Electric potential and potential difference

A sparrow lands on a high-voltage power line and nothing happens to it, yet a person standing on the ground who touched the same wire would be in mortal danger. The word “voltage” explains the difference: the bird’s two feet are at almost the same potential, while a person’s hand and feet are not. In the lesson «Electric charge, Coulomb’s law and the electric field» we described the field through force — the field strength. Now we describe it through energy: the work of the field, potential and potential difference. This is the second half of the DİM test-collection chapter «Electric charge. Electric field»; besides calculations it often asks about φ(r) and E(r) graphs, equipotential surfaces and I–II–III statements.

Work done by the electric field

Place a positive charge q in the uniform field between the plates of a parallel-plate capacitor. A constant force F = q · E acts on it. When the charge moves a distance d along a field line, the field does work A = F · d = q · E · d. If the charge travels a path s at an angle α to the field lines, the work is A = F · s · cos α, and s · cos α is the projection of the displacement onto the direction of the field lines. So in a uniform field only the distance covered along the field lines counts — just as only the change in height counts in the work of gravity.

Now take the charge from point 1 to point 2 along a straight line, a broken line or a curve. Split the path into tiny steps: on steps perpendicular to the field lines the work is zero, and the steps along the lines always add up to the same projection d. So the work of the field does not depend on the shape of the path — only on the positions of the start and end points. The same holds for the field of a point charge: radial steps involve work, steps along circular arcs around the charge do not. An important consequence: when a charge goes round a closed path and returns to its starting point, the total work of the field is zero.

Potential energy and potential

Because the work of the field does not depend on the path, every position of a charge in the field has a definite potential energy Wp — like the energy m · g · h in the gravitational field. Positive work of the field reduces the potential energy: like a falling stone, a positive charge left free moves towards lower potential energy. The zero level of potential energy is chosen by convention; for point charges it is usually taken infinitely far away.

The potential energy is proportional to the charge q₀ brought into the field: double the charge and the energy doubles. So the ratio Wp / q₀ no longer depends on that charge and describes only the given point of the field — this is the potential. Field strength is the force characteristic of a field, potential is its energy characteristic. Field strength is a vector; potential is a scalar: it has a sign but no direction.

In the field of several charges each charge creates its own potential, and the resulting potential is their algebraic sum — added as numbers with signs, not as vectors. This is much easier than finding the field strength: no directions to draw.

Potential difference, the electronvolt and equipotential surfaces

In practice what matters is not the potential itself but the difference between the potentials of two points, because the work of the field depends on it: A = Wp₁ − Wp₂ = q · φ₁ − q · φ₂ = q · (φ₁ − φ₂). The potential difference is also called voltage and is written U. In a circuit voltage is measured with a voltmeter; in electrostatics an electrometer is used: its case is earthed, its rod is connected to the body, and the deflection of the needle shows how much the body’s potential differs from the Earth’s (the Earth’s potential is usually taken as zero).

In atomic and particle physics energies are tiny, so instead of joules the electronvolt (eV) is used: the energy an electron gains from the field when it passes through a potential difference of 1 V. For larger energies keV (10³ eV), MeV (10⁶ eV) and GeV (10⁹ eV) are used.

For a uniform field we now have two expressions for the work: A = q · E · d and A = q · U. Setting them equal gives the link between field strength and voltage. It also explains the second unit of field strength: 1 N/C = 1 V/m. Moving along the field lines, the potential drops by E volts every metre.

The set of points with the same potential is called an equipotential surface. Moving a charge along such a surface gives U = 0, so the field does no work. That is only possible if the force is perpendicular to the surface: field lines are always perpendicular to equipotential surfaces.

• The equipotential surfaces of a point charge are concentric spheres centred on the charge (circles in the drawing).
• The equipotential surfaces of a uniform field are parallel planes perpendicular to the field lines.
• Field lines point towards decreasing potential; where surfaces drawn at equal steps (say every 10 V) are crowded, the field is strong.
• The surface (and the whole volume) of a conductor is equipotential.

Conductors and dielectrics in an electrostatic field

A conductor (a metal) contains free electrons. When it is brought into a field, the electrons shift until the resulting field inside the conductor is exactly zero (electrostatic induction; see the lesson «Electric charge, Coulomb’s law and the electric field»). If any field remained inside, the electrons would keep moving. Once equilibrium is reached:

1. The field strength inside the conductor is zero: E = 0.
2. All excess charge sits on the outer surface of the conductor; the inner surface of a hollow sphere stays uncharged.
3. All points of the conductor have the same potential: it is an equipotential body, and field lines meet its surface at right angles.
4. Charge gathers more densely on sharply curved, pointed parts of the surface — the pointed tip of a lightning rod works this way.

This gives the idea of electrostatic shielding: inside a metal box or a fine metal mesh (a Faraday cage) there is no external electrostatic field. Sensitive instruments are covered by metal screens, and people inside a metal-bodied car or a plane are protected when lightning strikes.

A dielectric has no free charges, but the field polarises its molecules: polar molecules (water, for example) turn along the field, while in non-polar molecules the electron clouds shift slightly. The bound charges that appear on the dielectric’s surfaces create a field opposite to the external one and weaken the total field — not down to zero as in a conductor, but by a factor of ε.

Motion of charged particles in an electric field

An electric field can speed up a charged particle or change its direction of motion — X-ray tubes, the cathode-ray tube of an oscilloscope and particle accelerators all work this way. The work of the field equals the change in the particle’s kinetic energy (the work–energy theorem). For electrons and protons gravity is billions of times weaker than the electric force, so it is neglected.

Now let the particle enter parallel to the plates, i.e. perpendicular to the field lines, with speed v₀. Along the plates no force acts on it, so that motion is uniform: x = v₀ · t. Perpendicular to the plates there is a constant acceleration a = |q| · E / m. It is exactly like a horizontally thrown body, with a in place of g, so the path is a parabola. After leaving the plates the particle flies in a straight line.

Two parallel plates carrying equal and opposite charges form a device that stores charge and field energy — a capacitor. Its capacitance, connections and energy are in the next lesson, «Capacitance and capacitors».