Q. 125.0( 1 Vote )

A charge Q is placed at the centre of an uncharged, hollow metallic sphere of radius a.

(a) Find the surface charge density on the inner surface and on the outer surface.

(b) If a charge q is put on the sphere, what would be he surface charge densities on the inner and the outer surfaces?

(c) Final the electric field inside the sphere at a distance x from the centre in the situations (a) and (b).

Answer :


Given:


Charge present at the centre of hollow metallic sphere=Q


Radius of sphere=a


Surface area of sphere=4πa2


We know that,



Electric fields at all points inside the conductor is zero


So,


If a charge q is placed within the cavity of a metallic sphere then taking the Gaussian surface S as shown in fig. electric field E =0 at all points on this surface and hence which ensures that charge contained in S is zero (by gauss’s law)



And if a charge +q is place din the cavity , there must be a charge -q on the inner surface of the conductor. If the conductor is neutral i.e. no charge is placed on it,a charge +q will appear on the outer surface.


So here,


Charge induced at inner surface =-Q


Charge induced at outer surface=+Q


Surface charge density is given by charge/total surface area


surface charge density on inner surface=


Surface charge density on outer surface=


Therefore surface charge densities on inner and outer surfaces are given by -Q/4πa2 and Q/4πa2 respectively.


(b) If a charge q is put on the sphere all the charge will move to the outer surface of the sphere(since, total charge enclosed by the sphere is zero)


Therefore inner charge density will remain same and outer charge density will increase as the new outer charge is Q+q


surface charge density on inner surface=


surface charge density on outer surface=


Therefore surface charge densities of inner and outer surfaces after putting charge q on the sphere is given by -Q/4πa2 and (Q+q)/4πa2 respectively


(c)


To find the electric field inside the sphere at a distance x from the sphere consider a spherical Gaussian surface of radius x


Applying Gauss law on this sphere





Where q is charge enclosed by the sphere


For situation(b) the charge enclosed by the Gaussian surface remains the same as the charge provided to the metallic sphere it will move to the outer surface of the sphere



Therefore for situations (a) and (b) the electric field inside the sphere at a distance x from the centre is same and given by Q/4πϵ0x2


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