An identical electric current is passed through two coils and it is found that both are producing equal magnetic field at their centres. If the ratio of the number of turns in the two coils is` 7:19`, then what should be the ratio of their radii?

Options

A. `19 : 7`
B. `1 : 1`
C. `7 : 19`
D. `1 : 4`

Explanation

To solve the problem, we need to derive the relationship between the magnetic fields produced by the two coils and the given ratio of the number of turns.

### Step-by-Step Solution:

1. **Understand the Magnetic Field Formula**:
The magnetic field \( B \) at the center of a circular coil carrying current \( I \) 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,
- \( I \) is the current, and
- \( r \) is the radius of the coil.

2. **Set Up the Equations for Both Coils**:
For the first coil (with radius \( r_1 \) and number of turns \( n_1 \)):
\[ B_1 = \frac{\mu_0 n_1 I}{2r_1} \]
For the second coil (with radius \( r_2 \) and number of turns \( n_2 \)):
\[ B_2 = \frac{\mu_0 n_2 I}{2r_2} \]

3. **Equate the Magnetic Fields**:
Since it is given that both coils produce equal magnetic fields at their centers, we can set \( B_1 = B_2 \):
\[ \frac{\mu_0 n_1 I}{2r_1} = \frac{\mu_0 n_2 I}{2r_2} \]

4. **Cancel Common Terms**:
The terms \( \mu_0 \), \( I \), and \( 2 \) are common on both sides and can be canceled out:
\[ \frac{n_1}{r_1} = \frac{n_2}{r_2} \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ \frac{r_1}{r_2} = \frac{n_1}{n_2} \]

6. **Substitute the Given Ratio**:
We are given that the ratio of the number of turns is \( \frac{n_1}{n_2} = \frac{7}{19} \). Therefore:
\[ \frac{r_1}{r_2} = \frac{7}{19} \]

7. **Final Result**:
The ratio of the radii of the two coils is:
\[ r_1 : r_2 = 7 : 19 \]