Question

A charged particle moving with a velocity \(\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}\) in a magnetic field \(\vec{B}\) experiences a force \(\vec{F} = F_1\hat{i} + F_2\hat{j}\). Here \(v_1, v_2, F_1, F_2\) are all constants. Then \(\vec{B}\) can be:

• A. \(\vec{B} = B_1\hat{i} + B_2\hat{j}\) with \(\frac{v_1}{v_2} = \frac{B_1}{B_2}\),
• B. \(\vec{B} = B_1\hat{i} + B_2\hat{j} + B_3\hat{k}\) with \(\frac{v_1}{v_2} = \frac{B_1}{B_2}\),
• C. \(\vec{B} = B_3\hat{j}\) with \(B_1 = B_2 = 0\),
• D. \(\vec{B} = B_1\hat{j} + B_2\hat{k}\) with \(\vec{B} = B_1\hat{i} + B_2\hat{j} + B_3\hat{k}\) and \(\frac{B_1}{B_2} = \frac{v_1}{v_2}\).

Hint:

Use the Lorentz force law to relate the velocity and magnetic field components, ensuring the force components match.