2.5 Electric potential

Definition of electric potential

Electric potential describes the potential energy per unit charge at a point in an electric field. It tells you how much energy a charge would have if placed at that location, without needing to know the specific charge involved. This makes it a scalar quantity, which is often easier to work with than the electric field vector.

Potential energy vs kinetic energy

Potential energy is stored energy due to a charge's position within an electric field. Kinetic energy is the energy of motion. As a charged particle moves through an electric field, it converts between these two forms.

• A positive charge released near another positive charge accelerates away, converting potential energy into kinetic energy.
• This conversion is exactly what happens in devices like particle accelerators and cathode ray tubes: a voltage sets up the field, and charges gain speed as they move through it.

Work and electric potential

Electric potential is defined as the work per unit charge required to move a test charge from a reference point (usually infinity) to a specific location:

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

• Work done against the electric field increases a charge's potential energy (like lifting a ball against gravity).
• Work done by the electric field decreases potential energy and increases kinetic energy.
• Units: joules per coulomb (J/C), which we call volts (V).

Electric potential difference

Electric potential difference, commonly called voltage, is the change in electric potential between two points. It's what drives current through circuits and determines how much energy is transferred to charges moving between those points.

Voltage concept

Voltage is the energy difference per unit charge between two locations:

$\Delta V = V_B - V_A$

Think of it like the height difference between two shelves. A ball on the higher shelf has more gravitational potential energy. Similarly, a charge at higher electric potential has more electric potential energy.

• Voltage can be positive, negative, or zero depending on the two points you're comparing.
• A positive charge naturally moves from high potential to low potential (just as a ball rolls downhill).

Units of electric potential

• Measured in volts (V), named after Alessandro Volta.
• One volt equals one joule per coulomb: $1 \text{ V} = 1 \text{ J/C}$
• Common multiples: millivolts (mV, $10^{-3}$ V) and kilovolts (kV, $10^{3}$ V).
• When a battery is rated at 9 V, that means it maintains a 9 V potential difference between its terminals.

Equipotential surfaces

Equipotential surfaces are regions where every point has the same electric potential. They're useful for visualizing how potential is distributed in space and for predicting how charges will move.

Equipotential lines in 2D

On a flat diagram, equipotential lines are contours of constant potential, much like elevation contour lines on a topographic map.

• They are always perpendicular to electric field lines. This has to be true because no work is done moving a charge along an equipotential (the potential doesn't change), and work is only zero when motion is perpendicular to the force.
• Where equipotential lines are packed closely together, the electric field is stronger (the potential is changing rapidly over a short distance).

Equipotential surfaces in 3D

In three dimensions, equipotential lines become surfaces:

• Point charge: equipotential surfaces are concentric spheres centered on the charge.
• Line charge: equipotential surfaces are coaxial cylinders.
• Uniform field (like between parallel plates): equipotential surfaces are flat planes perpendicular to the field.

No work is done moving a charge anywhere along an equipotential surface.

Calculating electric potential

To find the electric potential at a point, you either integrate the electric field along a path or sum contributions from individual charges. You always need a reference point where $V = 0$, which is typically at infinity (for point charges) or at ground.

Point charges

The potential at distance $r$ from a single point charge $q$ is:

$V = k\frac{q}{r}$

where $k = 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2$ is Coulomb's constant.

• Notice this is a $1/r$ relationship, not $1/r^2$ like the electric field. Potential falls off more slowly with distance than the field does.
• $V$ is positive for positive charges and negative for negative charges.
• Because potential is a scalar, you don't need to worry about direction when adding contributions from multiple charges.

Continuous charge distributions

For objects like charged rods, disks, or spheres, you integrate over infinitesimal charge elements $dq$:

$V = \int k\frac{dq}{r}$

1. Express $dq$ in terms of the appropriate charge density ($\lambda$ for linear, $\sigma$ for surface, $\rho$ for volume).
2. Write $r$ as the distance from each $dq$ to the point where you want the potential.
3. Exploit symmetry wherever possible to simplify the integral.

Superposition principle

Because potential is a scalar, the total potential at any point is simply the algebraic sum of potentials from all sources:

$V_{total} = V_1 + V_2 + V_3 + \ldots$

This is one of the biggest advantages of working with potential instead of the electric field. You just add numbers rather than adding vectors.

Electric field vs potential

Electric field and potential are two ways of describing the same electrostatic situation. The field tells you about forces; the potential tells you about energy. Knowing one lets you find the other.

Relationship between E and V

• The electric field points from regions of high potential toward regions of low potential.
• The magnitude of the field is related to how quickly the potential changes with distance. A large potential change over a short distance means a strong field.
• The work done by the electric field on a charge $q$ moving through a potential difference $\Delta V$ is:

$W = -q\Delta V$

Gradient of potential

The electric field is the negative gradient of the potential:

$\vec{E} = -\nabla V$

In Cartesian coordinates, this expands to:

$\vec{E} = -\left(\frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k}\right)$

In simpler cases (one dimension), this reduces to $E_x = -\frac{dV}{dx}$. If you know how potential varies with position, you can find the field by taking the derivative.

Potential energy of charge systems

The potential energy of a system of charges represents the total work required to assemble that configuration by bringing each charge in from infinity.

