Question

A rectangular loop of sides 25 cm and 10 cm carrying a current of 10 A is placed with its longer side parallel to a long straight conductor 10 cm apart carrying current 25 A. The net force on the loop is:

• A. \( 6.25 \times 10^{-5} \, {N} \)
• B. \( 5.5 \times 10^{-5} \, {N} \)
• C. \( 3.75 \times 10^{-5} \, {N} \)
• D. \( 8.75 \times 10^{-11} \, {N} \)

Hint:

To calculate the force on a current-carrying wire near another current-carrying conductor, use the formula for the magnetic field around a long straight wire and then apply the force formula \( F = I L B \).

Answer 1

The Correct Option is A

We are given a rectangular loop with current \( I = 10 \, {A} \) and dimensions of \( 25 \, {cm} \) and \( 10 \, {cm} \), placed parallel to a long straight conductor carrying a current \( I_{{wire}} = 25 \, {A} \), with the loop placed at a distance of \( d = 10 \, {cm} \) from the wire. The formula for the magnetic force on a current-carrying conductor due to a magnetic field is given by: \[ F = I L B \] where \( L \) is the length of the conductor and \( B \) is the magnetic field produced by the wire. 
Step 1: The magnetic field at a distance \( r \) from a long straight conductor carrying a current \( I_{{wire}} \) is given by Ampere's law: \[ B = \frac{\mu_0 I_{{wire}}}{2 \pi r} \] where \( \mu_0 = 4\pi \times 10^{-7} \, {T} \cdot {m/A} \) is the permeability of free space. For the given problem, the distance from the wire is \( r = 10 \, {cm} = 0.1 \, {m} \), and the current in the wire is \( I_{{wire}} = 25 \, {A} \). Substituting the values into the formula for the magnetic field: \[ B = \frac{(4\pi \times 10^{-7}) \times 25}{2 \pi \times 0.1} = \frac{10^{-7} \times 25}{0.1} = 2.5 \times 10^{-6} \, {T}. \] 
Step 2: Now, the force on the side of the loop of length \( L = 0.25 \, {m} \) (longer side of the rectangle) carrying current \( I = 10 \, {A} \) is: \[ F = I L B = 10 \times 0.25 \times 2.5 \times 10^{-6} = 6.25 \times 10^{-5} \, {N}. \] Thus, the net force on the loop is \( 6.25 \times 10^{-5} \, {N} \).