## Charged Particle In Electric Field

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# Motion of Charged Particle in Electric Field (Transverse Electric Field)

Let the charged particle be moving with an initial uniform velocity u in the X-direction and the constant electric field be applied in the Y-direction. Then at time t = 0, we havev

_{x}= u, v

_{y}= v

_{z}= 0 and x = y = z = 0

There is no acceleration along X and Z-direction (ax = az = 0).

a

_{y}= qEy/m = qE/m … (1)

Here, m is the mass and q is the charges of the particle.

But the particle has got an initial velocity in the X-direction. Hence, it will continue to move in the X-direction with the same velocity.

. : v

_{x}= u, v

_{y}= qE/m.t, v

_{z}= 0 … (2)

Integrating these equations w.r.t. time t, we get displacements along the three axes,

x = ut, y = ½ qE/m t

^{2}and z = 0 … (3)

Eliminating t between the equation of x and y.

y = ½ qE/m . (x/u)

^{2}

Or x

^{2}= 2mu

^{2}/qE.y = Ky … (4)

Where K is a constant. This equation represents a parabola.

The transverse displacement suffered by the particle during passage through the plate of length l is

y1 = 1/2at

^{2}= ½ . qE/m . (l/u)

^{2}= qEt

^{2}/2mu

^{2}… (5)

When the particle leaves the field, it follows a straight ling path which is tangent to the parabola. Its direction of travel after emerging from the field is inclined to the original of travel (x-axis) by an angle θ give by

Tan θ = vy/vx = qE/mu .t = qEl/mu

^{2}(. : t = l/u) … (6)

This principle is used in cathode ray oscillographs and in television pictures tubes.

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