Electric Potential and Electric Potential Energy (Chapter 25) – Study Notes

Electric Potential and Electric Potential Energy

Introduction to Electric Potential Energy

Electric potential energy is a form of energy stored by electric charges due to their positions in an electric field. When a charge is placed in an electric field, it experiences a force that can cause it to accelerate, converting potential energy into kinetic energy.

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

• Kinetic Energy (K): The energy of motion, given by $K = \frac{1}{2}mv^2$.

• Force on a Charge: $F = qE$ where q is the charge and E is the electric field.

• Acceleration: $a = \frac{q}{m}E$

• Energy Conversion: Electric potential energy is converted into kinetic energy as the charge moves in the field.

Example: A positive charge released in a uniform electric field accelerates toward the negative plate, increasing its kinetic energy as its potential energy decreases.

Work and Energy in Electric Fields

Work is required to move a charge within an electric field, and this work is related to changes in kinetic and potential energy.

• Work Done by a Force: $W = \int_A^B \vec{F} \cdot d\vec{s}$

• Change in Kinetic Energy: $W = K_f - K_i$

• Change in Potential Energy: $\Delta U = U_f - U_i = -W$

• Conservative Forces: The electric force is conservative, meaning the work done depends only on the initial and final positions, not the path taken.

Example: Moving a charge from point A to point B in a uniform field, the work done by the field equals the decrease in potential energy.

Work Done by a Constant Force

When a constant force acts on a particle, the work done depends on the displacement and the angle between the force and displacement vectors.

• Work Formula: $W = F \Delta r \cos \theta$

• Key Point: Maximum work is done when force and displacement are in the same direction ($\theta = 0$).

Electric Potential Energy in a Uniform Electric Field

In a uniform electric field, the potential energy of a charge changes as it moves between two points. The field does work on the charge, converting potential energy into kinetic energy.

• Work Done by Electric Field: $W = \vec{F} \cdot \Delta \vec{r} = F \Delta r$

• Change in Potential Energy: $\Delta U = qE(s_f - s_i)$

• Potential Energy at Position: $U = U_0 + qEs$

Example: A charge inside a parallel-plate capacitor 'falls' toward the negative plate, losing potential energy and gaining kinetic energy.

Comparing Potential Energy of Charges in a Capacitor

The potential energy of a charge in a capacitor depends on its position relative to the plates.

• Positive Charges: Higher potential energy closer to the positive plate; potential energy increases with distance from the negative plate.

• Negative Charges: Higher (less negative) potential energy closer to the negative plate.

Electric Potential Energy of Two Point Charges

The potential energy of a system of two point charges depends on their separation and the magnitude of the charges.

• Potential Energy Formula: $U_{elec} = k \frac{q_1 q_2}{r}$, where $k = \frac{1}{4\pi\varepsilon_0}$

• Work to Assemble Charges: The work required to bring two charges from infinity to a separation $r$ is equal to their potential energy.

• Like Charges: Potential energy is positive; work must be done against the repulsive force.

• Opposite Charges: Potential energy is negative; work is done by the attractive force.

Potential Energy of Multiple Point Charges

For a system of multiple point charges, the total potential energy is the sum of the potential energies for each pair of charges.

• Total Potential Energy: $U = \sum_{i

• Assembly Process: Bring charges one by one, each time calculating the work done against the field of the existing charges.

Electric Potential and Potential Difference

Electric potential is a scalar quantity that represents the potential energy per unit charge at a point in an electric field. The potential difference between two points is the work required to move a unit charge between them.

• Potential Difference: $\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{s}$

• Unit: 1 Volt (V) = 1 Joule/Coulomb (J/C)

• Relation to Electric Field: In a uniform field, $E = -\frac{\Delta V}{\Delta s}$

• Electron-Volt: 1 eV = $1.602 \times 10^{-19}$ Joules; energy gained by an electron moving through a potential difference of 1 V.

Electric Potential in a Uniform Field and Capacitors

In a parallel-plate capacitor, the potential difference between the plates is related to the electric field and the separation between the plates.

• Potential Difference: $V = Ed$

• Potential at a Point: Defined relative to infinity, where $V_{\infty} = 0$.

Electric Potential Due to Point Charges

The electric potential at a point due to a point charge is given by:

• Potential at Distance r: $V = k \frac{q}{r}$

• Superposition Principle: For multiple charges, $V_{net} = \sum_{i=1}^N k \frac{q_i}{r_i}$

Electric Potential for Continuous Charge Distributions

For a continuous distribution of charge, the total electric potential at a point is found by integrating the contributions from each infinitesimal charge element.

• Potential Due to Charge Element: $dV = k \frac{dq}{r}$

• Total Potential: $V = k \int \frac{dq}{r}$

Electric Potential of a Uniformly Charged Ring

The electric potential at the center of a uniformly charged ring (or arc) can be calculated by integrating over the charge distribution.

• Potential at Center: $V = k \frac{Q}{R}$, where $Q$ is the total charge and $R$ is the radius.

• For a semicircular ring: The potential is proportional to the total charge and inversely proportional to the radius.

Summary Table: Key Equations and Concepts

Concept	Equation	Description
Force on Charge	$F = qE$	Force experienced by a charge in an electric field
Kinetic Energy	$K = \frac{1}{2}mv^2$	Energy of motion
Work Done	$W = \int_A^B \vec{F} \cdot d\vec{s}$	Work to move a charge
Potential Energy (2 charges)	$U = k \frac{q_1 q_2}{r}$	Energy due to two point charges
Potential Difference	$\Delta V = -\int_A^B \vec{E} \cdot d\vec{s}$	Work per unit charge
Potential (point charge)	$V = k \frac{q}{r}$	Potential at distance r from a charge
Potential (continuous)	$V = k \int \frac{dq}{r}$	Potential due to a charge distribution
Potential (capacitor)	$V = Ed$	Potential difference across plates

Additional info: These notes expand on the original slides by providing full definitions, formulas, and context for each concept, ensuring a self-contained study guide suitable for college-level physics students.