Question

In a wire of radius 1 mm, a steady current of 2 A uniformly distributed across the cross-section of the wire is flowing. Then the magnetic field at a point 0.25 mm from the centre of the wire is

• A. \( 100 \, \mu\text{T} \)
• B. \( 200 \, \mu\text{T} \)
• C. \( 300 \, \mu\text{T} \)
• D. \( 400 \, \mu\text{T} \)

Hint:

Magnetic field due to a long straight wire carrying current I: - Inside the wire (\(r \le R\)): \( B_{in} = \frac{\mu_0 I r}{2\pi R^2} \) (current is uniformly distributed). - Outside the wire (\(r \ge R\)): \( B_{out} = \frac{\mu_0 I}{2\pi r} \). Here \(R\) is the radius of the wire, \(r\) is the distance from the centre. \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \). \( 1 \, \text{mm} = 10^{-3} \, \text{m} \). \( 1 \, \mu\text{T} = 10^{-6} \, \text{T} \).

Answer 1

The Correct Option is A

Radius of the wire \( R = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \).
Steady current \( I = 2 \) A.
Point distance from the centre \( r = 0.
25 \, \text{mm} = 0.
25 \times 10^{-3} \, \text{m} \).
Since \( r<R \), the point is inside the wire.
For a long straight wire carrying current uniformly distributed, the magnetic field \(B\) inside the wire (\(r \le R\)) is given by: \[ B_{in} = \frac{\mu_0 I r}{2\pi R^2} \] where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m A}^{-1} \) is the permeability of free space.
Substitute the values: \[ B = \frac{(4\pi \times 10^{-7}) \times (2 \, \text{A}) \times (0.
25 \times 10^{-3} \, \text{m})}{2\pi (1 \times 10^{-3} \, \text{m})^2} \] \[ B = \frac{4\pi \times 10^{-7} \times 2 \times 0.
25 \times 10^{-3}}{2\pi \times 1^2 \times (10^{-3})^2} \, \text{T} \] \[ B = \frac{4\pi \times 2 \times 0.
25}{2\pi \times 1} \times \frac{10^{-7} \times 10^{-3}}{(10^{-3})^2} \, \text{T} \] \[ B = \frac{2 \times 2 \times 0.
25}{1} \times \frac{10^{-10}}{10^{-6}} \, \text{T} \] \( 2 \times 0.
25 = 0.
5 \).
So \( 2 \times 2 \times 0.
25 = 2 \times 0.
5 = 1 \).
\[ B = 1 \times 10^{-10 - (-6)} \, \text{T} = 1 \times 10^{-10+6} \, \text{T} = 1 \times 10^{-4} \, \text{T} \] We need the answer in microtesla (\( \mu\text{T} \)).
\( 1 \, \mu\text{T} = 10^{-6} \, \text{T} \).
\[ 10^{-4} \, \text{T} = 10^{-4} \times 10^6 \, \mu\text{T} = 10^2 \, \mu\text{T} = 100 \, \mu\text{T} \] This matches option (1).