A particle of charge and mass is projected from origin with an initial velocity There exists a uniform magnetic field and a space varying

Question

A particle of charge $q$ and mass $m$ is projected from origin with an initial velocity $\vec{v} = (\frac{v_{0}}{\sqrt{2}} \hat{x} + \frac{v_{0}}{\sqrt{2}} \hat{y}) .$ There exists a uniform magnetic field $\vec{B} = B_{0} \hat{z}$ and a space varying electric field $\vec{E} = E_{0} e^{- λ x} \hat{x}$ within the region $0 \leq x \leq L$ . After travelling a distance such that $x$ -coordinate has changed from $x = 0$ to $x = L$ , the change in the kinetic energy is ________. [2026]

Smart Achievers · Accepted Answer

(1)

$Δ K = W_{E} + W_{M} = W_{E} + 0$

$W_{E} = \int_{0}^{L} q E d x$

$W_{E} = \int_{0}^{L} q E_{0} e^{- λ x} d x = q E_{0} {[\frac{e^{- λ x}}{- λ}]}_{0}^{L}$

$\Rightarrow W_{E} = \frac{q E_{0}}{λ} [1 - e^{- λ L}]$

$\Rightarrow Δ K_{E} = \frac{q E_{0}}{λ} (1 - e^{- λ L})$

Solution

1 $\frac{q E_{0}}{λ} [1 - e^{- λ L}]$

2 $(\frac{v_{0} q B_{0}}{2 λ}) [2 - e^{- 2 λ L}]$

3 $\frac{q E_{0}}{λ} [1 + e^{- λ L}]$

4 $q [\frac{E_{0} + v_{0} B_{0}}{λ}] [1 - e^{- λ L / 2}]$

Ans.
(1)

$Δ K = W_{E} + W_{M} = W_{E} + 0$

$W_{E} = \int_{0}^{L} q E d x$

$W_{E} = \int_{0}^{L} q E_{0} e^{- λ x} d x = q E_{0} {[\frac{e^{- λ x}}{- λ}]}_{0}^{L}$

$\Rightarrow W_{E} = \frac{q E_{0}}{λ} [1 - e^{- λ L}]$

$\Rightarrow Δ K_{E} = \frac{q E_{0}}{λ} (1 - e^{- λ L})$

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Solution

1 qE0λ[1-e-λL]

2 (v0qB02λ)[2-e-2λL]

3 qE0λ[1+e-λL]

4 q[E0+v0B0λ][1-e-λL/2]