Q11 of 46 Page 141

A charge Q is distributed uniformly within the material of a hollow sphere of inner and outer radii r1 and r2 figure. Find the electric field at a point P a distance x away from the centre for r1 < x < r2. Draw a rough graph showing the electric field as a function of x for 0 < x < 2r2 figure.



Given:


Amount of charge distributed within the hollow sphere=Q


Inner and outer radius of hollow sphere=r1 and r2


Volume charge density of hollow sphere is given by



….(i)


To find the electric field at a point P at a distance x away from the centre consider a spherical gaussian surface of radius x()



By symmetry all points on this surface are equivalent and electric field at all points have same magnitude and is in radial direction


Therefore flux through this surface is given by


..(ii)


Charge enclosed by this surface is given by




…(iii)


We know that,


By Gauss’s law, flux of net electric field (E⃗ ) through a closed surface S equals the net charge enclosed (qin) by the surface divided by ϵ0



Using gauss’s law and eqn.(ii) and (iii)




This electric field is directly proportional to x for r1<x<r2


For


Considering a similar gaussian spherical surface of radius x such that


Charge enclosed by this surface =Q


Total electric flux through this surface =


Using gauss law we can write



The graph showing electric field as a function of x is shown as follows



More from this chapter

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9

A spherical volume contains a uniformly distributed charge of density 2.0 × 10–4 C m–3. Find the electric field at a point inside the volume at a distance 4.0 cm from the centre.

 10

The radius of a gold nucleus (Z = 79) is about 7.0 × 10–15 m. Assume that the positive charge is distributed uniformly throughout the nuclear volume. Find the strength of the electric field at

(a) the surface of the nucleus and


(b) at the middle point of a radius.


Remembering that gold is a conductor, is it justified to assume that the positive charge is uniformly distributed over the entire volume of the nucleus and does not come to the outer surface?


 12

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).


 13

Consider the following very rough model of a beryllium atom. The nucleus has four protons and four neutrons confined to a small volume of radius 10–15 m. The two 1s electrons make a spherical charge cloud at an average distance of 1.3 × 1011 m from the nucleus, whereas the two 2s electrons make another spherical cloud at an average distance of 5.2 × 10–11 m from the nucleus. Find the electric field at

(a) a point just inside the 1s cloud and (b) a point just inside the 2s cloud.