Question

A large sphere with radius R, supported near the earth's surface as shown has charge density p(r) that varies as r^n (where n is 0,1,2..) for 0<r<R and reaches a max value of p as you get to r=R. a non conducting uncharged string of length L with a second tiny sphere of radius b, mass m, and excess charge q is suspended from the large sphere as shown. suppose the string is cut gently without otherwise disturbing the setup and the ball begins to move. Derive a formula for the instantaneous acceleration of the small ball just after the string is cut.

Answer 1

given large sphere radius R
charge density rho(r)
rho(r) = rhoo*r^n
at r = R
rhop = rhoo*R^n

rho(r) = rhop*(r/R)^n
string length = L
mass m, radius b, charge q

now, assuming the ball to be belolw the sphere

electric field due to the sphere at the location of the ball = E
E = kQ/(L + R)^2
now
dQ = rho(r)*4*pi*r^2*dr
dQ = rhop*4*pi*r^(n + 2)*dr/R^n

for n + 2 = -1
n = -3, rho is not defined at the center
hence
for n ! = -3
Q = rhop*4*pi*R^(n + 3)/(n + 3)R^n = 4*pi*rhop*R^3/(n + 3)

hence
E = 4*pi*rhop*R^3/4*pi*epsilon*(n + 3)(L + R)^2
E = rhop*R^3/epsilon*(n + 3)(L + R)^2

hence
initial acceleration = (qE + mg)/m = q*rhop*R^3/epsilon*(n + 3)(L + R)^2*m + g ( downwards)

for spehre above sphere
initial acceleration = (qE - mg)/m = q*rhop*R^3/epsilon*(n + 3)(L + R)^2*m - g ( upwards)