A solid metal sphere of radius a is charged to a potential V(initial)>0 and then

Question

A solid metal sphere of radius a is charged to a potential V(initial)>0 and then isolated from the charging source. It is then surrounded by joining two uncharged metal hemispherical shells of inner radius b and outer radius 2b, as shown above, without touching the inner sphere or any source of charge.
a. In terms of the given quantities and fundamental constant, determine the initial charge Q(initial) on the solid sphere before it was surrounded by the outer shell.
b. Indicate the induced charge on the following after the outer shell is in place
i. The inner surface of the shell
ii. The outer surface of the shell
c. Indicate the magnitude of the electric field as a function of r and the direction ( if any ) of the field for the regions indicated below. Write your answer on the appropriate lines.
i. r < a magnitude_________ Direction____________
ii. a <r < b magnitude_______ Direction____________
iii. b < r < 2b magnitude________ Direction__________
iv. 2b < r magnitude__________ Direction__________
d. Does the inner sphere exert a force on the uncharged hemisphere while the shell is being assembled ? why or why not ?

Explanation / Answer

a.the initial charge is Q=V(initial) X 4 X pi X o X a.
.this is because the potential on the surface of a charged solid metal sphere is given by

V=Q / (4 X pi X 0 X a)

b.i.The inner surface of the shell has charge q1=(-1)*Q.

b.ii.The outer surface of the shell has charge q2=Q.

c.i. magnitude = 0. direction = none.

c.ii. magnitude = Q/ (4 X pi X 0 X r2 ) , direction : radially away from the surface of solid sphere

c.iii. magnitude = 0 , direction = none .

c. iv. magnitude = Q/ (4 X pi X 0 X r2 ) , direction = radially outward from the surface of the shell.

d. While the shell is being assembled , the charged sphere applies an attractive force on the uncharged hemispheres.It is because of the fact that the +vely charged sphere induces a negative charge on the inside of the hollow hemispheres , and we know that unlike charges attract each other, hence there is an attractive force between the hemisphers and the solid sphere.

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