Electric Potential and Capacitance: Study Notes

Electric Potential

Definition and Relationship to Potential Energy

The electric potential at a point in space is a scalar quantity that represents the potential energy per unit charge at that location due to electric forces. It is related to the electric potential energy U by:

• Formula: $V = \frac{U}{q'}$

• Key Point: Only the source charge appears in this expression; the source charge creates the electric potential around it.

• Units: Volts (V), where 1 V = 1 J/C.

Electric Potential of a Point Charge and a Charged Sphere

The electric potential outside a charged sphere is the same as that outside a point charge. For a sphere of radius R and charge Q, the potential at a distance r > R from the center is:

• Formula: $V = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r}$

• Potential at the surface: $V_0 = \frac{1}{4\pi\varepsilon_0} \frac{Q}{R}$

• Charge for a sphere: $Q = 4\pi\varepsilon_0 R V_0$

• Potential outside: $V = V_0 \frac{R}{r}$

• Potential decreases inversely with distance from the center.

Electric Potential of Multiple Charges

When several point charges are present, the electric potential at a point is the algebraic sum of the potentials due to each charge:

• Formula: $V = \sum_i \frac{1}{4\pi\varepsilon_0} \frac{q_i}{r_i}$

• Where: $r_i$ is the distance from charge $q_i$ to the point in space.

Connecting Electric Potential and Electric Field

Geometry of Potential and Field

The relationship between electric field and electric potential is illustrated by equipotential surfaces:

• Electric field lines are always perpendicular to equipotential surfaces.

• Field direction: $\vec{E}$ points "downhill," in the direction of decreasing potential $V$.

• Field strength: Inversely proportional to the spacing $d$ between equipotential surfaces.

Arrangements of Charges and Equipotentials

Different charge configurations produce distinct patterns of electric field lines and equipotential surfaces:

• Point charge: Field lines radiate outward, perpendicular to equipotentials.

• Electric dipole: Field lines curve from positive to negative charge; equipotentials are more complex.

• Parallel-plate capacitor: Field is uniform and equipotential spacing is constant between plates.

Capacitance and Capacitors

Definition and Properties

A capacitor consists of two conductors (electrodes or plates) with equal but opposite charge. Capacitors store charge and are essential in electronic circuits.

• Capacitance (C): The constant of proportionality between charge and potential difference.

• Formula: $Q = C \Delta V$

• SI unit: Farad (F), where 1 F = 1 C/V.

• Capacitance depends on: Shape, size, and spacing of electrodes.

• Charging a capacitor: Requires moving charge from one electrode to the other, typically using a battery.

Charging a Capacitor

When a battery is connected to a capacitor, charge flows until the potential difference across the capacitor equals the battery voltage.

• Process: Charge flows from one plate, through the battery, to the other plate.

• Equilibrium: $\Delta V_C = \Delta V_{bat}$

• After battery removal: The capacitor remains charged, with $\Delta V_C$ still equal to the battery voltage.

Parallel-Plate Capacitor

A parallel-plate capacitor consists of two plates of area A separated by distance d. The capacitance is given by:

• Formula: $C = \varepsilon_0 \frac{A}{d}$

• Electric field inside: $E = \frac{\Delta V}{d}$

• Potential at distance x: $V_x = \Delta V \frac{x}{d}$

Energy Stored in Capacitors

Energy and Dielectrics

The energy stored in a capacitor is:

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

• Energy density in the electric field: $u = \frac{1}{2} k \varepsilon_0 E^2$

• Dielectric: Increases capacitance by a factor $k$ (dielectric constant).

Applications and Examples

Example: Proton in a Capacitor

A proton is shot through a parallel-plate capacitor charged to a potential difference. The farthest distance from the negative plate is determined by energy conservation:

• Known: $v_i = 2.0 \times 10^5$ m/s, $d = 2.00$ mm, $\Delta V = 500$ V

• Find: $x_f$ (distance where $v_f = 0$)

Electrocardiogram (ECG) and Electric Potential

The heart generates electric potentials that can be mapped as positive and negative regions, which are fundamental to ECG measurements.

• Positive and negative potentials: Represented by spatial distribution around the heart.

• Application: Used in medical diagnostics to monitor heart activity.

Summary Table: Capacitance and Related Quantities

Quantity	Symbol	SI Unit	Formula
Electric Potential	V	Volt (V)	$V = \frac{U}{q}$
Capacitance	C	Farad (F)	$C = \varepsilon_0 \frac{A}{d}$
Energy Stored	U	Joule (J)	$U = \frac{1}{2} C (\Delta V)^2$
Electric Field	E	Volt/meter (V/m)	$E = \frac{\Delta V}{d}$

Additional info: These notes expand on brief points and include academic context for clarity and completeness, suitable for exam preparation.