Consider a thick walled metallic spherical shell with inner radius a and outer radius b. No charge is given to the spherical shell. A point charge +Q is kept at the centre. Find the electric potential at a point distance x from the centre where `0lt x lt oo`.

Explanation

See the figure for charges induced on the inner and outer surfaces of a thick walled spherical shell. Charge induced on the inner surface is -Q and charge appearing on outer surface is +Q . These two spherical surfaces will act like two thin spherical shells of radius a and b respectively. In between these two faces, electric field intensity will remain zero. Hence, electric potential will remain constant for `a lt x lt b`. Moreover to proceed further we should be aware of the fact that elecrtric potential, due to a thin spherical shell for points outside it or on the surface, is same as that of a point charge as it it were kept at its centre and electric potential inside the thin spherical shell is same as that on its surface. We know that electric potential due to a point charge at a point that is at a distance x from it is given by the formula `V=(q)/(4pi epsilon_(0)x)` and now we can use the above stated information to write electric potential for different points due to the given system.
(i) `0 lt x lt a`
`V=(Q)/(4pi epsilon_(0)x)+(-Q)/(4pi epsilon_(0)a)+(Q)/(4pi epsilon_(0)b)`
`rArr V=(Q)/(4pi epsilon_(0)x)-(Q)/(4pi epsilon_(0)a)+(Q)/(4pi epsilon_(0)b)" " ...(i)`
(ii) `a lt x lt b`
`V=(Q)/(4pi epsilon_(0)x)+(-Q)/(4pi epsilon_(0)x)+(Q)/(4pi epsilon_(0)b)`
`rArr V=(Q)/(4pi epsilon_(0)b)" " ...(ii)`
Note that value of potential for x = a, is same from both the equations (i) and (ii). Hence, there is no discontinuity.
(iii) `b lt x lt oo`
`V=(Q)/(4pi epsilon_(0)x)+(-Q)/(4pi epsilon_(0)x)+(Q)/(4pi epsilon_(0)x)`
`rArr V=(Q)/(4pi epsilon_(0)x)" " ...(iii)`
Note that value of potential at x = b is same from both the equaiton (ii) and (iii). Hence, there is no discontinuity at this point also.

Electric potential due to the sbove system is defined for all the points except x = 0. At x = 0, potential approaches infinity.
Graph between electric potential and distance from the centre is shown in the figure.