The electric current in a circular coil of two turns produced a magnetic induction of 0.2 T at its centre. The coil is unwound and then rewound into a circular coil of four turns. If same current flows in the coil, the magnetic induction at the centre of the coil now is

Options

A. 0.2T
B. 0.4T
C. 0.6T
D. 0.8T

Explanation

To solve the problem, we need to determine the magnetic induction at the center of a circular coil after it has been rewound into a coil with a different number of turns. Let's break down the solution step by step.

### Step 1: Understand the relationship between magnetic induction, number of turns, and current.
The magnetic induction \( B \) at the center of a circular coil is given by the formula:
\[ B = \frac{\mu_0 n I}{2R} \]
where:
- \( B \) is the magnetic induction,
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns,
- \( I \) is the current flowing through the coil,
- \( R \) is the radius of the coil.

### Step 2: Set up the equations for the two scenarios.
For the first coil with \( n_1 = 2 \) turns, the magnetic induction \( B_1 \) is given as 0.2 T. Thus, we can write:
\[ B_1 = \frac{\mu_0 n_1 I}{2R_1} = 0.2 \, \text{T} \]

For the second coil with \( n_2 = 4 \) turns, we need to find the new magnetic induction \( B_2 \):
\[ B_2 = \frac{\mu_0 n_2 I}{2R_2} \]

### Step 3: Relate the lengths of wire used in both coils.
The length of the wire used in both coils remains the same. The length of the wire \( L \) for the first coil is:
\[ L = n_1 \cdot 2\pi R_1 = 2 \cdot 2\pi R_1 = 4\pi R_1 \]

For the second coil:
\[ L = n_2 \cdot 2\pi R_2 = 4 \cdot 2\pi R_2 = 8\pi R_2 \]

Since the lengths are equal, we can equate them:
\[ 4\pi R_1 = 8\pi R_2 \]
From this, we can simplify to find the relationship between \( R_1 \) and \( R_2 \):
\[ R_2 = \frac{R_1}{2} \]

### Step 4: Substitute \( R_2 \) into the equation for \( B_2 \).
Now we can substitute \( R_2 \) into the equation for \( B_2 \):
\[ B_2 = \frac{\mu_0 n_2 I}{2R_2} = \frac{\mu_0 \cdot 4 I}{2 \cdot \left(\frac{R_1}{2}\right)} = \frac{\mu_0 \cdot 4 I}{R_1} \]

### Step 5: Relate \( B_2 \) to \( B_1 \).
Now we can relate \( B_2 \) to \( B_1 \):
\[ B_1 = \frac{\mu_0 \cdot 2 I}{2R_1} = \frac{\mu_0 I}{R_1} \]
Thus, we can express \( B_2 \) in terms of \( B_1 \):
\[ B_2 = 4 \cdot B_1 \]

### Step 6: Calculate \( B_2 \).
Given that \( B_1 = 0.2 \, \text{T} \):
\[ B_2 = 4 \cdot 0.2 \, \text{T} = 0.8 \, \text{T} \]

### Final Answer:
The magnetic induction at the center of the coil with 4 turns is \( \boxed{0.8 \, \text{T}} \).

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