Electric Potential and Gauss’ Law – Study Notes for PHY 131

Electric Potential Energy and Electric Potential

Electric Potential Energy

Electric potential energy (Ue) is the energy stored in a system due to the arrangement of electric charges. This energy depends only on the positions of the charges, not on the path taken to assemble them, because the electric force is conservative.

• Reference Point: The potential energy is often defined as zero at infinity for two point charges.

• Formula for Two Point Charges: $U_e = \frac{k q_1 q_2}{r}$ where k is Coulomb's constant, q1 and q2 are the charges, and r is the separation.

• Change in Potential Energy: Bringing opposite charges closer together decreases potential energy; for like charges, potential energy increases and external work is required.

• Work and Energy: $\Delta U = -W = -\int \vec{F} \cdot d\vec{l}$

Example: Proton and Electron

• For a proton and electron: $U_e = -\frac{k e^2}{r}$

• Change in energy when brought closer: $\Delta U_e = U_{ef} - U_{ei} = k e^2 \left( \frac{r_f - r_i}{r_i r_f} \right) < 0$

• This is analogous to a mass falling in a gravitational field; the field does positive work.

Like Charges

• If both particles have the same sign, $\Delta U_e > 0$; work must be done by an external agent to bring them closer.

Potential Energy of Multiple Point Charges

To calculate the potential energy of a system of n point charges:

• Place the first charge; no work is required.

• Bring in each subsequent charge, calculating the work done against the electric field of the existing charges.

• General Formula: $U_e = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \sum_{j>i}^{n} \frac{q_i q_j}{r_{ij}}$

Example: Three Charges in a Right Triangle

• Sum the potential energies for each pair: $U_e = \frac{k q_1 q_2}{r_{12}} + \frac{k q_1 q_3}{r_{13}} + \frac{k q_2 q_3}{r_{23}}$

Electric Potential

Definition and Units

Electric potential (V) is the electric potential energy per unit charge. It is a scalar quantity measured in volts (V), where 1 V = 1 J/C.

• Formula: $V = \frac{U_e}{q_{test}}$

• For a point charge: $V = \frac{kQ}{r}$

• Electron-volt (eV): A convenient energy unit in atomic physics: $1 \text{ eV} = e \times 1 \text{ V} = 1.60 \times 10^{-19} \text{ J}$

Potential Difference and Energy Change

• When a charge q moves through a potential difference ΔV, its potential energy changes by: $\Delta U_e = q \Delta V$

Potential Due to Multiple Point Charges

• Formula: $V = V_1 + V_2 + ... + V_n = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{Q_i}{r_i}$

Example: Kinetic Energy Change

• Given points P and S at different potentials, a charge +2e moves from P to S: $K_f - K_i = U_i - U_f = -\Delta U = -q \Delta V = -q(V_s - V_p)$ $= -(+2e)(200.0 - 500.0) \text{ kV}$ $= +600 \text{ keV} = +9.6 \times 10^{-14} \text{ J}$

Example: Electron Accelerated by Potential Difference

• Electron accelerated from rest to $7.26 \times 10^6$ m/s: $K_f = -\Delta U = -q \Delta V$ $\frac{1}{2} m v_f^2 = -q \Delta V$ $\Delta V = \frac{m v_f^2}{2q} = \frac{(9.11 \times 10^{-31} \text{ kg})(7.26 \times 10^6 \text{ m/s})^2}{2(-1.60 \times 10^{-19} \text{ C})}$ $= +150 \text{ Volts}$ Note: The electron moves from low V to high V.

Finding Electric Potential from Electric Field

General Relationship

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

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

Uniform Electric Field

• For a uniform field along the y-axis: $V = -\int_0^y E \cdot dy = -Ey$ V = 0 when y = 0

Potential Due to a Distribution of Point Charges

Superposition Principle

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

• Formula: $V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{Q_i}{r_i}$

• Example: For charges at coordinates, find the potential at a specified point by summing contributions from each charge.

Summary Table: Key Formulas and Concepts

Concept	Formula	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 from E-field	$V_b - V_a = -\int_a^b \vec{E} \cdot d\vec{l}$	Relates field and potential
Potential Energy Change	$\Delta U_e = q \Delta V$	Energy change for charge q
Potential (multiple charges)	$V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{Q_i}{r_i}$	Sum of potentials from all charges

Additional info:

• These notes cover foundational concepts in electrostatics, including energy, potential, and the relationship between electric field and potential. They are essential for understanding Gauss' Law and the behavior of conductors, which are likely covered in subsequent sections.