First slide

Electric field

Question

An electron is projected from a distance d and with initial velocity u parallel to a uniformly charged flat conducting plate as shown. It strikes the plate after travelling a distance $l$ along the direction of projection. The surface charge density $σ$ of the conducting plate is.

Moderate

A

$σ = \frac{{dε}_{0} {mu}^{2}}{eℓ}$
B

$σ = \frac{2 {dε}_{0} {mu}^{2}}{{eℓ}^{2}}$
C

$σ = \frac{{dε}_{0} mu}{eℓ}$
D

$σ = \frac{2 {dε}_{0} mu}{eℓ}$

Solution

For the motion of the electron along Y-axis
$d = \frac{1}{2} \frac{eE}{m} t^{2}$ ………………….(1)
where E is the electric field due to the charged plate given by
$E = \frac{σ}{ε_{0}}$ ………………………….(2)
where $σ$ is the surface density of charge of the plate.
The time taken by the electron is
$t = \frac{ℓ}{u}$ …………………………..(3)
From equations (1), (2) and (3), we obtain
$\begin{array}{l} d = \frac{1}{2} \frac{eσ}{ε_{0} m} {(\frac{ℓ}{u})}^{2} \\ \Rightarrow σ = \frac{2 {dε}_{0} {mu}^{2}}{{eℓ}^{2}} \end{array}$

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