A charged particle with velocity $\hat{v} = x \hat{i} + y \hat{j}$ moves in a magnetic field $\vec{B} = y \hat{i} + x \hat{j}$ . The force acting on the particle has magnitude F. Which one of the following statements is/ are correct?

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No force will act on charged particle if x = y

lf $x > y, F \propto (x^{2} - y^{2})$

lf $x > y$ and the charge is positive, the force will act along z-axis

lf $y > x$ the force will act along y-axis

answer is A.

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Detailed Solution

If $x = y, then \vec{v} ∥ \vec{B}, i.e., {\vec{F}}_{mag} = 0$

Hence, (a) is correct.

$\begin{array}{l} {\vec{F}}_{mag} = q (\vec{v} \times \vec{B}) = q [(x \hat{i} + y \hat{j}) \times (y \hat{i} + x \hat{j})] \\ {\vec{F}}_{mag} = q (x^{2} - y^{2}) \hat{k} \end{array}$

If x > y, then the force is along z-axis.

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