The current passing through a circular coil of two rurns produces a magnetic firld of `4 mu T` as its centre. The coil is then rewound, so as to have four turns and the current passing through it is doubled. What is the new magnetic firld at the centre of the coil ?

Options

A. `8 muT`
B. `16 muT`
C. `24 muT`
D. `32 muT`

Explanation

To solve the problem, we need to determine the new magnetic field at the center of a circular coil after it has been rewound and the current has been doubled. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
- The initial coil has **2 turns** (N1 = 2).
- The magnetic field at the center of this coil is given as **4 µT** (B1 = 4 µT).
- The current through the coil is **I1**.

### Step 2: Determine the New Conditions
- The coil is rewound to have **4 turns** (N2 = 4).
- The current is doubled, so the new current is **I2 = 2 * I1**.

### Step 3: Use the Formula for Magnetic Field
The magnetic field (B) at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2\pi N I}{R} \]
Where:
- \( \mu_0 \) is the permeability of free space (constant).
- \( N \) is the number of turns.
- \( I \) is the current.
- \( R \) is the radius of the coil.

### Step 4: Compare the Magnetic Fields
For the first coil:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2\pi N_1 I_1}{R_1} \]
For the second coil:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{2\pi N_2 I_2}{R_2} \]

### Step 5: Analyze the Radius
Since the wire length remains constant, we can assume that the radius changes when the number of turns changes. The length of the wire is given by:
\[ L = 2\pi R_1 N_1 = 2\pi R_2 N_2 \]
From this, we can derive that:
\[ R_2 = \frac{N_1}{N_2} R_1 = \frac{2}{4} R_1 = \frac{1}{2} R_1 \]

### Step 6: Substitute Values into the Magnetic Field Formula
Now substituting \( N_2 \) and \( I_2 \) into the equation for \( B_2 \):
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{2\pi (4) (2 I_1)}{\frac{1}{2} R_1} \]
This simplifies to:
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{2\pi \cdot 8 I_1}{\frac{1}{2} R_1} \]
\[ B_2 = \frac{\mu_0}{4\pi} \cdot \frac{16\pi I_1}{R_1} \]

### Step 7: Relate \( B_2 \) to \( B_1 \)
Since:
\[ B_1 = \frac{\mu_0}{4\pi} \cdot \frac{2\pi N_1 I_1}{R_1} \]
We can express \( B_2 \) in terms of \( B_1 \):
\[ B_2 = 8 \cdot B_1 \]

### Step 8: Calculate the New Magnetic Field
Substituting \( B_1 = 4 \mu T \):
\[ B_2 = 8 \cdot 4 \mu T = 32 \mu T \]

### Final Answer
The new magnetic field at the center of the coil after rewinding and doubling the current is **32 µT**.

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