Electrostatic Potential and CapacitanceIntroduction
Riya lives on the fourth floor. Every morning she carries a bucket of water up the stairs. Sina prefers the lift, and on Sundays Riya takes the long sloping ramp at the back of the building instead. Three very different routes — yet the bucket always ends up at the same height above the ground, so it gains exactly the same gravitational potential energy. The formula mgh only asks how high you went, not which way.
You met this idea in Class 11. When an outside force lifts a body against gravity, or squeezes a spring, the work it does is stored as potential energy. Let go, and the body speeds up as that stored energy turns back into kinetic energy, while the total of the two stays the same. Forces that behave like this are called conservative forces; gravity and the spring force are two of them.
The electric force between two charges at rest is a third. That should not surprise you: Coulomb's law and Newton's law of gravitation are both inverse-square laws, and they differ mainly in what makes the force — charge instead of mass. So a charge placed in an electric field has electrostatic potential energy, just as the bucket has gravitational potential energy.
Try it. A large positive charge Q sits on the left. Pick a path, then push a small positive test charge from R to P and watch the work meter. Do all three paths.
The amber bar stopped at 120 µJ every time. On the arc it did not move at all, because the arc keeps the charge at the same distance from Q and moving across the green field lines costs nothing. On the wiggly path it went up and down, but it still finished at 120 µJ. The work done against an electrostatic force depends only on where you start and where you finish.
Where the Work Goes
Let us look closely at what you just did. A charge Q sits at the origin, and we bring a small test charge q from R to P against its repulsion. Fig. 2.1 of your textbook shows exactly this; every figure number in this chapter is the textbook's, so you can look each one up. Two things keep the picture clean:
- The test charge is so small that it does not disturb Q. (Or Q is held fixed by some force we do not care about.)
- We push with an external force that is just equal and opposite to the electric force, . Then there is no net force, the charge creeps along at a constant, tiny speed, and none of our work turns into kinetic energy.
So while the charge moves from R to P, the electric force does
All of that work is stored as potential energy of the charge. The work done by the external force from R to P is
and the potential energy difference between P and R is defined as exactly this work:
The electric potential energy difference between two points is the work an external force must do to move a charge, without accelerating it, from one point to the other.
Two remarks follow. First, the right-hand side depends only on the two end points — which is what your three paths showed, and what makes the force conservative. Second, only differences of potential energy mean anything. If we added 50 J to the potential energy at every single point, would change by
So we are free to choose where . The natural choice is infinity, where the charge feels no force at all. Taking R at infinity in Eq. (2.2) gives
The potential energy of a charge q at a point is the work done by an external force (equal and opposite to the electric force) in bringing q from infinity to that point.