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 will use the formula for the magnetic field at the center of a circular coil. The magnetic field \( B \) at the center of a coil is given by the formula:

\[ B = \frac{\mu_0 I}{2R} \]

where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

### Step 1: Write the magnetic field equations for both coils

For the first coil:
\[ B = \frac{\mu_0 I}{2R_1} \]

For the second coil:
\[ 2B = \frac{\mu_0 (2I)}{2R_2} \]

### Step 2: Simplify the equations

From the first coil:
\[ B = \frac{\mu_0 I}{2R_1} \quad \text{(1)} \]

From the second coil:
\[ 2B = \frac{\mu_0 (2I)}{2R_2} \implies 2B = \frac{\mu_0 I}{R_2} \quad \text{(2)} \]

### Step 3: Set up the ratio of the two equations

Now, we will divide equation (1) by equation (2):
\[ \frac{B}{2B} = \frac{\frac{\mu_0 I}{2R_1}}{\frac{\mu_0 I}{R_2}} \]

### Step 4: Cancel out common terms

Canceling out \( B \) and \( \mu_0 I \) from both sides, we have:
\[ \frac{1}{2} = \frac{R_2}{2R_1} \]

### Step 5: Rearrange the equation to find the ratio of radii

Cross-multiplying gives:
\[ R_2 = R_1 \]

Thus, the ratio of the radii \( R_1 \) to \( R_2 \) is:
\[ \frac{R_1}{R_2} = 1 \]

### Final Answer
The ratio of the radii \( R_1 : R_2 \) is \( 1 : 1 \).

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