Question

Two parallel plates (2 cm by 2.5 cm) generate an electric field. An electron is fired with a speed of 4.4 x 107 m/s between the plates, perpendicular to the electric field direction. (See figure). The plates are charged to +/- 25 nC. a. Which plate must be positive, the top or the bottom plate? b. If the plates are 2 cm long in the direction the electron is travelling, the electron's trajectory when it leaves the plates will be at what angle relative to the horizontal?

Answer 1

Force on a charge is given by F = Eq where E is the electric field and q is the charge.

So, for an electron of charge - e , F = - Eq

Note the negative sign which means the force on an electron acts in the direction opposite to the direction of the electric field.

So, if the electron should deviate upwards, the electric field should point from the top plate to the bottom plate.

Since the electric field lines come OUT of positive charge and go INTO the negative charge, we conclude that the top plate is positively charged.

b.) Note that the horizontal velocity of the electron does not change throughout since there is no force in the horizontal direction.

The time taken by electron to cross the plates is t = horizontal distance / horizontal speed

t = 0.02 m / (4.4 x 10⁷ m/s ) = 4.545454545 x 10^-10 s

Electric field inside a parallel plate capacitor E = σ / ε

where ε is the permittivity and σ is the charge density

σ = charge / area = 25 x 10^-9 / (0.02 x 0.025) = 5 x 10^-5 C / m²

ε = 8.85418782 × 10^-12 m^-3 kg^-1 s⁴ A²

E = 5 x 10^-5 / 8.85418782 × 10^-12 = 5647045.332 N /C

Force F = Eq = - Ee = - 5647045.332 x 1.60217662 × 10^-19 = 9.047564003 x 10^-13 N

acceleration of the electron = F / m_e =  9.047564003 x 10^-13 / 9.10938356 × 10^-31 = 9.9321364 x 10¹⁷ m/s²

vertical velocity of the electron as it crosses the plates be v

v = at = 9.9321364 x 10¹⁷ x 4.545454545 x 10^-10 = 451460745.4 m/s

if θ is the angle made with the horizontal

tan θ = vertical velocity / horizontal velocity

tan θ = 451460745.4 / 4.4 x 10⁷ = 10.26047149

θ = tan^-1 10.26047149 = 84.43345325 degrees