A small conducting spherical shell with inner radius a and outer radius b is con
ID: 1406907 • Letter: A
Question
A small conducting spherical shell with inner radius a and outer radiusb is concentric with a larger conducting spherical shell with inner radius c and outer radius d. The inner shell has a total charge of -2qand the outer shell has a total charge of +4q.
Select True or False for the following statements.
True/False? The total charge on the outer surface of the large shell is +2q.
True/False? The electric field in the region r > d is zero.
True/False? The total charge on the inner surface of the large shell is -6q.
True/False? The electric field in the region c < r < d is zero.
True/False? The radial component of the electric field in the region r < a is given by +2q/(4??0r2).
True/False? The total charge on the inner surface of the small shell is zero.
True/False? The total charge on the outer surface of the small shell is -2q.
Explanation / Answer
Given that,
A small conducting spherical shell with inner radius a and outer radiusb is concentric with a larger conducting spherical shell with inner radius c and outer radius d. The inner shell has a total charge of -2q and the outer shell has a total charge of +4q.
so if we put a spherical surface inside the inner surface of the sphere we will get 0 flux so that means there is 0 CHARGE INSIDE THE FIRST sphere which is obvious..
if we put a spherical surface between the inner surface and the outer surface of the inner sphere we must have a flux of 0 because its a conducting surface... and these are static charges... so again the inner surface must have a charge of 0
True , The total charge on the outer surface of the large shell is +2q because the inner surface of the inner sphere has a charge of 0 that means the outer surface of the inner sphere must have a charge of + 2q because the sphere itself must have a charge of +2q
now we have the inner sphere complete
The total charge on the inner surface of the large shell is -6q.
FALSE because the inner sphere has a charge of +2q we can have the surface anywhere between the two surfaces and get a flux related to +2q this is because the inner sphere will be inside any surface that is generated between the inner sphere and the outer sphere..
we can put a surface between the inner surface of the outer sphere and the outer surface of the sphere and the flux must be 0 due to its inside the conducting sphere with static charges..
this means that since we KNOW we have +2q charges from the first sphere in order to have 0 flux we must have 0 charges total inside the gausian sphere we made..
Qnet = +2q + Qinnersurface
since Qnet = 0
0 = +2q + Qinnersurface
Qinnersurface = -2q
so this shows that the inner surface of the outer sphere must be a charge of -2q
so we have a charge of +2q on the inner spheres outer surface and -2q on the outer spheres inner surface..
this adds up to 0 total and this makes 0 flux inside the outer sphere
===============
final part we put a gaussian sphere outside the outer sphere..
since the net charges = +2q and +4q were given that means the total net charge inside the system is +6q
this means we need a net charge on the outersphere of +4q but we have a charge of -2q on the inner surface of it ...
mathematically
Qnet = +4q
Qinner = -2q
Qouter = ?
Qnet = Qinner + Qouter
+4q = -2q + Qouter
Qouter = +4q +2q
= +6q
so the total charge on the outersurface of the outer sphere is +6q
that creates the necessary +4q charge on the outer sphere
that makes the necessary charge of +6 q for the total system of the two spheres.
and that is the semi easy but can be complicated way to find out what the charges are ..
to summarize
Qa = 0 inner side
Qb = +2q outer side
Qc = -2q innerside
Qd = +6q outer side
Hence by this,
The electric field in the region r > d is zero. FALSE
The total charge on the outer surface of the small shell is -2q. TRUE
The electric field in the region c < r < d is zero. TRUE
The radial component of the electric field in the region r < a is given by -2q/(40r2). FALSE
The total charge on the inner surface of the large shell is +2q. TRUE
The total charge on the outer surface of the large shell is +2q. TRUE
The total charge on the inner surface of the small shell is -2q. FALSE