Electric Potential

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Textbook

University Physics Volume 2: Chapter 7

Theory

Background: The Electric Force is a Conservative Force

The electric force is a conservative force, meaning that the work done by or against the electric force depends only on the initial and final positions, not on the path taken. This allows us to define the electric potential energy $U$ in an electric field as the negative of the work $W$ done by the electric force:

$U=-W$

where $W$ is the work done by the electric force to move a charge within the field.

Electric Potential and Electric Potential Energy

Electric Potential Energy

Electric potential energy is the energy stored in a system of charges due to their positions in an electric field. This is similar to gravitational potential energy, where the position of an object in a gravitational field determines its potential energy.

Electric Potential (Voltage)

The electric potential $V$ at a point is defined as the electric potential energy per charge:

$V={\frac {U}{q}}$

Electric potential, also known as voltage, is measured in joules per coulomb (J/C).

Finding the Electric Potential Energy from the Electric Potential

If the electric potential $V$ at a point is known, the electric potential energy $U$ of a charge $q$ placed at that point can be calculated as:

$U=q\cdot V$

Electric Potential due to a Point Charge at Rest

The electric potential $V$ at a distance $r$ from a point charge $Q$ is given by: $V=k_{e}{\frac {Q}{r}}$ where $k_{e}$ is the electrostatic constant.

Potential Energy of a System of Charged Particles

The total electric potential energy of a system of charges is the sum of the potential energies between all pairs of charges in the system. For example, for three charges, $q_{1},q_{2},$ and $q_{3}$, the potential energy is: $U=k_{e}\left({\frac {q_{1}q_{2}}{r_{12}}}+{\frac {q_{1}q_{3}}{r_{13}}}+{\frac {q_{2}q_{3}}{r_{23}}}\right)$

Electric Potential and the Electric Field - Equipotential Lines

Equipotential lines represent points of equal electric potential in an electric field. They are always perpendicular to electric field lines. Moving along an equipotential line requires no work, as the electric potential remains constant.

Electric Potential, Current and Power

Electric potential difference, or voltage, causes current to flow in a conductor.

Current $I$ is defined as the rate at which electric charge flows through a conductor: $I={\frac {Q}{t}}$ where $Q$ is the charge in coulombs and $t$ is time in seconds.

Power $P$ is the rate of doing work, or work done per unit time. In an electric circuit, power represents the rate at which electrical energy is transferred or converted: $P={\frac {W}{t}}$ where $W$ is the work done in joules and $t$ is the time in seconds.

The power $P$ delivered by an electric current $I$ due to a potential difference $V$ can be expressed as:

$P=V\cdot I$

Example Problems

Projectile Motion

Projectile motion problems involve analyzing the effects of gravity and may also consider electric forces in certain cases where charged particles move in an electric field.

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