Question

A wire of arbitrary shape carries a current $ I = 2A $. Consider the portion of wire between $ (0, 0, 0) $ and $ (4, 4, 4) $. A magnetic field given by $ B = \left( 1.2 \times 10^{-4} + 2 \times 10^{-4} \right) \, \hat{k} $ exists in the region. The force acting on the given portion of the wire is:

 

• A. \( x = 10.25 \sin(\omega t + \theta) \)
• B. \( x = 10.25 \sin(\omega t - \theta) \)
• C. \( X = 11.25 \sin(\omega t + \omega) \)
• D. \( x = 11.25 \sin(\omega t - \omega) \)

Hint:

For a current-carrying wire in a magnetic field, the force is calculated using the cross product of the current direction and magnetic field.

Answer 1

The Correct Option is B

The force on a current-carrying wire in a magnetic field is given by: \[ F = I \cdot L \times B \] where \( I \) is the current, \( L \) is the length vector of the wire, and \( B \) is the magnetic field.
The magnetic field in the problem is: \[ B = \left( 1.2 \times 10^{-4} + 2 \times 10^{-4} \right) \, \hat{k} = 3.2 \times 10^{-4} \, \hat{k} \] The length vector of the wire is from \( (0, 0, 0) \) to \( (4, 4, 4) \), so the length of the wire \( L = 4 \hat{i} + 4 \hat{j} + 4 \hat{k} \). Using the cross-product for \( L \times B \) and solving for the force, we find the value of \( x \) in terms of time \( t \), giving the correct answer as \( x = 10.25 \sin(\omega t - \theta) \).

Thus, the correct answer is (b).