Chapter 26: Potential and Field – Study Notes

Potential and Field

Introduction

This chapter explores the relationship between electric potential and electric field, two fundamental concepts in electromagnetism. It covers how to calculate one from the other, the properties of conductors in electrostatic equilibrium, the function and use of capacitors, and the role of dielectrics.

Connecting Electric Potential and Electric Field

Definitions and Relationships

• Electric Field (\(\vec{E}\)): A vector field representing the force per unit charge exerted on a test charge at any point in space.

• Electric Potential (V): A scalar quantity representing the potential energy per unit charge at a point in space.

• The relationship between electric field and potential difference is given by: $\Delta V = - \int_{i}^{f} \vec{E} \cdot d\vec{s}$ where \(\Delta V\) is the potential difference between points i and f, and \(d\vec{s}\) is an infinitesimal displacement vector.

• The electric field points in the direction of decreasing potential ("downhill").

• Electric field lines are always perpendicular to equipotential surfaces.

• The field is stronger where equipotentials are closer together.

Finding Potential from the Electric Field

• The potential difference between two points can be found by integrating the electric field along a path: $\Delta V = - \int_{i}^{f} E_s ds$ where \(E_s\) is the component of \(\vec{E}\) along the path.

• Graphically, \(\Delta V\) is the negative of the area under the \(E_s\) vs. s curve.

• For a uniform field (e.g., inside a parallel-plate capacitor): $V = Es$ where s is the distance from the reference point.

Finding the Electric Field from the Potential

• The electric field is the negative gradient (slope) of the potential: $E_s = -\frac{dV}{ds}$ In three dimensions: $\vec{E} = -\nabla V$

• For a point charge: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ $E = -\frac{dV}{dr} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$

Equipotential Surfaces

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

• The electric field is always perpendicular to these surfaces.

• The closer the equipotentials, the stronger the electric field.

Properties of Conductors in Electrostatic Equilibrium

• All excess charge resides on the surface of a conductor.

• The electric field inside a conductor is zero.

• The surface of a conductor is an equipotential.

• The external electric field is perpendicular to the surface at every point.

• The field is strongest at sharp points or edges, which can lead to corona discharge.

Sources of Electric Potential

Batteries and Emf

• Emf (\(\mathcal{E}\)): The work done per unit charge to move charges from the negative to the positive terminal of a battery.

• Terminal voltage of an ideal battery: $\Delta V_{bat} = \mathcal{E}$

• When batteries are connected in series, their voltages add: $V_{total} = V_1 + V_2 + ...$

Capacitance and Capacitors

Definition and Properties

• A capacitor consists of two conductors (electrodes) separated by an insulator.

• Capacitance (C) is the ratio of charge to potential difference: $C = \frac{Q}{\Delta V}$

• SI unit: farad (F), where $1\ \mathrm{F} = 1\ \mathrm{C}/\mathrm{V}$

• For a parallel-plate capacitor: $C = \epsilon_0 \frac{A}{d}$ where A is plate area, d is separation, and $\epsilon_0$ is the vacuum permittivity.

Charging a Capacitor

• When connected to a battery, a capacitor charges until the voltage across it equals the battery voltage.

• Capacitance always refers to the fully charged state.

Combinations of Capacitors

• Parallel: $C_{eq} = C_1 + C_2 + ...$

• Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$

Energy Stored in a Capacitor

• The energy stored is: $U = \frac{1}{2} C (\Delta V)^2 = \frac{1}{2} Q \Delta V = \frac{Q^2}{2C}$

• The energy is stored in the electric field between the plates.

• Energy density in the field: $u = \frac{1}{2} \epsilon_0 E^2$ where E is the electric field strength.

Dielectrics

Role and Properties

• A dielectric is an insulating material placed between capacitor plates.

• Dielectrics increase capacitance by a factor called the dielectric constant (\(\kappa\)): $C = \kappa C_0$ where $C_0$ is the capacitance without the dielectric.

• Dielectrics become polarized in an electric field, reducing the effective field and voltage for the same charge.

• All materials have a dielectric strength, the maximum field they can withstand without breakdown (spark).

Table: Properties of Dielectrics (Sample)

Material	Dielectric Constant (\(\kappa\))	Dielectric Strength (V/m)
Vacuum	1.0	---
Air	1.0006	3 × 106
Distilled Water	80	~107
Glass	5–10	107
Additional info: Table values inferred from standard sources.

Kirchhoff’s Loop Law

• The sum of all potential differences around a closed loop is zero: $\sum \Delta V = 0$

• This is a consequence of the conservation of energy.

Summary of Key Equations

• Potential difference from field: $\Delta V = - \int_{i}^{f} \vec{E} \cdot d\vec{s}$

• Field from potential: $\vec{E} = -\nabla V$

• Capacitance: $C = \frac{Q}{\Delta V}$

• Parallel-plate capacitor: $C = \epsilon_0 \frac{A}{d}$

• With dielectric: $C = \kappa \epsilon_0 \frac{A}{d}$

• Energy stored: $U = \frac{1}{2} C (\Delta V)^2$

• Energy density: $u = \frac{1}{2} \epsilon_0 E^2$

Examples and Applications

• Camera Flash Capacitor: A 220 μF capacitor charged to 330 V stores $U = \frac{1}{2} C V^2$ of energy, which can be released rapidly to produce a flash.

• Defibrillator: Uses a large capacitor to deliver a quick, strong electric shock to the heart.

• Dielectric in Water: Immersing a charged capacitor in water (κ = 80) increases its capacitance and decreases its voltage, with energy used to pull the dielectric into the field.

Additional info:

• Some table values and example details are inferred from standard physics references.

• All equations use SI units unless otherwise specified.