Q. 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.

Answer :


Given:


Radius of nucleus=10-15m


Radius of 1-s charge cloud=1.3× 10-11m=r1


Radius of 2-s charge cloud =5.2× 10-11m=r2


(a)for calculating electric field at a point just inside 1s cloud


Consider a gaussian spherical surface of radius just equal to the radius of 1s cloud=r1


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 passing through this surface is given by


…(i)


We know that,


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



Charge enclosed by this sphere =total charge on the nucleus=q


Now,


q= C


C


Using gauss’s law and eqn.(i)




Putting the values of r1 ,q and ϵ0


N/C


N/C


Therefore electric field at a point just inside the 1s cloud is given by 3.4× 1013N/C


(b)for calculating electric field at a point just inside the 2s cloud


Consider a gaussian spherical surface of radius just equal to the radius of 2s cloud=r2


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 passing through this surface is given by


…(i)


We know that,


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



Charge enclosed by this sphere =total charge on the nucleus+ charge of 2 1s electrons=q


Now,


q= C


C


Using gauss’s law and eqn.(i)




Putting the values of r1 ,q and ϵ0


N/C


N/C


Therefore electric field at a point just inside the 2s cloud is given by 1.1× 1012N/C


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