Electric Potential, Potential Energy, and Electric Field: Study Notes for Physics

Electric Potential Energy

Definition and Properties

Electric potential energy is the energy stored due to the interaction between a charged particle and an electric field. It is a form of potential energy that arises from the position of a charge within an electric field.

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

• Work and Path Independence: The work done by the electric force when moving a charge between two points is independent of the path taken, because the electric force is conservative.

• Uniform Electric Field: In a uniform electric field, the work done to move a charge q over a distance d is given by: $W = qEd$

Example: Moving a charge between two parallel plates in a capacitor.

Electric Potential Energy of Point Charges

Two Point Charges

The electric potential energy between two point charges depends on their magnitudes, separation, and the reference point chosen for zero potential energy.

• Formula: $U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r}$ where $q_1$ and $q_2$ are the charges, $r$ is the distance between them, and $\varepsilon_0$ is the vacuum permittivity.

• Reference Point: Potential energy is often defined as zero when the charges are infinitely far apart ($r \to \infty$).

• Multiple Charges: For a system of more than two charges, the total potential energy is the sum of the potential energies for each pair.

• Sign of Energy: Electric potential energy can be negative, depending on the sign of the charges.

Example: Calculating the energy of a system of three charges.

Energy and Conservation of Energy

Mechanical Energy and Conservation

Mechanical energy is the sum of kinetic and potential energies in a system. Conservation of energy states that if no external work is done, the total mechanical energy remains constant.

• Kinetic Energy: $K = \frac{1}{2}mv^2$

• Conservation of Energy: If $W_{ext} = 0$, then $E_{mech, initial} = E_{mech, final}$

Example: A charged particle moving in an electric field.

Electric Potential Energy of a Dipole

Electric Dipole and Dipole Moment

An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment is a vector pointing from the negative to the positive charge.

• Dipole Moment: $\vec{p} = q\vec{d}$

• Potential Energy in an Electric Field: $U = -\vec{p} \cdot \vec{E} = -pE\cos\theta$ where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$.

Example: Rotating a water molecule (a permanent dipole) in an electric field.

Electric Potential

Definition and Properties

Electric potential is the electric potential energy per unit charge. It is a scalar field assigning a value to every point in space.

• Formula: $V = \frac{U}{q}$

• Units: Joules per Coulomb (J/C), or Volt (V)

• Relation to Electric Field: $1\ \text{V/m} = 1\ \text{N/C}$

• Potential Difference (Voltage): The difference in electric potential between two points.

• Analogy: Electric potential is analogous to gravitational potential energy ($mgh$).

Example: Calculating the speed of a proton moving through a potential difference.

Electric Potential of Continuous Charge Distributions

Calculating Potential

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

• General Formula: $V = \frac{1}{4\pi\varepsilon_0} \int \frac{dQ}{r}$ where $dQ$ is the infinitesimal charge and $r$ is the distance from $dQ$ to the point of interest.

• Procedure:

  1. Divide the total charge $Q$ into small pieces $dQ$.

  2. Express $dQ$ in terms of the geometry and charge density.

  3. Integrate over the object, using appropriate limits.

Example: Finding the potential on the axis of a uniformly charged ring or disk.

Relationship Between Electric Field and Potential

Connection and Calculations

The electric field and electric potential are related through calculus. The electric field is the negative gradient of the electric potential.

• Formula: $\vec{E} = -\nabla V$ In one dimension: $E_x = -\frac{dV}{dx}$

• Direction: The electric field points "downhill" in potential.

Example: Calculating $E$ from a graph of $V$ versus position.

Equipotential Surfaces

Definition and Properties

An equipotential surface is a surface where the electric potential is constant at every point. These are often represented as curves in 2D diagrams.

• Properties:

  • Equipotential surfaces are always perpendicular to electric field lines.

  • Closely spaced equipotentials indicate a strong electric field.

  • Equipotential surfaces never cross.

Example: Mapping equipotential lines around point charges.

Conductors in Electrostatic Equilibrium

Properties of Conductors

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

• Excess charge resides on the surface.

• The electric field inside the conductor is zero.

• The entire surface is at the same electric potential (equipotential).

• The electric field just outside the surface is perpendicular and proportional to the surface charge density.

• Charge density is highest at points of greatest curvature (sharp edges).

Example: Calculating the field near a charged conducting sphere.

Capacitance

Definition and Properties

Capacitance is the ability of a system to store electric charge per unit voltage. It depends only on the geometry of the system.

• Formula: $C = \frac{Q}{V}$

• Units: Farad (F), where $1\ \text{F} = 1\ \text{C}/\text{V}$

• Parallel-Plate Capacitor: $C = \varepsilon_0 \frac{A}{d}$, where $A$ is plate area and $d$ is separation.

Example: Calculating the capacitance and charge stored in a parallel-plate capacitor.

Capacitors in Parallel and Series

Combining Capacitors

Capacitors can be connected in parallel or series, affecting the total capacitance.

• Parallel: $C_{eq} = C_1 + C_2 + ...$ (same voltage across each)

• Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$ (same charge on each)

Example: Calculating equivalent capacitance for a network of capacitors.

Energy Stored in a Capacitor

Formulas and Applications

The energy stored in a capacitor is related to its capacitance and the voltage across it.

• Formula: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$

• Energy Density: $u = \frac{1}{2}\varepsilon_0 E^2$

• Power: Power is the rate at which energy is used or transferred: $P = \frac{dU}{dt}$

Example: Calculating energy stored and power dissipated in a camera-flash capacitor.

Dielectrics and Capacitance

Role of Dielectrics

Inserting a dielectric material between capacitor plates increases the capacitance by reducing the effective electric field.

• Dielectric Constant (κ): $C = \kappa \varepsilon_0 \frac{A}{d}$

• Dielectric Strength: Maximum electric field a material can withstand before breakdown.

Example: Calculating capacitance and energy for a capacitor filled with a dielectric.

Summary Table: Key Formulas

Concept	Formula	Units
Electric Potential Energy (point charges)	$U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r}$	J
Electric Potential	$V = \frac{U}{q}$	V (J/C)
Electric Field from Potential	$\vec{E} = -\nabla V$	V/m or N/C
Capacitance (parallel plates)	$C = \varepsilon_0 \frac{A}{d}$	F
Energy Stored in Capacitor	$U = \frac{1}{2}CV^2$	J
Capacitance with Dielectric	$C = \kappa \varepsilon_0 \frac{A}{d}$	F

Additional info: Some formulas and context have been expanded for clarity and completeness.