Question

In the figure particles 1 and 2 are fixed in place on an x axis, at a separation of L = 6.50 cm. Their charges are q1 = +e and q2 = -64e. Particle 3 with charge q3 = +5e is to be placed on the line between particles 1 and 2, so that they produce a net electrostatic force F3 net on it. (a) At what coordinate should particle 3 be placed to minimize the magnitude of that force? (b) What is that minimum magnitude?

Answer 1

Let, the third charge is placed at x distance from charge q1.

Force on q3 due to q1,

F31 = k*5e*e / x^2

F31 = 5ke^2 / x^2

Force on q3 due to q2,

F32 = k*5e*64e / (L - x)^2

F32 = 320ke^2 / (L - x)^2

Net force = F31 + F32 = 0

Fnet = ke^2 * [(5 / x^2) + (320 / (L-x)^2)]

For minimize the magnitude of that force on q3,

d(Fnet) / dx = 0

d(Fnet) / dx = ke^2 * [(-10 / x^3) + (640 / (L - x)^3)] = 0

1 / x^3 = 64 / (L - x)^3

1 / x = 4 / (L - x)

x = L / 5 = 6.50 / 5

x = 1.3 cm

coordinate for minimize the magnitude of force on q3= (1.3 , 0) cm

(b)

Fmin =  ke^2 * [(5 / x^2) + (320 / (L-x)^2)]

Fmin = 9*10^9*(1.6*10^(-19))^2 * [5 / (0.013)^2 + 320 / (0.052)]

Fmin = 3.41*10^(-23) N