Potential and voltage

A field pushes a charge and can transfer energy when it moves. Can we mark an energy map of space in advance? Such a map is more convenient than force when finding work between two points.

At one point, doubling a test charge doubles both the force and its potential energy in an external field. Divide energy W_{\text{п}} by charge q, keeping its sign: \varphi=W_{\text{п}}/q. This electric potential describes the field point for a chosen energy zero. Its unit is the volt, 1\,\text{В}=1\,\text{Дж/Кл}. It describes energy, not a direction: one number φ labels the point, whereas force requires vector E.

Like height above a chosen surface, potential needs a reference zero. Add the same constant to every value: differences between points remain unchanged, as does the work between them. This is freedom to choose a reference, not energy physically added by a device. The height analogy highlights differences but has a limit: electric charge can be negative, so its energy need not increase with potential.

Consider transfer from point 1 to point 2. The difference U_{12}=\varphi_1-\varphi_2 is the voltage in this direction, measured in volts. Reversing start and end gives U_{21}=-U_{12}. Potential refers to one point relative to the chosen zero; voltage refers to a pair and their order. Equal potentials give zero voltage, even if the individual values are large or negative.

Work done by an electrostatic field equals the decrease in potential energy: A=q(\varphi_1-\varphi_2)=qU_{12}, hence U_{12}=A/q. Doubling q in the same field between the same points doubles the work; J/C gives V. For a positive charge moving to lower potential, the field does positive work. Negative q reverses the sign. Keep the charge sign, and distinguish work by the field from work by an external device moving the charge.

In a uniform field, force \vec F=q\vec E is constant. Work depends only on the displacement component along E: A=qEd, where d is its signed projection in metres. Perpendicular motion has d = 0 and zero work even over a long path. Motion against the field has negative d. Doubling the along-field displacement doubles work. Around any closed path, net work by an electrostatic field is zero.

Between wide parallel plates, far from their edges, the field is nearly uniform. Moving distance d along it gives voltage U = Ed, so the field magnitude is E=U/d. Here U and d are positive magnitudes: voltage and plate separation. At fixed U, doubling the gap halves E; at a fixed gap, doubling U doubles E. From 1\,\text{В}=1\,\text{Дж/Кл} follows V/m = N/C, connecting the energy and force descriptions.

Charge can also be measured by how much passes through a conductor's cross-section. A steady current of one ampere transfers one coulomb each second: 1\,\text{Кл}=1\,\text{А}\cdot\text{с}. Twice the time at the same current transfers twice the charge. Amperes measure the transfer rate; coulombs measure the transferred charge. For now this relates units; the current lesson will explain sustained charge motion and the source's role.

For point source Q with zero at infinity, potential is \varphi=kQ/(\varepsilon r). Here r is distance, k the Coulomb constant, and ε the relative permittivity of a uniform linear isotropic medium. At twice the distance, |φ| halves, whereas E would fall by a factor of four. The sign of φ matches Q. At infinity it approaches the chosen zero; near a real charged object, check that the point model applies.

Place charge q_2 in the field of q_1. Multiplying the source potential by the second charge gives pair energy W_{\text{п}}=kq_1q_2/(\varepsilon r), with zero at infinite separation. Unlike charges have negative energy: energy must be supplied to separate the attracting pair to the zero level. This is a consequence of the chosen zero, not an impossible energy store. Bringing like charges closer raises their positive energy, which can be released as they separate.

For several sources, add potentials algebraically: \varphi=\varphi_1+\varphi_2+\ldots, using a consistent zero for every contribution. At the midpoint between equal opposite charges, potentials cancel even though the fields add and E is nonzero. Zero potential therefore does not mean zero force. Potential is a scalar energy value, whereas electric field strength is a directional measure of its spatial change.

A +2 μC charge moves from a point at 50 V to one at 20 V. Find the work done by the field.

Voltage in the transfer direction: U₁₂=50−20=30 V.

The charge q=2·10⁻⁶ C is positive.

Work A=qU₁₂=2·10⁻⁶×30=6·10⁻⁵ J.

Check: a positive charge moves to lower potential, so interaction energy decreases and the field does positive work.

Work done by the field: +60 μJ.

Voltage is the potential difference between chosen points; a common shift of zero does not change work A = qU.

Work's sign depends on both transfer direction and charge sign. In a uniform field, use the displacement component along E.

Potentials add algebraically; fields add as vectors. Zero potential alone does not imply zero field.

Next, apply voltage to energy storage in a capacitor.