Electric Potential: Concepts, Calculations, and Applications

Electric Potential

Electric Potential Energy versus Electric Potential

Electric potential energy is the energy stored due to the position of a charge in an electric field, analogous to gravitational potential energy. Electric potential, also known as voltage, is the potential energy per unit charge at a point in space.

• Electric Potential Energy (U): The energy a charge has due to its position in an electric field.

• Electric Potential (V): Defined as $V = \frac{U}{q}$, where U is electric potential energy and q is the charge.

• Units: Electric potential energy is measured in joules (J), electric potential in volts (V), where $1\,\text{V} = 1\,\text{J}/\text{C}$.

• Example: A charge near an infinitely large sheet of charge experiences a uniform electric field and thus a change in electric potential energy as it moves.

Recall: Work and Path Integration

Work is the energy transferred to or from an object via the application of force along a displacement. In the context of electric fields, work is done when a charge moves in the field.

• Work Formula: $W = \vec{F} \cdot \vec{s} = F s \cos \theta$

• Path Integration: For variable force, $W = \int_{i}^{f} \vec{F} \cdot d\vec{s}$

• Work in Electric Fields: The work done depends on the path taken and the direction of force relative to displacement.

• Example: If the force is perpendicular to displacement, no work is done.

Gravitational vs. Electric Potential Energy

Gravitational potential energy is a familiar analogy for electric potential energy. Both depend on position within a field.

• Gravitational Potential Energy: $U_g = mgh$

• Electric Potential Energy: $U_e = qEd$ for a uniform field

• Example: A mass raised above the ground stores gravitational potential energy; a charge moved in an electric field stores electric potential energy.

Calculating the Potential from the Field

The electric potential difference between two points is related to the electric field and the path taken between those points.

• Potential Difference: $V_f - V_i = -\int_{i}^{f} \vec{E} \cdot d\vec{s}$

• Uniform Field: $\Delta V = -E \Delta s$

• Interpretation: Moving a charge against the field increases potential energy; moving with the field decreases it.

Potential due to a Point Charge

The electric potential at a distance r from a point charge q is given by:

• Formula: $V(r) = \frac{kq}{r}$, where $k = \frac{1}{4\pi\epsilon_0}$

• Sign: Positive for $q > 0$, negative for $q < 0$

• Example: The potential decreases with distance from the charge.

Equipotential Surfaces

Equipotential surfaces are regions where the electric potential is constant. No work is required to move a charge along an equipotential surface.

• Properties: Equipotentials are always perpendicular to electric field lines.

• Work: $W = 0$ when moving along an equipotential.

• Example: Spherical surfaces around a point charge are equipotentials.

Calculating the Field from the Potential

The electric field is related to the rate of change of electric potential (the gradient).

• Formula: $\vec{E} = -\nabla V$

• Interpretation: The field points in the direction of greatest decrease of potential.

Potentials on, within, and near Conductors

Conductors in electrostatic equilibrium have special properties regarding electric potential.

• Inside a Conductor: The electric field is zero, and the potential is constant throughout.

• On the Surface: The surface of a conductor is an equipotential.

• Example: A conducting shell maintains the same potential inside and on its surface.

Potential due to a Group of Point Charges

The total electric potential at a point due to multiple point charges is the algebraic sum of the potentials from each charge.

• Formula: $V = \sum_{i} \frac{kq_i}{r_i}$

• Superposition Principle: Potentials add as scalars, not vectors.

• Example: At the midpoint between two equal charges, the potential is the sum from both.

Potential due to a Continuous Charge Distribution

For continuous distributions, the potential is found by integrating over the charge distribution.

• Line Distribution: $V = \int \frac{k dq}{r}$, where $dq = \lambda dx$

• Surface Distribution: $dq = \sigma dA$

• Volume Distribution: $dq = \rho dV$

• Example: The potential on the axis of a charged ring or disk is calculated by integrating contributions from each element of charge.

Summary Table: Key Formulas and Concepts

Concept	Formula	Units
Work	$W = \vec{F} \cdot \vec{s}$	J (joules)
Gravitational Potential Energy	$U_g = mgh$	J (joules)
Electric Potential Energy	$U_e = qEd$	J (joules)
Electric Potential (Point Charge)	$V = \frac{kq}{r}$	V (volts)
Potential Difference	$\Delta V = -\int \vec{E} \cdot d\vec{s}$	V (volts)
Electric Field from Potential	$\vec{E} = -\nabla V$	N/C (newtons per coulomb)

Additional info:

• These notes expand on the provided slides and images, filling in academic context and definitions for clarity.

• Examples and formulas are provided for both discrete and continuous charge distributions, as well as for conductors and equipotential surfaces.