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Question

A particle of mass m and charge q is projected from the origin with a horizontal speed v into a space having electric field of magnitude E directed downwards as shown in the figure. If the effect due to gravity can be neglected, then choose the correct statement(s) among the following.


A
The equation of trajectory is y=12(qE2xmv2)
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B
The kinetic energy after a displacement y is qEy
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C
The horizontal and vertical components of acceleration are ax=0, ay=qEm
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D
The horizontal and vertical displacements x and y after a time t are x=vt and y=12ayt2
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Solution

The correct option is D The horizontal and vertical displacements x and y after a time t are x=vt and y=12ayt2
Since, there is no force acting on the mass in the x direction, Fx=0

ax=0

Now, since the electric field acts downwards, the force acting on the particle in the y direction, Fy=qE

Using Fy=may

ay=qEm

So, option (b) is the correct answer.

Now considering the second equation of motion, x=uxt+12axt2

ux=v and ax=0

x=vt .....(1)
Similarily, for y=uyt+12ayt2

uy=0 and ay=qEm

y=12ayt2=12(qEm)t2 .....(2)

So, option (d) is the correct answer.

We, have from (1), t=xv

Substituting t=xv in (2), we get

y=12(qEx2mv2)

So, option (c) is the correct answer.

Now, kinetic energy KE is,

KE=12m(v2x+v2y)

where, vx=v and vy=ayt=qEmt

KE=12m(v2+q2E2t2m2)

So, option (a) is the incorrect answer.

Hence, options (b) , (c) and (d) are the correct answers.

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