Question 4
to find the electric potential at the surface of the sphere we have to correlate it with the electric field .
as we found out that electric field outside a sphere is the same electric field found due to a point charge, then we have to find the electric potential due to a point charge in the center of the sphere.
[BIMG]http://up4.m5zn.com/9bjndthcm6y53q1w0kvpz47xgs82rf/2009/12/2/10/pt0w9y3t0.png[/BIMG]
by symmetry, and also by looking at the figure above that V is a function of r only, where r is the radial distance from the origin.
The x-component of the electric field generated along this axis takes the form
[BIMG]http://up4.m5zn.com/9bjndthcm6y53q1w0kvpz47xgs82rf/2009/12/2/10/5on1w09mp.png[/BIMG]
Both the Y- and Z-components of the field are zero.
Both variables are related via [BIMG]http://up4.m5zn.com/9bjndthcm6y53q1w0kvpz47xgs82rf/2009/12/2/10/qfpeahtot.png[/BIMG]
then by integrating we will get [BIMG]http://up4.m5zn.com/9bjndthcm6y53q1w0kvpz47xgs82rf/2009/12/2/10/2tmiwizev.png[/BIMG]
where V0 is an arbitrary constant. Finally, making use of the fact that V = V(r)
[BIMG]http://up4.m5zn.com/9bjndthcm6y53q1w0kvpz47xgs82rf/2009/12/2/10/zmqv5qa19.png[/BIMG]
that is it this a way that we can find the potential at the surface which is the same as we get due to point charge.
I hope this is correct please check first then write it in the answer sheet.
Good luck