A charge Q is distributed over a metal sphere of radius R.What is the electric field and potential at the centre?

According to the question the charge distributed on the metal sphere is on the surface of  metal sphere and  from property of electric field we know that there is no electric field inside the metal conducting sphere hence electric field inside is zero E = 0. and the electric field at centre is also zero,  now at inner point inside the sphere it is considered that the whole charge is concentrated at one point hence potential inside the sphere at any point is 
V =Vsurface  = kQR volt.
hence the value of electric field is zero and the potential at centre is equal to potential at surface of metal sphere and it is not zero.

 

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