Question

1a) If a capacitor were made from two coaxial, concentric spheres with the inner sphere having a radius of r, and the outer sphere having a radius of 1.5r, what would be the capacitance of this setup? Remember, capacitance is defined as C = Q/V . You may use r=1.5cm

1b) What would your answer to the previous question be if the space between the spheres were filled with styrofoam, κ = 4 ?

Answer 1

1a)

Let the charge on the inner sphere be Q, then the field at a distance r from the center of the sphere is

ε₀*4*π*r²*E = Q {from Gauss law}

Or, E = Q/(ε₀*4*π*r²)

Now, The potential difference between the spheres is found by integrating E from r to 1.5r

So, V= ∫E*dr [from r to 1.5r] =  ∫Q/(ε₀*4*π*r²)dr [from r to 1.5r] = (1/ε₀*4*π) * Q * ∫ (1/r²)dr [from r to 1.5r]

So, V= (1/ε₀*4*π) * Q ( 1/r - 1/1.5r )

So, C= Q/V= ε₀*4*π*r*1.5r/(1.5r-r) = ε₀*4*π*r*1.5r/0.5r = ε₀*4*π*3r = 12ε₀*π*r

When r= 1.5cm= 0.015m, then

C= 12ε₀*π*0.015= 5*10^-12 F

1b)

When k= 4, E= Q/(ε₀*4*π*r²) / 4 = Q/(ε₀*16*π*r²)

Also, V= (1/ε₀*16*π) * Q ( 1/r - 1/1.5r )

So, C= Q/V= ε₀*16*π*r*1.5r/(1.5r-r) = ε₀*16*π*r*1.5r/0.5r = ε₀*16*π*3r = 48ε₀*π*r

So, C= 2*10^-11F