Electric Potential and Its Relationship to Electric Field

Chapter 25: The Electric Potential

Electric Potential and Electric Potential Energy

The concept of electric potential is central to understanding how energy is stored and transferred in electric systems. Electric potential energy is the energy a charge possesses due to its position in an electric field.

• Electric Potential Energy (U): For two point charges, $q_1$ and $q_2$ separated by distance $r$, the potential energy is given by: $U = \frac{k q_1 q_2}{r}$ where $k$ is Coulomb's constant.

• Multiple Point Charges: The total potential energy is the sum over all unique pairs: $U_{total} = \sum_{iwhere$r_{ij}$is the distance between charges$q_i$and$q_j$.

• Electric Potential (V): Defined as the potential energy per unit charge: $V = \frac{U}{q}$ The unit is the volt (V), where $1\,\text{V} = 1\,\text{J}/\text{C}$.

• Electric Potential of a Point Charge: $V = \frac{k q}{r}$ where $q$ is the source charge and $r$ is the distance from the charge.

• Electric Potential of Many Charges: The potential at a point is the sum of the potentials due to each charge: $V = \sum_{i} \frac{k q_i}{r_i}$ where $r_i$ is the distance from charge $q_i$ to the point.

• Electric Potential of a Charged Sphere: Outside a uniformly charged sphere of radius $R$ and charge $Q$: $V = \frac{k Q}{r}$ for $r \geq R$ If the surface potential $V_0$ is known: $V = V_0 \frac{R}{r}$ for $r \geq R$

• Electric Potential of a Dipole: For a dipole in a uniform electric field, the change in potential energy is: $\Delta U = -p E \cos\theta$ where $p$ is the dipole moment, $E$ is the electric field strength, and $\theta$ is the angle between $p$ and $E$.

Example: A proton released from the surface of a charged sphere will accelerate away due to the repulsive force, converting potential energy into kinetic energy.

Energy Conservation in Electric Fields

As a charged particle moves through a changing electric potential, the total energy is conserved.

• Relationship: $\Delta K + \Delta U = 0$ where $K$ is kinetic energy and $U$ is potential energy.

• Application: If a positive charge moves to a region of higher potential, its kinetic energy decreases.

Example: A proton moving through a potential difference of $100\,\text{V}$ will gain or lose kinetic energy depending on the direction of motion relative to the electric field.

Electric Potential Inside a Parallel-Plate Capacitor

A parallel-plate capacitor creates a uniform electric field between its plates, resulting in a linear change in electric potential.

• Electric Field: $E = \frac{\Delta V}{d}$ where $\Delta V$ is the voltage across the plates and $d$ is the separation.

• Electric Potential: $V(s) = E s$ where $s$ is the distance from the negative electrode.

• Units: Electric field can be expressed as $\text{V}/\text{m}$ or $\text{N}/\text{C}$, which are equivalent.

Example: The potential increases linearly from $V = 0$ at the negative plate to $V = Ed$ at the positive plate.

Chapter 26: Potential and Field

Connecting Electric Potential and Electric Field

The electric field and electric potential are closely related. The electric field is the spatial rate of change of the electric potential.

• Potential Difference: The potential difference between two points is: $\Delta V = - \int_{s_i}^{s_f} \vec{E} \cdot d\vec{s}$

• Finding Potential from Field: If the electric field is known, the potential difference can be found by integrating the field along a path.

• Finding Field from Potential: The electric field is the negative gradient of the potential: $\vec{E} = -\nabla V$ In one dimension: $E = -\frac{dV}{ds}$

Example: If $V(x)$ is plotted versus $x$, the electric field $E_x$ is the negative slope of the graph.

Equipotential Surfaces

Equipotential surfaces are regions where the electric potential is constant. The electric field is always perpendicular to these surfaces.

• Properties:

  • Equipotential surfaces never intersect.

  • The electric field points from higher to lower potential.

  • The spacing of equipotential surfaces indicates the strength of the field: closer surfaces mean stronger fields.

Example: Near a charged conductor, equipotential surfaces match the shape of the conductor.

Conductors in Electrostatic Equilibrium

When a conductor is in electrostatic equilibrium, several important properties emerge.

• All excess charge resides on the surface.

• The surface is an equipotential.

• The electric field inside the conductor is zero.

• The external electric field is perpendicular to the surface at the surface.

• The electric field is strongest at sharp points or corners.

Example: A corona discharge occurs at pointed metal tips where the electric field is very strong.

Summary Table: Key Relationships

Quantity	Formula	Description
Electric Potential (Point Charge)	$V = \frac{k q}{r}$	Potential at distance $r$ from charge $q$
Electric Potential (Many Charges)	$V = \sum_{i} \frac{k q_i}{r_i}$	Sum of potentials from all charges
Potential Difference	$\Delta V = - \int \vec{E} \cdot d\vec{s}$	Change in potential along a path
Electric Field from Potential	$\vec{E} = -\nabla V$	Field is negative gradient of potential
Potential Energy (Two Charges)	$U = \frac{k q_1 q_2}{r}$	Energy due to interaction of two charges
Potential Energy (Dipole)	$U = -p E \cos\theta$	Energy of dipole in uniform field

Additional info:

• Energy conservation is a fundamental principle in electric systems: as a charge moves through a potential difference, its kinetic and potential energies change, but their sum remains constant.

• Equipotential surfaces are useful for visualizing electric fields and understanding the behavior of conductors.

• In practice problems, always apply energy conservation and the relationships between field and potential to solve for unknowns.