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 have
        v_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 t²     and     z = 0                    … (3)

Eliminating t between the equation of x and y.
            y = ½ qE/m . (x/u)²
Or            x² = 2mu²/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 = 1/2at² = ½ . qE/m . (l/u)² = qEt²/2mu²                             … (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/mu²     (. : t = l/u)                    … (6)

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



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