A thin wire of length `l` is carrying a constant current. The wire is bent to form a circular coil. If radius of the coil, thus formed, is equal to R and number of turns in it is equal to `n`, then which of the following graphs represent (s) variation of magnetic field induction (B) at centre of the coil

Options

A.
B.
C.
D.

Answer

To solve the problem step by step, we will analyze the relationship between the magnetic field induction (B) at the center of a circular coil formed by bending a wire of length \( l \) carrying a constant current \( I \).

### Step 1: Determine the relationship between the wire length, radius, and number of turns

When the wire of length \( l \) is bent into a circular coil with \( n \) turns, the circumference of the coil can be expressed as:
\[ n \times 2\pi R = l \]
From this equation, we can express the radius \( R \) in terms of the length of the wire and the number of turns:
\[ R = \frac{l}{2\pi n} \]

### Step 2: Write the formula for the magnetic field at the center of the coil

The magnetic field \( B \) at the center of a circular coil with \( n \) turns carrying a current \( I \) is given by the formula:
\[ B = \frac{n \mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Substitute the expression for \( R \)

Now, substituting the expression for \( R \) from Step 1 into the formula for \( B \):
\[ B = \frac{n \mu_0 I}{2 \left(\frac{l}{2\pi n}\right)} = \frac{n \mu_0 I \cdot 2\pi n}{2l} = \frac{\mu_0 I n^2 \pi}{l} \]
This shows that the magnetic field \( B \) is directly proportional to the square of the number of turns \( n \):
\[ B \propto n^2 \]

### Step 4: Analyze the relationship between B and R

From the earlier expression for \( R \), we can see that as \( n \) increases, \( R \) decreases. Since \( B \) is inversely proportional to \( R \) (as \( R \) is in the denominator), we can conclude that:
\[ B \propto \frac{1}{R} \]
This means that if \( R \) increases, \( B \) decreases, and vice versa.

### Step 5: Determine the type of graph

- Since \( B \) is directly proportional to \( n^2 \), the graph of \( B \) versus \( n \) will be a parabola opening upwards.
- Since \( B \) is inversely proportional to \( R \), the graph of \( B \) versus \( R \) will be a hyperbola, indicating that as \( R \) increases, \( B \) decreases.

### Conclusion

Based on the analysis, the correct graph representing the variation of magnetic field induction \( B \) at the center of the coil with respect to the number of turns \( n \) is a parabola opening upwards. The graph with respect to the radius \( R \) will be a hyperbola.