A particle of charge q and mass m is projected from origin with an initial velocity v = ((v₀)/(√(2))x + (v₀)/(√(2))y). There exists a uniform magnetic field B = B₀z and a space varying electric field E = E₀ e-λ xx within the region 0 x L. After travelling a distance such that x -coordinate has changed from x = 0 to x = L, the change in the kinetic energy is ______. (a) (qE₀)/(λ)[1 - e-λ L] (b) ((v₀ qB₀)/(2λ))[2 - e-2λ L] (c) (qE₀)/(λ)[1 + e-λ L] (d) q((E₀ + v₀ B₀)/(λ))[1 - e-λ L/2]

Answer: (a). The magnetic force does no work, so only E changes the kinetic energy. Δ K = ∫₀L qE₀ e-λ x dx = (qE₀)/(λ)[1 - e-λ L] So the answer is (a).

A particle of charge q and mass m is projected from origin with an… | JEE Main 2026 (5 Apr, Shift 2)

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Moving Charge in a Magnetic Field

A particle of charge $q$ and mass $m$ is projected from origin with an initial velocity $\vec{v} = \left(\frac{v_0}{\sqrt{2}}\hat{x} + \frac{v_0}{\sqrt{2}}\hat{y}\right)$ . There exists a uniform magnetic field $\vec{B} = B_0\hat{z}$ and a space varying electric field $\vec{E} = E_0 e^{-\lambda x}\hat{x}$ within the region $0 \leqslant x \leqslant L$ . After travelling a distance such that $x$ -coordinate has changed from $x = 0$ to $x = L$ , the change in the kinetic energy is ______.

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Answer: (a)

The magnetic force does no work, so only $\vec{E}$ changes the kinetic energy.
$\Delta K = \int_0^L qE_0 e^{-\lambda x}\,dx = \dfrac{qE_0}{\lambda}\left[1 - e^{-\lambda L}\right]$
So the answer is (a).

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