Electric Potential, Potential Energy, and Field: Study Notes for Chapters 25 & 26

Chapter 25: Work and Energy in Electrostatics

25.1 Electric Potential Energy

Electric potential energy is the energy associated with the configuration of charged objects in an electric field. It is a measure of the work required to assemble a system of charges.

• Definition: The electric potential energy $U_{elec}$ of a charge $q$ in a uniform electric field $E$ at a distance $s$ from the negative plate is given by:

$U_{elec} = qEs$

• Application: In a parallel-plate capacitor, moving a charge between plates changes its potential energy.

• Example: A positive charge moves towards the negative plate, gaining kinetic energy and losing potential energy.

25.2 The Potential Energy of Point Charges

The potential energy stored in a system of two point charges depends on their separation and magnitudes.

• Formula for two charges:

$U_{elec} = \frac{K q_1 q_2}{r}$

• For a system of many charges:

$U_{elec} = \sum_{\text{all pairs}} \frac{K q_i q_j}{r_{ij}}$

• Bound Charges: A charge is bound if its kinetic energy is less than the magnitude of its potential energy, causing it to reverse direction at a maximum distance.

• Example: If $K_2 < |\frac{K q_1 q_2}{r_{12}}|$, the particle turns around (is bound).

25.3 The Potential Energy of a Dipole

An electric dipole in a uniform electric field has potential energy depending on its orientation relative to the field.

• Formula:

$U_{elec} = -pE \cos \phi = -\vec{p} \cdot \vec{E}$

• Where: $\vec{p}$ is the dipole moment, $\vec{E}$ is the electric field, and $\phi$ is the angle between them.

• Comparison: The potential energy is lowest when the dipole aligns with the field.

25.4 The Electric Potential

Electric potential is a scalar quantity representing the potential energy per unit charge at a point in space due to source charges.

• Formula:

$U_{elec} = qV$

• Unit: The volt (V), where $1\text{ V} = 1\text{ J}/1\text{ C}$.

• Sign of Potential: The sign depends on the source charge: positive for positive charges, negative for negative charges.

25.5 Electric Potential Inside a Parallel-Plate Capacitor

In a parallel-plate capacitor, the electric potential varies linearly between the plates.

• Formula:

$V_{cap} = Es$

• If $\Delta V$ is the potential difference between plates:

$V_{cap} = \frac{s}{d} \Delta V$

• Equipotential Surface: A surface where the potential is constant at all points; can be visualized as a contour map.

25.6 The Electric Potential of a Point Charge

The electric potential due to a point charge decreases with distance from the charge.

• Formula for a point charge:

$V_{point} = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$

• For a sphere of radius $R$ with charge $Q$:

$V_{sphere} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$ for $r \geq R$

25.7 The Electric Potential of Many Charges

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

• Formula:

$V = \sum_i \frac{q_i}{4\pi\epsilon_0 r_i}$

• For continuous charge distributions:

For a ring: $V_{ring\ on\ axis} = \frac{Q}{4\pi\epsilon_0 \sqrt{R^2 + z^2}}$ For a disk: $V_{disk\ on\ axis} = \frac{Q}{2\pi\epsilon_0 R^2} \left( \sqrt{z^2 + R^2} - |z| \right)$

Chapter 26: Potential and Field

26.1 Connecting Potential and Field

The electric potential and electric field are related; both describe how source charges affect the space around them.

• Relationship:

$\Delta V = V_f - V_i = - \int \vec{E} \cdot d\vec{s}$

• The change in potential is the negative of the work done by the electric field per unit charge.

26.2 Finding the Electric Field from the Potential

The electric field can be calculated from the spatial variation of the electric potential.

• Formula for field in s-direction:

$E_s = -\frac{dV}{ds}$

• General formula:

$\vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right)$

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

Applications and Examples

• Faraday Cage: A metal enclosure that shields its interior from external electric fields. Used in practical applications to protect sensitive equipment.

• Electron Gun: Electrons are accelerated by a potential difference in a parallel-plate capacitor. The final speed can be calculated using energy conservation:

$K = \frac{1}{2}mv^2 = q(V_{cap})$ $\Rightarrow v = \sqrt{\frac{2qV_{cap}}{m}}$

• Electrostatic Precipitator: Removes smoke particles from waste gas by charging them and attracting them to collecting plates using a high potential difference.

Summary Table: Key Equations

Concept	Equation	Description
Potential Energy (Uniform Field)	$U_{elec} = qEs$	Charge $q$ in field $E$ at distance $s$
Potential Energy (Point Charges)	$U_{elec} = \frac{K q_1 q_2}{r}$	Two point charges separated by $r$
Potential Energy (Dipole)	$U_{elec} = -pE \cos \phi$	Dipole in uniform field
Electric Potential (Point Charge)	$V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$	At distance $r$ from charge $q$
Electric Field from Potential	$E = -\frac{dV}{ds}$	Field in direction $s$
General Electric Field	$\vec{E} = -\nabla V$	Gradient of potential

Additional info:

• These notes cover topics from Chapter 25 (Work and Energy in Electrostatics) and Chapter 26 (Potential and Field), including practical demonstrations and quiz solutions.

• Examples and applications such as Faraday cages, electron guns, and electrostatic precipitators illustrate the real-world relevance of these concepts.