Part A A charge Q = -610 nC is uniformly distributed on a ring of 2.4-m radius.

Question

Part A A charge Q = -610 nC is uniformly distributed on a ring of 2.4-m radius. A point charge q = +480 nC is fixed at the center of the ring, as shown in the figure. An electron is projected from infinity toward the ring along the axis of the ring. This electron comes to a momentary halt at a point on the axis that is 5.0 m from the center of the ring. What is the initial speed of the electron at infinity? (e = 1.60 x 10^-19 C, k = 1/4pi epsilon 0 = 8.99 x 10^9 N. m^2/c^2, m el = 9.11 x 10^31 kg) q = +480 nC Q = -610 nC 2.2 x 10^6 m/s 6.6 x 10^6 m/s 1.1 x 10^6 m/s 4.5 x 10^6 m/s 3.4 x 10^6 m/s

Explanation / Answer

The electric potential is not necessary to solve this problem.
Let k (Coulomb's constant) = 9*10^9
r = 2.4 [m]
d = 5 [m]
e (charge on electron) = -1.6*10^-19
m of electron = 9.1*10^-31 [kg]
Conserving the energy of the system ->
E_initial = E_final
U_electrical_i + K_mechanical_i = U_electrical_f + K_mechanical_f
At infinity, electric potential energy is 0, at final position, velocity is 0 (therefore K = 0)
At final position, there is electric potential energy due to the ring and due to the point charge; the potential due to the ring is positive and the potential due to the charge is negative because an electron is negatively charged.
The distance from the ring to the electron is equal at points and is given by ?d^2 + r^2
This gives
0.5*m*u^2 = | ( k*e*Q / ?d^2+r^2 ) - ( k*q*e / d^2 ) |
Solving for u
u = 6.649 [Mm/s] or 6.649*10^6 [m/s] <-------- ANS

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