Two coils are having magnetic field B and 2B at their centres and current i and 2i then the ratio of their radius is

Options

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

Answer

To solve the problem, we need to analyze the magnetic fields produced by the two coils and relate them to their respective currents and radii.

### Step-by-Step Solution:

1. **Understanding Magnetic Field in a Coil:**
The magnetic field \( B \) at the center of a circular coil carrying current \( i \) is given by the formula:
\[ B = \frac{\mu_0 i}{2R} \]
where \( \mu_0 \) is the permeability of free space and \( R \) is the radius of the coil.

2. **Setting Up the Given Information:**
- For the first coil, the magnetic field is \( B \) and the current is \( i \).
- For the second coil, the magnetic field is \( 2B \) and the current is \( 2i \).

3. **Applying the Formula for Both Coils:**
- For the first coil:
\[ B = \frac{\mu_0 i}{2R_1} \]
- For the second coil:
\[ 2B = \frac{\mu_0 (2i)}{2R_2} \]

4. **Simplifying the Second Coil's Equation:**
From the second coil's equation, we can simplify:
\[ 2B = \frac{\mu_0 (2i)}{2R_2} \implies 2B = \frac{\mu_0 i}{R_2} \]

5. **Relating the Two Magnetic Fields:**
Now we have two equations:
- \( B = \frac{\mu_0 i}{2R_1} \)
- \( 2B = \frac{\mu_0 i}{R_2} \)

6. **Expressing \( R_1 \) and \( R_2 \):**
From the first equation, we can express \( R_1 \):
\[ R_1 = \frac{\mu_0 i}{2B} \]
From the second equation, we can express \( R_2 \):
\[ R_2 = \frac{\mu_0 i}{2B} \]

7. **Finding the Ratio of Radii:**
Now, we find the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{\frac{\mu_0 i}{2B}}{\frac{\mu_0 i}{2(2B)}} = \frac{1}{1} \]

8. **Conclusion:**
Therefore, the ratio of the radii \( R_1 : R_2 \) is:
\[ R_1 : R_2 = 1 : 1 \]

### Final Answer:
The ratio of their radii is \( 1 : 1 \).