Assume that a ball of charged particles has a uniformly distributed negative cha

Question

Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the tunnel or outside the ball. Let FR be the magnitude of the electrostatic force on the proton when it is located at the ball's surface, at radius R. As a multiple of R, how far from the surface is there a point where the force magnitude is 0.57FR if we move the proton (a) away from the ball and (b) into the tunnel?

Explanation / Answer

let Q be the negative charge on ball

on the surface

Electric field E1 = k*Q/R^2

electric force FR = k*Q*q/R^2

at a distance r away from surface

Electric field E2 = k*Q/r^2

electric force E2 = k*Q*q/r^2 = 0.57*FR

k*Q*q/r^2 = 0.57*k*Q*q/R^2

r = R/sqrt0.57 = 1.32R

R + x = 1.32R

x = 0.32 R <<<<-------ANSWER

-------------------------

Inside the ball

ELEtric field E3 = k*Q*r/R^3 = 0.57*FR

k*Q*r/R^3 = 0.57*k*Q*/R^2

r = 0.57*R

R-x = 0.57R

x = 0.43R <<<<-------ANSWER

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