Question

Q3 q1 q2 [In all questions, if a variable isn't needed, you may omit it. 1. What is the electric potential, at the point where qi is located, due to Qs and q2? 2. What is the electric field, at the point where qi is located, due to Qs and q2? Provide magnitude and direction. 3. In terms of qi, r, and R, what is the direction and magnitude of the electrostatic force on qı? Assume q2-2qi, but Q3 -3qi is negative. 4. In terms of q1, r, and R, what is the total electrostatic flux through a Gaussian surface, just outside the sphere in this example? 5. In terms of q1, r, and R, what is the electric potential on the left edge of the surface (Radius R)

Answer 1

1. The potential at q1 due to charge Q3 is

= $\frac{Q_3}{4\pi \epsilon _oR}$ , It is the potential at the center of uniformly charge sphere of radius R. we assume it is a thin shell as inner radius is not given.

The potential anywhere inside the sphere is same as the electric field is 0 inside the sphere due to Q3 as we show next

potential at q1 due to charge q2

= $\frac{Q_2}{4\pi \epsilon _or}$

Total potential at q1 due to q2 and q3

$\frac{1}{4\pi \epsilon _or}(q_3/R + q2/r)$

2. construct a spherical Gaussian surface of radius r concentric with q 2 and passing through q1. The field every where due to the charge enclosed inside q2 is uniform

E = $\frac{q_2}{4\pi \epsilon _or^2}$

field at q1 due to q3 is 0 as the source is not enclosed inside the surface.

3. q2 = 2 q1

Electric field at q1 = E

force on q1 = Eq1 =  $\frac{q_1^2}{2\pi \epsilon _or^2}$ , the direction of the force is away from the center (q2) as both charges are of same sign

4. Total charge enclosed inside the Gaussian surface q= q1 +2q1 -3q1 = 0

flux outside the Gaussian surface = q/$\epsilon _o$ =0

5.

Potential from q3.

potential any where outside the sphere is same as if the total charge is concentrated at the center

= q3/4$\pi \epsilon _o$R = -3q₁ / 4$\pi \epsilon _o$R

potential due to q2 = q₂ /24$\pi \epsilon _o$R = 2q₁/4$\pi \epsilon _o$R

potential due to q1 = q₁/4$\pi \epsilon _o$(R -r)

total potential at the left edge outside the sphere

= -3q₁ / 4$\pi \epsilon _o$R + 2q₁ / 4$\pi \epsilon _o$R + q₁ / 4$\pi \epsilon _o$(R -r)

= q₁/ 4$\pi \epsilon _o$ ( -1/R + 1/R-r )

= q₁r / 4$\pi \epsilon _o$(R -r)