Positive charge Q is distributed uniformly over each of two

Positive charge Q is distributed uniformly over each of two spherical volumes with radius R. One sphere of charge is centered at the origin and the other at x = 2R. Find the magnitude and direction of the net electric field due to these two distributions of charge at the following points on the x-axis. (Use any variable or symbol stated above along with the following as necessary: ₀.)

(a) x=0

(b) x= R/2

(c) x=R

(d) x=3R

Solution

(A)

at x = 0, the electric field is,

      E = E1 + E2

         = kQr/R^3 + kQ/r^2

         = kQ(0)/R^3 + kQ/(2R)^2

         = 0 + kQ/4R^2                         [since, k = 1/4*pi*e₀]

         = Q / 16*pi*e₀*R^2, along the x-axis

(b)

at x = R/2, the electric field is,

    E = E1-E2

       = kQr/R^3 - kQ/r^2

      = kQ(R/2)/R^3 - kQ/(3R/2)^2

      = kQ/18R^2

      = Q/ 72*pi*e⁰*R^2

(c)

at x = R, the electric field is,

    E = E1-E2 = E1-E1 = 0

(d)

at x = 3R, the electric field is,

E = E1 + E2

         = kQ/r^2 + kQ/r^2

         = kQ/(3R)^2 + kQ/R^2

         = (kQ/R^2)[(1/9) + 1]

         = [10/9]kQ/R^2

         = [10/9]Q/4*pi*e₀*R^2

         = 5Q/ 18*pi*e₀*R^2    , in +x-axis