Uniform magnetic fields of different strengths (B_1. and B_2), both...

Uniform magnetic fields of different strengths

\left(B_1\right.

and B_2), both normal to the plane of the paper exist as shown in the figure. A charged particle of mass m and charge q , at the interface at an instant, moves into the region 2 with velocityv

and returns to the interface. It continues to move into region 1 and finally reaches the interface. What is the displacement of the particle during this movement along the interface?<br/><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/GJUcZpO6YRncsChthnpFYEw-AUdwPrN0tPsqTp0K-is.original.fullsize.png"/><br/>(Consider the velocity of the particle to be normal to the magnetic field and

B_2 \gt B_1

)

Options

A $\frac{\mathrm{mv}}{\mathrm{qB}_1}\left(1-\frac{\mathrm{B}_2}{\mathrm{~B}_1}\right) \times 2$
B $\frac{\mathrm{mv}}{\mathrm{qB}_1}\left(1-\frac{\mathrm{B}_1}{\mathrm{~B}_2}\right)$
C $\frac{\mathrm{mv}}{\mathrm{qB}_1}\left(1-\frac{\mathrm{B}_2}{\mathrm{~B}_1}\right)$
D $\frac{\mathrm{mv}}{\mathrm{qB}_1}\left(1-\frac{\mathrm{B}_1}{\mathrm{~B}_2}\right) \times 2$

Correct answer

D. $\frac{\mathrm{mv}}{\mathrm{qB}_1}\left(1-\frac{\mathrm{B}_1}{\mathrm{~B}_2}\right) \times 2$

Step-by-step solution

\overrightarrow{\mathrm{v}}

\perp

\overrightarrow{\mathrm{B}}

, so charge particle will move in circular path, whose radius is given by

\mathrm{R}=\frac{\mathrm{mv}}{\mathrm{qB}}

Uniform magnetic fields of different strengths (B_1. and B_2), both normal to the plane of the paper exist as shown in

Starting point

\rightarrow \mathrm{A}

Ending point

\rightarrow \mathrm{C}

\therefore

Net displacement

=\mathrm{AC}

\begin{aligned} & \mathrm{AC}=\mathrm{CD}-\mathrm{AD} \\ & \mathrm{AC}=\frac{2 \mathrm{mv}}{\mathrm{qB}_1}-\frac{2 \mathrm{mv}}{\mathrm{qB}_2} \\ & \mathrm{AC}=\frac{2 \mathrm{mv}}{\mathrm{qB}_1}\left[1-\frac{\mathrm{B}_1}{\mathrm{~B}_2}\right]\end{aligned}

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