Physics electricity - electric force, potential, field, PE

Question

True statements about the relationship between the electric field and electric potential include which of the following?
    I. If the electric field at a certain point is zero, then the electric potential at the same point is also zero.
   II. If the electric potential at a certain point is zero, then the electric field at the same point is also zero.
  III. The electric potential is inversely proportional to the strength of the electric field.
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) None of the above

How do you approach this question? Since electric field  = potential energy *q/r ( since E (electric field) = kQ/r2 and U (potential energy) = kQq/r ), if either is 0 shouldn't the other one be 0 as well? Using the same logic for the 2nd one, I thought that both the 1st and the 2nd would be right. Is this wrong? Thanks :)

 

Answer 1

This may best be understood by looking at examples:

 

Statement I is incorrect.  Consider a field point exactly in the middle between 2 equal charges q separated by a distance 2r.  The electric field at that point is zero, but the electric potential is kq/r + kq/r, which is not zero.  The electric field is the negative gradient of the electric potential, so the field being zero only means the spatial derivatives of the potential are zero, while the potential itself may be non-zero at that same point.  (I do not know if you took a calculus-based physics; if not, then ignore the statement about the gradient).

 

Statement II is incorrect.  Consider a field point in the center of an electric dipole with distance 2r between +q and -q.  The potential is kq/r - kq/r = 0, but there is a net electric force (hence a field) that accelerates a positive test charge at that point toward -q and away from +q.

 

Statement III is incorrect.  The electric potential at a field point at a distance r from a source charge q is V = kq/r.  The magnitude of the electric field at the test point is E = kq/r^2.  You can see V is not inversely proportional to E.  The correct relation between the vector E and the scalar V is E = -grad(V) (the latter from calculus-based physics).  

 

Bottom line answer:  None of the statements are correct, so answer (E) is the correct answer.

 

One more thing:  Please be careful not to confuse electric potential with electric potential energy.  The electric potential energy of 2 charges q and Q separated by a distance r is:

 

U = kqQ/r

 

But the electric potential at a field point located a distance r from a source charge Q is:

 

V = kQ/r

 

It then follows that U = qV if there is a test charge q a distance r from a source charge Q.

 

I hope this helps.