Electric potential

Electric potential, also known as the electrostatic potential, is the difference in electric potential energy per unit of electric charge between two points in a static electric field. It tells us how much work is needed to move a tiny test charge from one point to another in the electric field, without disturbing the field itself.

In simple terms, electric potential is like the electric potential energy of a charged particle divided by its charge. This means it describes a property of the electric field itself, not of the particle. We usually measure it in units called volts, which are the same as joules per coulomb (J⋅C⁻¹).

Electric potential is very important in both static and changing electric fields. In changing fields, it works together with the magnetic vector potential to describe the electric field. This helps scientists understand how electric and magnetic fields are connected, especially when they change over time.

Introduction

Classical mechanics studies ideas like force, energy, and potential. Force and potential energy are linked. When a force pushes an object, the object accelerates. As an object moves with the force, its potential energy goes down. For example, a cannonball at the top of a hill has more gravitational potential energy than at the bottom. As it rolls down, this energy turns into motion – kinetic energy.

We can describe the potential of some force fields so that an object’s potential energy depends only on where it is. Two such fields are a gravitational field and an electric field. These fields affect objects because of what they are made of (like mass or electric charge) and where they are placed.

An object can have electric charge. Since an electric field pushes charged objects, a positive charge feels a push in the direction of the electric field vector, while a negative charge feels a push in the opposite direction.

Electrostatics

Main article: Electrostatics

Electric potential is a way to understand how electric fields work. It tells us how much energy we would need to move a tiny test charge from one place to another in an electric field. This helps us predict how charges will behave without having to calculate the whole field each time.

When we talk about electric potential, we often think of a single point charge, like a small charged ball. The potential around this charge depends on how far away we are from it. The farther we are, the lower the potential, making it easier to move a charge in that area. This idea helps us understand more complex situations with many charges, whether they are points or spread out over an area.

Generalization to electrodynamics

Main article: Gauge fixing

When magnetic fields change over time, the electric field isn't simple anymore. This happens because changing electric fields and magnetic fields are linked. To handle this, we use two kinds of potentials: a scalar potential and a magnetic vector potential.

The magnetic vector potential helps us describe the magnetic field. Even though there are many ways to choose these potentials, they all give the same electric and magnetic fields. Different choices can make the math easier or harder depending on what we're studying.

Units

The SI unit for electric potential is the volt, named after Alessandro Volta. It is written as V. This is why we call the difference in electric potential between two points a voltage. Some older units exist, but they are rarely used today.

Galvani potential versus electrochemical potential

Main articles: Galvani potential, Electrochemical potential, and Fermi level

In metals and other materials, the energy of an electron is influenced by both the electric potential and the surrounding atoms. When a voltmeter connects two different metals, it shows a potential difference that accounts for these atomic differences. This measured value is called the electrochemical potential or Fermi level. The pure electric potential, without adjustments, is known as the Galvani potential. The words "voltage" and "electric potential" can mean either of these depending on the situation.

Common formulas

Charge configuration	Figure	Electric potential
Infinite wire		V = − λ 2 π ε 0 ln ⁡ x , {\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,} where λ {\displaystyle \lambda } is uniform linear charge density.
Infinitely large surface		V = − σ x 2 ε 0 , {\displaystyle V=-{\frac {\sigma x}{2\varepsilon _{0}}},} where σ {\displaystyle \sigma } is uniform surface charge density.
Infinitely long cylindrical volume		V = − λ 2 π ε 0 ln ⁡ x , {\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,} where λ {\displaystyle \lambda } is uniform linear charge density.
Spherical volume		V = Q 4 π ε 0 x , {\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},} outside the sphere, where Q {\displaystyle Q} is the total charge uniformly distributed in the volume.	V = Q ( 3 R 2 − r 2 ) 8 π ε 0 R 3 , {\displaystyle V={\frac {Q(3R^{2}-r^{2})}{8\pi \varepsilon _{0}R^{3}}},} inside the sphere, where Q {\displaystyle Q} is the total charge uniformly distributed in the volume.
Spherical surface		V = Q 4 π ε 0 x , {\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},} outside the sphere, where Q {\displaystyle Q} is the total charge uniformly distributed on the surface.	V = Q 4 π ε 0 R , {\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}R}},} inside the sphere for uniform charge distribution.
Charged Ring		V = Q 4 π ε 0 R 2 + x 2 , {\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}{\sqrt {R^{2}+x^{2}}}}},} on the axis, where Q {\displaystyle Q} is the total charge uniformly distributed on the ring.
Charged Disc		V = σ 2 ε 0 [ x 2 + R 2 − x ] , {\displaystyle V={\frac {\sigma }{2\varepsilon _{0}}}\left[{\sqrt {x^{2}+R^{2}}}-x\right],} on the axis, where σ {\displaystyle \sigma } is the uniform surface charge density.
Electric Dipole		V = 0 , {\displaystyle V=0,} on the equatorial plane.	V = p 4 π ε 0 x 2 , {\displaystyle V={\frac {p}{4\pi \varepsilon _{0}x^{2}}},} on the axis (given that x ≫ d {\displaystyle x\gg d} ), where x {\displaystyle x} can also be negative to indicate position at the opposite direction on the axis, and p {\displaystyle p} is the magnitude of electric dipole moment.