Answer to Question #134278 in Electricity and Magnetism for Rohan

Question

The electric field in an electrostatic situation is denoted by the function E and the potential by the function V in a certain volume. Inside this volume,

(a) If E is given, V can be uniquely written.

(b) If V is given, E can be uniquely written.

(c) If E is same everywhere in the volume, V cannot change in this volume.

(d) If V is same everywhere in the volume, E cannot change in this volume.

(Note:- This question have one or more than one correct choice(s) out of the four given choices. Any number of options may be correct.)

Answer 1

We know that for an electrostatic field

$E=-\nabla V.$

But if we add a constant V_{0 }to V, we'll obtain

$E=-\nabla(V+V_0) = - \nabla V.$

Therefore, if E is given, we cannot uniquely determine V because of a unknown constant. So (a) is not true. But if we know V, we take gradient of it and get E uniquely, so (b) is true.

Next, if V is constant elsewhere, it's gradient will be 0, so E is also a constant (0), so (d) is true.

If E is non-zero constant, then V can linearly depend on radius, so it can be not the same is every point.