PHYC10004 Chap.3 Electric Potential and Equipotential Surfaces
Electric Potential and Equipotential Surfaces
Electric potential is the scalar that makes electrostatics easier. Because the electrostatic field is conservative, the work done moving a charge depends only on its start and end points, so each point can be given a potential, the potential energy per unit charge, defined through ΔV = −∫E · ds.
For a point charge V = q/4πε₀r, and the potentials of several charges simply add as signed numbers.
This chapter, drawn from Weeks 2 and 3, connects potential to the field in both directions: integrate the field to get a potential difference, or take the slope of the potential to recover the field.
It introduces equipotential surfaces, which always cross field lines at right angles, and uses them to explain why a conductor is at one potential throughout, why charge concentrates on sharp points, and how a charged particle gains kinetic energy when it moves through a potential difference. The potential of continuous distributions such as rings and rods is treated as a scalar integral.
What this chapter covers
- 01
Conservative Fields
- 02
Potential Difference
- 03
Potential of Point Charges
- 04
Equipotential Surfaces
- 05
Field From Potential
- 06
Potential Energy of Charges
- 07
Conductors at One Potential
- 08
The Electronvolt
Worked example · free
Find the zero-potential points of two unequal charges
- 1V = 0 where the two terms cancel: 4.0/d₁ = 2.0/d₂, so the distance to the positive charge must be twice the distance to the negative charge.
- 1Between the charges, d₁ = x and d₂ = 0.30 − x: x = 2(0.30 − x), giving x = 0.20 m.
- 1Beyond the negative charge, d₁ = x and d₂ = x − 0.30: x = 2(x − 0.30), giving x = 0.60 m.
- 1To the left of the positive charge the larger charge is always closer, so no zero exists there.
Key terms
- Electric Potential
- Electric potential is the potential energy per unit charge at a point, measured in volts, with zero usually taken at infinity.
- Potential Difference
- Potential difference is the change in potential between two points, equal to minus the line integral of the field from the first point to the second.
- Equipotential Surface
- An equipotential surface joins points at the same potential. No work is done moving a charge along it, and field lines cross it at right angles.
- Conservative Field
- A conservative field is one in which the work done moving a charge between two points does not depend on the path taken; every electrostatic field is conservative.
- Electronvolt
- The electronvolt is the energy gained by a charge e moving through a potential difference of one volt, equal to 1.6022 × 10⁻¹⁹ J.
- Electric Potential Energy
- Electric potential energy is the work needed to assemble a set of charges from infinite separation, equal to qV for a single charge at potential V.
Electric Potential and Equipotential Surfaces FAQ
What is the difference between potential and potential energy?
Potential is a property of a point in space, energy per unit charge in volts. Potential energy belongs to a charge placed at that point and equals qV in joules. The lab notes warn the two are easy to confuse, so keep track of which one a question asks for.
Can the potential be zero where the field is not?
Yes. Midway between equal and opposite charges the potential contributions cancel, but both fields point towards the negative charge and add. The reverse also happens, so zero-potential points and zero-field points are generally found in different places.
Why is a conductor at the same potential throughout?
In electrostatic equilibrium the field inside a conductor is zero, so no work is done moving a charge between any two points within it. Its whole volume, including the surface, therefore sits at one potential.
Which way does a charge move in a potential difference?
A positive charge released from rest moves towards lower potential, gaining kinetic energy equal to q times the potential drop. A negative charge such as an electron moves towards higher potential, and its kinetic energy rises by the same rule with its own sign.
Exam move
Build a two-column habit: for every arrangement you meet, find the field at a point by vector addition and the potential at the same point by scalar addition, and compare the effort. Practise the uniform-field chain of work, potential difference, field and force, and the energy statement qΔV + ΔK = 0 for protons and electrons.
Sketch equipotentials around a point charge, a dipole and parallel plates, always drawing field lines at right angles to them. Finally, work a charged-sphere problem in which you find the surface field, the surface potential and the energy of a released electron, paying attention to every sign.
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