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Question

A particle of mass $$m$$ and negative charge $$q$$ is thrown in a gravity free space with speed $$u$$ from the point $$A$$ in the large nonconducting charged sheet with surface charge density $$\sigma $$, as shown in figure. The maximum distance from $$A$$ on sheet where the particle can strike is $$\cfrac { x{ \epsilon}_{ 0 }{ u }^{ 2 }m }{ q\sigma  } $$. Find $$x$$.

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Solution

Electric potential due to large non conducting sheet = $$\dfrac {\sigma} {2 \epsilon_{o}}$$

Force on charge = qE = $$\dfrac {\sigma q} {2 \epsilon_{o}}$$ towards sheet
acceleration, a = F/m = $$\dfrac {\sigma q} {2m \epsilon_{o}}$$

initial velocity perpendicular to sheet = $$u \sin\theta$$
velocity parallel to sheet = $$u \cos \theta$$

Time of flight = $$\dfrac {2 u \sin \theta}{a}$$
Distance traveled parallel to sheet = $$\dfrac {2 u \sin \theta u \cos \theta }{a}$$ = $$\dfrac {u^{2} \sin (2\theta)}{a}$$
So distance travelled will be maximum for $$\theta =45^\circ$$ and equal to
$$\dfrac{u^{2} 2 m \epsilon_{o}}{\sigma q}$$ =$$\dfrac{2u^{2}  m \epsilon_{o}}{\sigma q}$$ 

therefore x = 2

Physics

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