Two-charge systems

For two point charges $q_1$ and $q_2$ separated by distance $r$:

$U = k\frac{q_1 q_2}{r}$

• Like charges (both positive or both negative): $U > 0$. You had to do positive work to push them together against their repulsion.
• Opposite charges: $U < 0$. The charges attract, so energy was released as they came together.

Multiple-charge systems

For three or more charges, sum the potential energy of every unique pair:

$U_{total} = \frac{1}{2}\sum_{i}\sum_{j \neq i} k\frac{q_i q_j}{r_{ij}}$

The factor of $1/2$ corrects for the fact that each pair gets counted twice in the double sum. For a system of three charges, you'd have three pairs to add up. For four charges, six pairs, and so on.

Conductors and electric potential

Conductors have free-moving charge carriers (electrons), which gives them special electrostatic properties.

Equipotential nature of conductors

In electrostatic equilibrium (no charges moving), the entire conductor is at a single potential. This includes both the surface and the interior.

Why? If there were a potential difference between two points inside the conductor, charges would flow until that difference disappeared. The consequence: the electric field inside a conductor is zero in equilibrium. This is the principle behind electrostatic shielding.

Charge distribution on conductors

• All excess charge resides on the surface of a conductor. None sits in the interior.
• Charge density is highest where the surface curves most sharply. A pointed tip concentrates charge and creates a very strong local field. This is the lightning rod effect.
• A hollow conductor shields its interior from external fields regardless of what's happening outside. This is how a Faraday cage works.

Capacitance and potential

Capacitance measures how much charge a system can store per volt of potential difference applied. It's central to energy storage in circuits.

Definition of capacitance

$C = \frac{Q}{V}$

• $Q$ is the magnitude of charge on either plate (they carry equal and opposite charges).
• Measured in farads (F): $1 \text{ F} = 1 \text{ C/V}$. One farad is enormous; real capacitors are typically in the picofarad (pF) to microfarad ($\mu$F) range.
• Capacitance depends only on the geometry of the capacitor and the material between the plates, not on $Q$ or $V$.

Parallel plate capacitor

Two parallel conducting plates of area $A$ separated by distance $d$:

$C = \frac{\epsilon_0 A}{d}$

where $\epsilon_0 = 8.85 \times 10^{-12} \text{ F/m}$ is the permittivity of free space.

• Larger plates (bigger $A$) store more charge at the same voltage.
• Closer plates (smaller $d$) also increase capacitance.
• The electric field between the plates is approximately uniform, which makes this geometry especially useful for calculations and for understanding more complex capacitor shapes.

Applications of electric potential

Electron volt as energy unit

The electron volt (eV) is the energy gained by a single electron accelerated through a potential difference of 1 volt:

$1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$

This unit is far more practical than joules at the atomic scale. For example, the ionization energy of hydrogen is about 13.6 eV, which is much easier to work with than $2.18 \times 10^{-18}$ J.

Cathode ray tubes

Cathode ray tubes (CRTs) demonstrate electric potential in action:

1. A heated cathode emits electrons (thermionic emission).
2. A large voltage (thousands of volts) accelerates the electrons toward the screen.
3. Electric or magnetic fields deflect the beam to specific positions.
4. Electrons strike a phosphor screen, producing visible light.

The kinetic energy gained by each electron equals $q\Delta V$, directly linking voltage to particle speed.

Particle accelerators

Particle accelerators use electric potentials to push charged particles to very high energies:

• Linear accelerators pass particles through a series of accelerating gaps, each adding energy via a potential difference.
• Circular accelerators (cyclotrons, synchrotrons) combine magnetic fields for steering with oscillating electric fields for repeated acceleration.
• These machines are used in particle physics research, materials science, and medical treatments like proton therapy for cancer.

Electrostatic potential energy

Conservation of energy

In an isolated electrostatic system, total energy is conserved:

$K_i + U_i = K_f + U_f$

A charge moving through an electric field trades potential energy for kinetic energy (or vice versa). This lets you calculate the speed of a particle at any point if you know the potential there, without needing to track forces along the entire path.

Potential energy in electric fields

For a point charge $q$ placed at a location where the potential is $V$:

$U = qV$

This is the energy of that charge due to the external field. For a positive charge at a positive potential, $U > 0$. For continuous charge distributions, the energy is found by integrating:

$U = \int \rho V \, dV$

where $\rho$ is the volume charge density and $dV$ is a volume element.

Dielectrics and electric potential

A dielectric is an insulating material placed between capacitor plates. It modifies the electric field and increases the capacitor's ability to store charge.

Effect on capacitance

Inserting a dielectric increases capacitance by a factor of $\kappa$ (the dielectric constant):

$C = \kappa C_0$

where $C_0$ is the capacitance without the dielectric. This happens because the dielectric's molecules polarize in the field, partially canceling the internal electric field and allowing more charge to accumulate on the plates for the same voltage.

• $\kappa = 1$ for vacuum, about 3.4 for paper, and around 80 for water.

Energy storage in dielectrics

The energy stored in a capacitor is:

$U = \frac{1}{2}CV^2 = \frac{1}{2}\kappa C_0 V^2$

With a dielectric, you can store more energy at the same voltage. However, every dielectric has a dielectric strength, which is the maximum electric field it can withstand before it breaks down and conducts. This sets the upper limit on the voltage you can safely apply, and therefore the maximum energy density of the capacitor.