Question

A particle of charge \( 2 C \) is moving with a velocity \( (3\hat{i} + 4\hat{j}) \) m/s in the presence of magnetic and electric fields. If the magnetic field is \( ( \hat{i} + 2\hat{j} + 3\hat{k} ) \) T and the electric field is \( (-2\hat{k}) \) NC\(^{-1}\), then the Lorentz force on the particle is:

• A. \( 50 N \)
• B. \( 20 N \)
• C. \( 30 N \)
• D. \( 40 N \)

Hint:

Lorentz force accounts for both electric and magnetic effects on a charged particle, calculated as \( q(\vec{E} + \vec{v} \times \vec{B}) \).

Answer 1

The Correct Option is C

The Lorentz force is given by: \[ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}) \] First, calculate the cross product \( \vec{v} \times \vec{B} \): \[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
3 & 4 & 0
1 & 2 & 3 \end{vmatrix} \] \[ = \hat{i} (4 \times 3 - 0 \times 2) - \hat{j} (3 \times 3 - 0 \times 1) + \hat{k} (3 \times 2 - 4 \times 1) \] \[ = \hat{i} (12) - \hat{j} (9) + \hat{k} (6 - 4) \] \[ = 12\hat{i} - 9\hat{j} + 2\hat{k} \] Now, adding \( \vec{E} \): \[ \vec{E} + \vec{v} \times \vec{B} = (12\hat{i} - 9\hat{j} + 2\hat{k}) + (-2\hat{k}) \] \[ = 12\hat{i} - 9\hat{j} + 0\hat{k} \] Multiplying by \( q = 2 C \): \[ \vec{F} = 2 (12\hat{i} - 9\hat{j} + 0\hat{k}) = 24\hat{i} - 18\hat{j} \] Magnitude: \[ |\vec{F}| = \sqrt{(24)^2 + (-18)^2} \] \[ = \sqrt{576 + 324} = \sqrt{900} = 30 N \]