of two concentric coils having same number of turns are 10 cm and 20 cm respectively. Equal currents are passed through them first in same direction and then in opposite direction. In these two conditions the ratio of resultant magnetic fields at the centre will be -

Options

A. `3 : 1`
B. `2 : 1`
C. `3 : 2`
D. ` 1 : 1`

Answer

To solve the problem, we need to find the ratio of the resultant magnetic fields at the center of two concentric coils when equal currents are passed through them in the same direction and then in opposite directions.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the inner coil, \( R_1 = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Radius of the outer coil, \( R_2 = 20 \, \text{cm} = 0.2 \, \text{m} \)
- Number of turns in both coils, \( n \) (same for both)
- Current through both coils, \( I \) (same for both)

2. **Magnetic Field Due to a Single Coil:**
The magnetic field 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.

3. **Calculate Magnetic Fields for Both Cases:**
- **Case 1: Currents in the Same Direction**
- Magnetic field due to the inner coil:
\[ B_1 = \frac{\mu_0 n I}{2R_1} = \frac{\mu_0 n I}{2 \times 0.1} \]
- Magnetic field due to the outer coil:
\[ B_2 = \frac{\mu_0 n I}{2R_2} = \frac{\mu_0 n I}{2 \times 0.2} \]
- Resultant magnetic field when currents are in the same direction:
\[ B_{\text{net}} = B_1 + B_2 = \frac{\mu_0 n I}{2 \times 0.1} + \frac{\mu_0 n I}{2 \times 0.2} \]

4. **Simplify the Resultant Magnetic Field:**
\[ B_{\text{net}} = \frac{\mu_0 n I}{2} \left( \frac{1}{0.1} + \frac{1}{0.2} \right) = \frac{\mu_0 n I}{2} \left( 10 + 5 \right) = \frac{\mu_0 n I}{2} \times 15 \]
\[ B_{\text{net}} = \frac{15 \mu_0 n I}{2} \]

5. **Case 2: Currents in Opposite Directions**
- Magnetic field due to the inner coil remains \( B_1 \) (same direction).
- Magnetic field due to the outer coil will be in the opposite direction:
\[ B_2 = -\frac{\mu_0 n I}{2R_2} \]
- Resultant magnetic field when currents are in opposite directions:
\[ B'_{\text{net}} = B_1 - B_2 = \frac{\mu_0 n I}{2R_1} - \frac{\mu_0 n I}{2R_2} \]

6. **Simplify the Resultant Magnetic Field for Opposite Directions:**
\[ B'_{\text{net}} = \frac{\mu_0 n I}{2} \left( \frac{1}{0.1} - \frac{1}{0.2} \right) = \frac{\mu_0 n I}{2} \left( 10 - 5 \right) = \frac{\mu_0 n I}{2} \times 5 \]
\[ B'_{\text{net}} = \frac{5 \mu_0 n I}{2} \]

7. **Calculate the Ratio of the Resultant Magnetic Fields:**
\[ \text{Ratio} = \frac{B_{\text{net}}}{B'_{\text{net}}} = \frac{\frac{15 \mu_0 n I}{2}}{\frac{5 \mu_0 n I}{2}} = \frac{15}{5} = 3 \]

### Final Answer:
The ratio of the resultant magnetic fields at the center when the currents are in the same direction to when they are in opposite directions is **3:1**.