A current - carrying wire of certain length is bent to form an arc that substends an angle `theta` at the centre. Let B be the magnetic field at the centre due to this wire. The correct graph between B and `theta` is

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to derive the relationship between the magnetic field \( B \) at the center of the arc and the angle \( \theta \) subtended by the arc. Here are the steps to find the correct graph between \( B \) and \( \theta \):

### Step 1: Understand the relationship between length of the wire and angle
The length of the arc \( L \) can be expressed in terms of the radius \( r \) and the angle \( \theta \) (in radians) as:
\[ L = r \theta \]

### Step 2: Express the radius in terms of length and angle
From the equation above, we can rearrange it to find the radius \( r \):
\[ r = \frac{L}{\theta} \]

### Step 3: Use the formula for magnetic field due to an arc
The magnetic field \( B \) at the center of an arc of radius \( r \) carrying a current \( I \) and subtending an angle \( \theta \) is given by:
\[ B = \frac{\mu_0 I \theta}{4 \pi r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 4: Substitute the expression for \( r \)
Now, substituting the expression for \( r \) into the magnetic field equation:
\[ B = \frac{\mu_0 I \theta}{4 \pi \left(\frac{L}{\theta}\right)} = \frac{\mu_0 I \theta^2}{4 \pi L} \]

### Step 5: Analyze the relationship between \( B \) and \( \theta \)
From the equation \( B = \frac{\mu_0 I}{4 \pi L} \theta^2 \), we can see that \( B \) is proportional to \( \theta^2 \). This indicates that as \( \theta \) increases, \( B \) increases quadratically.

### Step 6: Identify the graph type
The relationship \( B \propto \theta^2 \) suggests that the graph of \( B \) versus \( \theta \) will be a parabola opening upwards. Therefore, the correct graph will show \( B \) increasing as \( \theta \) increases.

### Conclusion
The correct graph between \( B \) and \( \theta \) is a parabolic curve, which corresponds to the option that represents a quadratic relationship.

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