Two concentric metal spherical shells have radii r_1= 10 and r_2= 50 cm. The inn

Question

Two concentric metal spherical shells have radii r_1= 10 and r_2= 50 cm. The inner shell has charge +2 C and the outer shell has charge -2 C. What is the potential V(r) between the spheres? What is the energy density in the field between the spheres (as a function of r)? What is the total energy stored in the electric field between the spheres? Calculate the capacitance using energy considerations, then compare to the value found using the usual formula for a spherical capacitor. In the Rutherford scattering experiment, an alpha particle (helium nucleus) is fired at the nucleus of a gold atom. The distance of closest approach of the alpha particle to the gold nucleus is 1 Angstrom (10^-10 m). What was the initial speed of the alpha particle?

Explanation / Answer

3. radius of outer shell , r2 = 50 cm
radius of inner shell, r1 = 10 cm
Charge on inner shell = q = 2 C
Cherge on outer shell = -q = -2 C
a. consider any point at radius r form the centre of shell such that r2<r<r1
then from gauss' law, electric field at this point is
E*4*pi*r^2 = q/epsilon
E = kq/r^2
now, electric potential V and electric field are related as under
V = -[integral]Edr
so, V = kq/r + C
Here, C is a constant that can be found as under
C is the constant contribution of potential to the net potential that is provided by the outer shell. The outer shell provides a constant potential inside it, so as to maintain a 0 Electric field due to any external charge inside the shell
so, V(r) = kq/r - kq/r2 = 8.98*10^9*2[1/r - 1/0.5] = 1.796*10^10[1/r - 2] V
b. Energy density in electric field E is epsilon*E^2/2 [ where epsilon is permittivity of free space]
E(r) = kq/r^2
Energy density = epsilon*k^2q^2/2r^4 = epsilon*q^2/r^2*16*pi^2*epsilon^2 = q^2/16r^2*pi^2*epsilon
c. now, dE/dV = q^2/16r^2*pi^2*epsilon = energy density
dE = q^2*dV/16r^4*pi^2*epsilon
now, V = 4pi*r^3/3
dV = 4pir^2dr
dE = 4piq^2*r^2dr/16r^4*pi^2*epsilon = q^2*dr/4*pi*epsilon*r^2 = kq^2dr
E = integrate from r = r1 to r = r2 = kq^2*(1/r1 - 1/r2)
d. Energy of a capacitor = 0.5CV^2
but q = CV
so, E = 0.5q^2/C = kq^2*(1/r1 - 1/r2)
C = 1/k*(1/r1 - 1/r2)
From formyla C for spehirical capacitor = 1/k(1/r1 - 1/r2)

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