Question

The figure below shows a spherical shell with uniform volume charge density ρ = 2.00 nC/m3, inner radius a = 5.0 cm, and outer radius b = 2.60a.

What is the magnitude of the electric field at the following radial distances:
(a) r = 0:


(b) r = a/2.00:


(c) r = a:


(d) r = 1.50a:


(e) r = b:


(f) r = 3.00b:

Answer 1

inner radius of the spherical shell a =5 cm =5*10^-2 m

outer radius of the spherical shell b=2.6a =13*10^-2 m

volume charge density ρ =2 nC/m^3 =2*10^-9 C/m^3

let the radius of the Gaussian surface is r.

at a ≤ r ≤ b, the enclosed charge q =(ρ)(4/3)(π)[(r^3-a^3)]   ...... (1)

therefore electric field is given by

E =k[q/r^2]   ......... (2)

where, k =1/4πε_0

eq (2) becomes

E =[ρ/3ε_0][(r^3-a^3)/r^2]     .......... (3)

...........................................................

a)

given r =0

so, the charge q lie on the surface of the shell is equal to zero.

then the electric field E =0 N/C

.............................................................

b)

given r =a/2

at this point the charge is also zero.

therefore electric field E =0 N/C

..........................................................

c)

similarly at r=a , the charge q =0 C

hence E =0 N/C

.....................................................

d) given r =1.5a

and permitivity of the free space ε_0 =8.85*10^-12 C^2/N.m^2

electric field E =[ρ/3ε_0][(r^3-a^3)/r^2]

E =3.97 N/C

...................................................

e)

given r =b =13*10^-2 m

electric field E =[ρ/3ε_0][(b^3-a^3)/r^2]

E =9.23 N/C

....................................................

f)

given r =3b

electric field E =[ρ/3ε_0][b^3-a^3)/(3b)^2]

E =1.026 N/C