Question

Force on a charge \( q = 10^{-9} \, \text{C} \) in uniform electric and magnetic fields is \( \vec{F} = (2 \times 10^{-10} \hat{i} + 3 \times 10^{-10} \hat{j}) \, \text{N} \).

Find the velocity (in m/s) of the charge if the electric field is \( 0.4 \hat{j} \, \text{V/m} \) and the magnetic field is \( 2 \times 10^{-3} \hat{k} \, \text{T} \).

• A. \( \vec{V} = 50 \hat{i} + 100 \hat{j} \, \text{m/s} \)
• B. \( \vec{V} = 100 \hat{i} + 100 \hat{j} \, \text{m/s} \)
• C. \( \vec{V} = 50 \hat{i} + 50 \hat{j} \, \text{m/s} \)
• D. \( \vec{V} = 50 \hat{i} + 10 \hat{j} \, \text{m/s} \)

Hint:

The Lorentz force equation can be used to find the velocity of a charge in the presence of both electric and magnetic fields. The cross product \( \vec{v} \times \vec{B} \) is crucial in calculating the force due to the magnetic field.

Answer 1

The Correct Option is A

Step 1: Understanding the Lorentz Force.
The Lorentz force on a charged particle is given by:
\[ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}) \] Where:
- \( q = 10^{-9} \, \text{C} \) (charge),
- \( \vec{E} = 0.4 \hat{j} \, \text{V/m} \) (electric field),
- \( \vec{B} = 2 \times 10^{-3} \hat{k} \, \text{T} \) (magnetic field),
- \( \vec{v} \) is the velocity of the charge.

Step 2: Applying the formula.
The total force is given by:
\[ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}) \] Substituting the known values: \[ (2 \times 10^{-10} \hat{i} + 3 \times 10^{-10} \hat{j}) = (10^{-9}) \left( 0.4 \hat{j} + \vec{v} \times 2 \times 10^{-3} \hat{k} \right) \]
Step 3: Finding the velocity.
Now we solve for \( \vec{v} \). Using the cross product \( \vec{v} \times \vec{B} \), we get the components of velocity. After solving the equations, we get:
\[ \vec{v} = 50 \hat{i} + 100 \hat{j} \, \text{m/s} \]
Step 4: Conclusion.
Therefore, the velocity of the charge is \( \vec{V} = 50 \hat{i} + 100 \hat{j} \, \text{m/s} \).
Final Answer: \( \vec{V} = 50 \hat{i} + 100 \hat{j} \, \text{m/s} \)