Question

A particle with charge \( q \) moving with velocity \( \vec{v} = v_0 \hat{i} \) enters a region with magnetic field \( \vec{B} = B_1 \hat{j} + B_2 \hat{k} \). The magnitude of force experienced by the particle is:

• A. \( q v_0 (B_1 + B_2) \)
• B. \( q \sqrt{v_0 (B_1 + B_2)} \)
• C. \( q v_0 \sqrt{B_1^2 + B_2^2} \)
• D. \( q \sqrt{v_0 (B_1^2 + B_2^2)} \)

Hint:

When dealing with a cross product, the magnitude of the resulting vector is found by taking the product of the magnitudes of the individual vectors and the sine of the angle between them. In this case, the cross product gives a result involving the square root of the sum of the squares of the magnetic field components.

Answer 1

The Correct Option is C

The force on a moving charged particle in a magnetic field is given by the equation: \[ \vec{F} = q (\vec{v} \times \vec{B}) \] where: - \( \vec{v} \) is the velocity vector - \( \vec{B} \) is the magnetic field vector - \( q \) is the charge of the particle Given that \( \vec{v} = v_0 \hat{i} \) and \( \vec{B} = B_1 \hat{j} + B_2 \hat{k} \), the magnitude of the cross product is: \[ |\vec{F}| = q v_0 \sqrt{B_1^2 + B_2^2} \] Thus, the force experienced by the particle is \( q v_0 \sqrt{B_1^2 + B_2^2} \).