Uniform magnetic fields of different strengths (B₁ and B₂), both… | JEE Main 2025 (7 Apr, Shift 1)

Q25JEE Main 2025 · 7 APR · SHIFT 1Difficult

Moving Charge in a Magnetic Field

Uniform magnetic fields of different strengths $(B_1$ 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 velocity $v$ 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?

(Consider the velocity of the particle to be normal to the magnetic field and $B_2 > B_1$ )

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Answer: (a)

Radii: $r_1 = \dfrac{mv}{qB_1}$ , $r_2 = \dfrac{mv}{qB_2}$ , with $r_2 < r_1$ since $B_2 > B_1$ .
In region 2 the particle makes a semicircle and returns to the interface shifted by $2r_2$ in one direction.
Both fields point the same way, so the sense of rotation is the same; in region 1 it makes a semicircle that shifts it by $2r_1$ in the opposite direction.
Net displacement $= 2(r_1 - r_2) = \dfrac{2mv}{qB_1}\left(1 - \dfrac{B_1}{B_2}\right)$
So the answer is (a).

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