The magnetic field at the centre of a circular coil of radius `r` carrying current `l` is `B_(1)`. The field at the centre of another coil of radius `2r` carrying same current `l` is `B_(2)`. The ratio `(B_(1))/(B_(2))` is

Options

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

Answer

To find the ratio of the magnetic fields \( \frac{B_1}{B_2} \) at the center of two circular coils with different radii but carrying the same current, we can follow these steps:

### Step 1: Write the formula for the magnetic field at the center of a circular coil.
The magnetic field \( B \) at the center of a circular coil of radius \( r \) carrying a current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Calculate \( B_1 \) for the first coil.
For the first coil with radius \( r \) and current \( I \):
\[ B_1 = \frac{\mu_0 I}{2r} \]

### Step 3: Calculate \( B_2 \) for the second coil.
For the second coil with radius \( 2r \) and the same current \( I \):
\[ B_2 = \frac{\mu_0 I}{2(2r)} = \frac{\mu_0 I}{4r} \]

### Step 4: Find the ratio \( \frac{B_1}{B_2} \).
Now, we can find the ratio of the two magnetic fields:
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0 I}{2r}}{\frac{\mu_0 I}{4r}} \]

### Step 5: Simplify the ratio.
When we simplify this expression, we get:
\[ \frac{B_1}{B_2} = \frac{\mu_0 I}{2r} \times \frac{4r}{\mu_0 I} = \frac{4}{2} = 2 \]

### Final Answer:
Thus, the ratio \( \frac{B_1}{B_2} \) is:
\[ \frac{B_1}{B_2} = 2 \]

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