Electric Fields, Potential, and Capacitors: Study Notes

Electric Fields and Potential

Electric Field of a Point Charge

The electric field produced by a point charge radiates outward (or inward for negative charges) and its strength decreases with distance from the charge.

• Definition: The electric field E at a distance r from a point charge q is given by:

$E = \frac{k|q|}{r^2}$

• Direction: Radially outward for positive charges, inward for negative charges.

• Magnitude: Decreases as $1/r^2$ with distance.

Uniform Electric Field

A uniform electric field has the same magnitude and direction at every point.

• Represented by equally spaced, parallel field lines.

• Commonly produced between two parallel plates with opposite charges.

• Field strength is constant: $E = \text{constant}$

Analogy with Gravity

Electric fields and gravitational fields share similar properties regarding potential energy and work.

• Potential Energy (PE): Changes as a charge moves in an electric field, similar to how mass changes PE in a gravitational field.

• Work: Work is done when the force has a component along the direction of displacement.

• Relation: $W = F \cdot d$ (work is force times displacement in the direction of the force).

Work in a Gravitational Field (Analogy)

• Work done by gravity: $\Delta U = U_{\text{final}} - U_{\text{initial}} = -\Delta KE$

• Only the component of force in the direction of displacement does work.

Electric Potential and Potential Energy

Potential Difference in a Uniform Electric Field

The potential difference between two points in a uniform electric field is given by:

$\Delta V = V_b - V_a = -E d \cos \theta$

• Potential (V): Scalar quantity, measured in volts (V).

• Potential Energy (U): For a charge q, $U = qV$.

• Potential difference depends on the field strength, distance, and angle between field and displacement.

Potential vs. Potential Energy

• Potential (V): Independent of the charge, property of the field.

• Potential Energy (U): Depends on both the charge and the potential: $U = qV$.

• Potential energy decreases as a positive charge moves in the direction of the electric field.

Electric Potential Due to a Point Charge

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

$V = \frac{kq}{r}$

• k: Coulomb's constant, $k = 8.99 \times 10^9 \ \text{N} \cdot \text{m}^2/\text{C}^2$

• Potential decreases with distance from the charge.

Example Problem

• A 20 nC charge is moved from a point where $V = 150$ V to a point where $V = -50$ V. The work done by the force that moves the charge is:

$W = q \Delta V$

Capacitors

Definition and Properties

A capacitor is a device that stores electric charge and energy in the electric field between two conductors (plates).

• When uncharged, $E = 0$ and $V = 0$ between the plates.

• When charged, a uniform electric field exists between the plates.

Capacitance

• Definition: The capacitance C is the ratio of the charge stored to the potential difference between the plates.

$C = \frac{q}{V}$

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

• For a parallel-plate capacitor:

$C = \varepsilon_0 \frac{A}{d}$

• $\varepsilon_0$ is the permittivity of free space: $\varepsilon_0 = 8.85 \times 10^{-12} \ \text{C}^2/\text{N} \cdot \text{m}^2$

• A: Area of one plate; d: separation between plates.

Energy Stored in a Capacitor

The energy stored in a charged capacitor is:

$U_{\text{stored}} = \frac{1}{2}CV^2 = \frac{1}{2}qV = \frac{q^2}{2C}$

• Capacitors store energy in the electric field between their plates.

Medical Applications

• Defibrillator: Uses capacitors to deliver a controlled electric shock to the heart.

• Pacemaker: Uses capacitors to regulate heartbeats.

Capacitor Combinations

• Capacitors can be combined in series or parallel to achieve desired capacitance values.

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

• Parallel: $C_{\text{eq}} = C_1 + C_2 + \cdots$

Summary Table: Key Equations and Concepts

Concept	Equation	Unit
Electric Field (point charge)	$E = \frac{k|q|}{r^2}$	N/C
Electric Potential (point charge)	$V = \frac{kq}{r}$	V
Potential Difference (uniform field)	$\Delta V = -Ed \cos \theta$	V
Capacitance (parallel plates)	$C = \varepsilon_0 \frac{A}{d}$	F
Energy in Capacitor	$U = \frac{1}{2}CV^2$	J

Additional info: The notes also reference example problems and figures (e.g., Figure P21.21) that illustrate the application of these concepts, such as calculating potential at a point due to multiple charges and determining work done by electric forces.