Electric Potential and Gauss’ Law – Study Notes

Electric Potential and Gauss’ Law

Overview

This chapter covers the fundamental concepts of electric potential energy, electric potential, equipotential surfaces, and the application of Gauss’ Law in electrostatics. These principles are essential for understanding the behavior of electric fields and potentials in various charge distributions and conductor configurations.

Electric Potential Energy

Definition and Properties

• Electric potential energy (Ue) is the energy stored in a system of charges due to their positions relative to each other.

• This energy depends only on the configuration of the charges, not on the path taken to assemble them (the electric force is conservative).

• The reference point for potential energy is often chosen so that Ue = 0 at infinity.

For two point charges:

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

• For opposite charges (e.g., proton and electron): $U_e = -\frac{k e^2}{r}$

• Bringing charges closer together (decreasing r) changes the potential energy:

$\Delta U_e = U_{ef} - U_{ei} = k e^2 \left( \frac{r_f - r_i}{r_i r_f} \right)$

• If both charges are of the same sign, $U_e > 0$ and work must be done by an external agent to bring them closer.

Potential Energy of Multiple Point Charges

• To find the total potential energy of a system of n point charges:

$U_e = \frac{1}{4\pi\varepsilon_0} \sum_{i=1}^n \sum_{j>i}^n \frac{q_i q_j}{r_{ij}}$

• Each unique pair of charges is counted once.

Example: For three charges at the vertices of a right triangle, sum the potential energies for each pair using their respective distances.

Electric Potential

Definition and Units

• Electric potential (V) is the electric potential energy per unit charge:

$V = \frac{U_e}{q_{test}}$

• Unit: volt (V), where 1 V = 1 J/C.

• In atomic and subatomic physics, the electron-volt (eV) is commonly used: 1 eV = $1.60 \times 10^{-19}$ J.

Potential Due to Point Charges

• For a point charge Q:

$V = \frac{kQ}{r}$

• For a collection of point charges:

$V = \sum_{i=1}^n V_i = \frac{1}{4\pi\varepsilon_0} \sum_{i=1}^n \frac{Q_i}{r_i}$

Potential Difference and Energy Change

• When a charge q moves through a potential difference ΔV, its potential energy changes by:

$\Delta U_e = q \Delta V$

• Example: If a charge of +2e moves from a point at 500.0 kV to 200.0 kV, the change in kinetic energy is:

$\Delta K = -q(V_s - V_p) = -(+2e)(200.0 - 500.0)\text{ kV} = +600\text{ keV} = +9.6 \times 10^{-14}\text{ J}$

Example: Electron Acceleration

• An electron accelerated from rest through a potential difference ΔV reaches speed vf:

$\frac{1}{2} m v_f^2 = -q \Delta V$

$\Delta V = -\frac{m v_f^2}{2q}$

• For $v_f = 7.26 \times 10^6$ m/s, $\Delta V = +150$ V (electron moves from low to high V).

Finding Electric Potential from Electric Field

General Relationship

• If the electric field E is known, the potential difference between points a and b is:

$V_b - V_a = -\int_a^b \vec{E} \cdot d\vec{l}$

• This integral is path-independent for electrostatic fields.

Uniform Electric Field

• For a uniform field E along the y-direction:

$V = -\int_0^y E \cdot dy = -Ey$

• Potential decreases linearly with distance in the direction of the field.

Potential from a Distribution of Point Charges

• Sum the contributions from each charge:

$V = \sum_{i=1}^n \frac{1}{4\pi\varepsilon_0} \frac{Q_i}{r_i}$

• Example: Find the potential at a point due to three charges arranged on a coordinate plane.

Equipotential Surfaces

Definition and Properties

• Equipotential surfaces are surfaces where the electric potential is constant.

• No work is required to move a charge along an equipotential surface.

• The electric field is always perpendicular to equipotential surfaces and points in the direction of decreasing potential.

Gauss’ Law

Statement and Application

• Gauss’ Law relates the electric flux through a closed surface to the charge enclosed:

$\Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$

• Useful for calculating electric fields in cases with high symmetry (spherical, cylindrical, planar).

• Choose a Gaussian surface so that E and dA are parallel and E is constant over the surface.

Electric Flux

• Electric flux through a surface is a measure of the number of electric field lines passing through that surface:

$\Phi_E = \int \vec{E} \cdot d\vec{A}$

• For a closed surface, the direction of the area vector is outward.

Examples of Gauss’ Law Applications

• Point charge: Use a spherical Gaussian surface to find the field at distance r from the charge.

• Infinite line of charge: Use a cylindrical Gaussian surface to find the field at distance r from the line.

• For finite objects or non-symmetric cases, Gauss’ Law is still valid but may not simplify calculations.

Properties of Conductors

Key Properties

• All excess charge resides on the surface of an isolated conductor.

• The electric field inside a conductor is zero in electrostatic equilibrium.

• The surface of a conductor is an equipotential.

• The electric field just outside a conductor is perpendicular to the surface.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Electric Potential Energy (2 charges)	$U_e = \frac{k q_1 q_2}{r}$	Energy due to two point charges
Electric Potential (point charge)	$V = \frac{kQ}{r}$	Potential at distance r from charge Q
Potential Difference	$\Delta U_e = q \Delta V$	Energy change for charge q moving through ΔV
Potential from E-field	$V_b - V_a = -\int_a^b \vec{E} \cdot d\vec{l}$	Potential difference from field
Gauss’ Law	$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\varepsilon_0}$	Relates flux to enclosed charge

Additional info:

• For continuous charge distributions, integrals replace sums in the formulas for potential and energy.

• Equipotential surfaces and field lines provide a visual understanding of electric fields and potentials.