Two concentric coplanar circular loops of radii `r_(1)` and `r_(2)` carry currents of respectively `i_(1)` and `i_(2)` in opposite direction (one clockwise and the other anticlockwise). The magnetic induction at the centre of the loops is half that due to `i_(1)` alone at the centre. if `r_(2)=2r_(1)`. the value of `i_(2)//i_(1)` is

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Magnetic Field Due to Each Loop
The magnetic field at the center of a circular loop carrying current \( i \) is given by the formula:
\[ B = \frac{\mu_0 i}{2r} \]
where \( \mu_0 \) is the permeability of free space, \( i \) is the current, and \( r \) is the radius of the loop.

### Step 2: Calculate the Magnetic Field for Each Loop
For the first loop with radius \( r_1 \) and current \( i_1 \):
\[ B_1 = \frac{\mu_0 i_1}{2r_1} \]

For the second loop with radius \( r_2 \) and current \( i_2 \):
\[ B_2 = \frac{\mu_0 i_2}{2r_2} \]

### Step 3: Substitute the Given Relationship Between the Radii
We are given that \( r_2 = 2r_1 \). Thus, we can substitute \( r_2 \) in the equation for \( B_2 \):
\[ B_2 = \frac{\mu_0 i_2}{2(2r_1)} = \frac{\mu_0 i_2}{4r_1} \]

### Step 4: Determine the Net Magnetic Field at the Center
The net magnetic field at the center of the loops, considering that the currents are in opposite directions, is given by:
\[ B_{\text{net}} = B_1 - B_2 \]
Substituting the expressions for \( B_1 \) and \( B_2 \):
\[ B_{\text{net}} = \frac{\mu_0 i_1}{2r_1} - \frac{\mu_0 i_2}{4r_1} \]

### Step 5: Simplify the Expression for the Net Magnetic Field
Factoring out \( \frac{\mu_0}{r_1} \):
\[ B_{\text{net}} = \frac{\mu_0}{r_1} \left( \frac{i_1}{2} - \frac{i_2}{4} \right) \]

### Step 6: Set Up the Given Condition
According to the problem, the net magnetic field is half of the magnetic field due to \( i_1 \) alone:
\[ B_{\text{net}} = \frac{1}{2} B_1 = \frac{1}{2} \left( \frac{\mu_0 i_1}{2r_1} \right) = \frac{\mu_0 i_1}{4r_1} \]

### Step 7: Equate the Two Expressions for the Net Magnetic Field
Setting the two expressions for \( B_{\text{net}} \) equal to each other:
\[ \frac{\mu_0}{r_1} \left( \frac{i_1}{2} - \frac{i_2}{4} \right) = \frac{\mu_0 i_1}{4r_1} \]

### Step 8: Cancel Out Common Terms
Since \( \mu_0 \) and \( r_1 \) are common on both sides, we can simplify:
\[ \frac{i_1}{2} - \frac{i_2}{4} = \frac{i_1}{4} \]

### Step 9: Solve for \( i_2 \)
Rearranging gives:
\[ \frac{i_1}{2} - \frac{i_1}{4} = \frac{i_2}{4} \]
\[ \frac{2i_1}{4} - \frac{i_1}{4} = \frac{i_2}{4} \]
\[ \frac{i_1}{4} = \frac{i_2}{4} \]
Thus, we find:
\[ i_2 = i_1 \]

### Step 10: Calculate the Ratio \( \frac{i_2}{i_1} \)
Finally, we compute the ratio:
\[ \frac{i_2}{i_1} = 1 \]

### Final Answer
The value of \( \frac{i_2}{i_1} \) is \( 1 \).

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