Electric Potential, Field, and Capacitance: Study Notes ch 26

Electric Potential Inside a Parallel-Plate Capacitor

Uniform Electric Field and Potential

In a parallel-plate capacitor, the electric field E between the plates is uniform, and the electric potential at a distance s from the negative plate is given by:

• Formula: $V_{cap} = Es$

• If $\Delta V$ is the potential difference between the plates, $\Delta V = V_+ - V_-$, then $V_{cap} = \frac{s}{d} \Delta V$

Equipotential Surfaces

• Definition: An equipotential surface is a surface where the electric potential is the same at all points.

• Equipotential surfaces are not physical surfaces; they are conceptual and help visualize potential differences.

• Equipotential surfaces can be represented as contour maps.

Applications: Electron Gun and Electrostatic Precipitator

Electron Gun

• An electron gun uses a parallel-plate capacitor to accelerate electrons using a potential difference.

• Example: If an electron is released from rest above the negative plate, its speed when reaching the positive plate is determined by energy conservation:

• Key Equations:

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

  • Potential energy change: $U_{elec} = q(V_- - V_+)$

  • Final speed: $v = \sqrt{\frac{2qV_{cap}}{m}}$

• Application: Electron guns were used in cathode ray tube TVs to accelerate electrons and produce colored light on the screen.

Electrostatic Precipitator

• Removes smoke particles from waste gas in power plants using a large potential difference (40–100 kV).

• Smoke particles acquire a negative charge and are attracted to collecting plates, which are then knocked to remove the particles.

The Electric Potential of a Point Charge

Point Charge Potential

The electric potential at a distance r from a point charge q is:

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

Sphere of Charge

• For a sphere of radius R with charge Q uniformly distributed, the potential at distance $r \geq R$ is:

• $V_{sphere} = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$

The Electric Potential of Many Charges

Superposition Principle

• The total electric potential at a point due to multiple charges $q_1, q_2, ...$ is:

• $V = \sum_i \frac{1}{4\pi\epsilon_0} \frac{q_i}{r_i}$

Ring and Disk of Charge

• Potential on axis of a ring: $V_{ring\ on\ axis} = \frac{1}{4\pi\epsilon_0} \frac{Q}{\sqrt{R^2 + z^2}}$

• Potential on axis of a disk: $V_{disk\ on\ axis} = \frac{Q}{2\pi\epsilon_0 R^2} (\sqrt{z^2 + R^2} - |z|)$

Connecting Potential and Field

Relationship Between Electric Field and Potential

• The electric potential difference $\Delta V$ between two points is related to the electric field $\vec{E}$ by:

• $\Delta V = V_f - V_i = -\int_i^f \vec{E} \cdot d\vec{s}$

• Electric field is the negative gradient of potential.

Finding the Electric Field from Potential

• In one dimension: $E_s = -\frac{dV}{ds}$

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

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

Equipotential Lines and Electric Field

Electric Field from Equipotential Lines

• The magnitude of the electric field between equipotential lines is $E = \frac{\Delta V}{\Delta s}$, where $\Delta s$ is the distance between lines.

• Where equipotential lines are closest, the electric field is strongest.

Direction of Electric Field

• The electric field direction is perpendicular to equipotential lines, pointing from higher to lower potential.

Conductors in Electrostatic Equilibrium

Properties of Conductors

• Inside a conductor at electrostatic equilibrium, $\vec{E} = 0$.

• All points inside the conductor are at the same electric potential.

Capacitance and Capacitors

Definition and Formula

• A capacitor consists of two electrodes with equal and opposite charge.

• Potential difference $\Delta V_C$ is related to charge $q$ and capacitance $C$ by $q = C \Delta V_C$.

• SI unit: farad (F), $1\ ext{F} = 1\ ext{C}/\text{V}$.

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

Capacitor Combinations

• Parallel: $C_{eq} = C_1 + C_2 + C_3 + ...$

• Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + ...$

Energy Stored in a Capacitor

Energy Formulas

• Potential energy stored: $U_C = \frac{q^2}{2C} = \frac{1}{2}q\Delta V_C = \frac{1}{2}C(\Delta V_C)^2$

• Energy density (per unit volume): $u_E = \frac{1}{2}\epsilon_0 E^2$

Conceptual Questions and Answers

Disconnected Capacitor: Q vs d

• If a capacitor is disconnected from a battery, the charge on the plates remains the same when the plate separation changes.

Disconnected Capacitor: E vs d

• Electric field between plates remains the same if charge is constant and plate separation changes.

Disconnected Capacitor: ΔV vs d

• Potential difference doubles if plate separation doubles, since $\Delta V = Ed$ and $E$ is constant.

Disconnected Capacitor: ΔV vs A

• Potential difference doubles if plate area halves, since capacitance halves and $\Delta V = q/C$.

Capacitor and Battery: Q vs d

• If the battery remains connected, the charge halves when plate separation doubles, since $C$ halves and $q = C\Delta V$.

Energy in Capacitor

• If the plate separation doubles (disconnected from battery), the energy stored doubles, $U_f/U_i = 2$.

Applications of Capacitors

Application	Description
Capacitive Keyboard	Spacing between plates changes as keys are pressed, altering capacitance.
Capacitor Microphone	Sound waves change plate distance, producing a signal.
Camera Flash	Capacitor stores charge for rapid release in flash units.
Defibrillator	Capacitor releases energy to restart heart rhythm.

Summary Table: Key Formulas

Concept	Formula (LaTeX)
Potential (point charge)	$V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$
Potential (many charges)	$V = \sum_i \frac{1}{4\pi\epsilon_0} \frac{q_i}{r_i}$
Capacitance (parallel plate)	$C = \epsilon_0 \frac{A}{d}$
Energy stored	$U_C = \frac{1}{2}C(\Delta V_C)^2$
Electric field from potential	$E = -\frac{dV}{ds}$

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