Question

A particle having charge \( 10^{-9} \, \text{C} \) moving in \( x \)-\( y \) plane in fields of \( 0.4 \hat{j} \, \text{N/C} \) and \( 4 \times 10^{-3} \hat{k} \, \text{T} \) experiences a force of \( \left( 4 \hat{i} + 2 \hat{j} \right) \times 10^{-10} \, \text{N} \). The velocity of the particle at that instant is _______ m/s.

• A. \( 50 \hat{i} + 50 \hat{j} \)
• B. \( 100 \hat{i} + 50 \hat{j} \)
• C. \( -50 \hat{i} + 100 \hat{j} \)
• D. \( 50 \hat{i} + 100 \hat{j} \)

Answer 1

The Correct Option is D

Step 1: Use the Lorentz force law.
The force \( F \) on a charged particle moving in an electric and magnetic field is given by the Lorentz force equation: \[ F = q(E + v \times B) \] where \( q = 10^{-9} \, \text{C} \) is the charge, \( E = 0.4 \hat{j} \, \text{N/C} \) is the electric field, and \( B = 4 \times 10^{-3} \hat{k} \, \text{T} \) is the magnetic field.
Step 2: Solve for the velocity.
The force experienced by the particle is: \[ F = (4 \hat{i} + 2 \hat{j}) \times 10^{-10} \, \text{N} \] We can equate the two expressions for force and solve for the velocity vector \( v \): \[ q(E + v \times B) = (4 \hat{i} + 2 \hat{j}) \times 10^{-10} \] By solving this vector equation, we find the velocity of the particle is \( v = 50 \hat{i} + 100 \hat{j} \) m/s.
Final Answer: \( 50 \hat{i} + 100 \hat{j} \, \text{m/s} \)