Consider a charged, hollow metal sphere. Hovering inside at the center is charge

Question

Consider a charged, hollow metal sphere. Hovering inside at the center is charged solid metal sphere. The image below is a cross section of this structure, where the dark parts are metal. The inner sphere has total charge Q_3 and the outer sphere has a total charge Q_3. a. Using Gauss's law and your knowledge of conductors, find an expression of the electric field in terms of the distance from the center of the spheres r, for each of the following locations: i. inside the inner sphere ii. between the two spheres. iii. inside the metal of the outer sphere, and iv. outside the entire structure b. If the inner has a total charge of +30 nC and a radius of 1.5 cm and the outer sphere has a total charge of +15 nC and a radius of 3.5 cm, what is the magnitude of the electric field at 2.5 cm from the center of the structure?

Explanation / Answer

2.a)

i) Since the smaller sphere is a metal, total charge on it Q1 will be resides on its surface. Hence no charge available inside.

Thus Qinside = 0 N

By Guass’s law,

inside = Einside*A =   Qinside/0

Einside =   Qinside/A0 = 0 N/C

ii) Between two spheres, Qbet = Q1

Ebet =   Qbet/A0 = Q1/A0 = Q1/4r20

iii) Inside the metal of outer sphere,

charge on the inner surface of outer sphere = - Q1

Net charge on the inside of outer sphere = Q1   - Q1 = 0 N/C

Ebet =   Qinside/A0 = Qinside/A0 = 0 N/C

iv) Outside the structure,

Qout = Q1 + Q2 =

Eout =   Qout/A0 = (Q1 + Q2)/40r2

b) from iii,

Inside the metal of outer sphere,

charge on the inner surface of outer sphere = - Q1

Net charge on the inside of outer sphere = Q1   - Q1 = 0 N/C

Ebet =   Qinside/A0 = Qinside/A0 = 0 N/C

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