An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density $+\sigma$. The electron at $\mathrm{t}=0$ is at a distance of 1 m from S and has a speed of $1 \mathrm{~m/s}$. The maximum value of $\sigma$ if the electron strikes S at $\mathrm{t}=1 \mathrm{~s}$ is $\alpha\left[\frac{\mathrm{m} \epsilon_{0}}{\mathrm{e}}\right] \frac{\mathrm{C}}{\mathrm{m}^{2}}$, the value of $\alpha$ is $\_\_\_\_$.

JEE Main · 2024 · 29 Jan 2024 · Shift 1 · Electrostatics

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet S having surface charge density

+ σ

. The electron at

t = 0

is at a distance of 1 m from S and has a speed of

1 m/s

. The maximum value of

σ

if the electron strikes S at

t = 1 s

α [\frac{m ϵ _{0}}{e}] \frac{C}{m ^{2}}

, the value of

α

____

Answer: 8

Field of an infinite charged sheet:

E = \frac{σ}{2 ε _{0}}

, so the electron (charge

- e

) is pulled toward the sheet with acceleration

a = \frac{e σ}{2 ε _{0} m}

.

Take the direction away from the sheet as positive. The electron starts

1 m

away moving at

u = + 1 m/s

and returns to the sheet (net displacement

- 1 m

) at

t = 1 s

Δ x = u t - \frac{1}{2} a t^{2} ⟹ - 1 = (1) (1) - \frac{1}{2} (\frac{e σ}{2 ε _{0} m}) (1)^{2}

\frac{e σ}{4 ε _{0} m} = 2 ⟹ σ = 8 (\frac{m ε _{0}}{e})

α = 8

.

Key confirmed: 8.

JEE Main · 2024 · 29 Jan 2024 · Shift 1 · Integer answer