Physics 207: Circuits, Capacitance, and DC Circuits Study Notes

Capacitance and Capacitors

Concept and Definition of Capacitance

Capacitance is a measure of a system's ability to store electric charge. It is defined as the ratio of the charge stored on the plates of a capacitor to the voltage across the plates.

• Capacitance (C): $C = \frac{Q}{V}$, where Q is the charge and V is the voltage.

• Unit: The SI unit of capacitance is the farad (F).

• Example: A parallel plate capacitor consists of two conductive plates separated by an insulator (dielectric). When connected to a battery, charge accumulates on the plates, creating a voltage difference.

Parallel Plate Capacitor

The parallel plate capacitor is a common configuration used to illustrate the principles of capacitance.

• Electric Field: $E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}$

• Voltage: $V = Ed = \frac{Qd}{\varepsilon_0 A}$

• Capacitance: $C = \frac{\varepsilon_0 A}{d}$, where A is the plate area and d is the separation distance.

• Example: Increasing the plate area or decreasing the separation increases the capacitance.

Cylindrical Capacitor

For two coaxial cylindrical conductors separated by vacuum, the capacitance per unit length is:

• $\frac{C}{\ell} = \frac{2\pi \varepsilon_0}{\ln\left(\frac{r_b}{r_a}\right)}$

• Application: Used in cable and transmission line design.

Capacitors in Series and Parallel

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

• Series: The reciprocal of the total capacitance is the sum of reciprocals of individual capacitances.

• $\frac{1}{C_{\text{eff}}} = \sum_i \frac{1}{C_i}$

• Parallel: The total capacitance is the sum of individual capacitances.

• $C_{\text{eff}} = \sum_i C_i$

• Example: Combining three capacitors of 3 μF, 11 μF, and 4 μF in parallel yields $C_{\text{eff}} = 18$ μF.

Effective Capacitance of a Network

Complex networks of capacitors can be reduced systematically by identifying series and parallel combinations.

Configuration	Formula	Example Calculation
Series	$\frac{1}{C_{\text{eff}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$	$\frac{1}{C_{\text{eff}}} = \frac{1}{11} + \frac{1}{12} + \frac{1}{6}$ $C_{\text{eff}} = 4$ μF
Parallel	$C_{\text{eff}} = C_1 + C_2 + \cdots$	$C_{\text{eff}} = 3 + 11 + 4 = 18$ μF

Energy Stored in a Capacitor

When a capacitor is charged, it stores electrical potential energy.

• Work to charge: $dW = v dq = \frac{q dq}{C}$

• Total energy stored: $W = \int_0^Q \frac{q dq}{C} = \frac{Q^2}{2C} = \frac{1}{2}CV^2 = \frac{1}{2}QV$

• Application: Capacitors are used in energy storage, filtering, and timing circuits.

Energy Density in a Capacitor

Energy density is the energy per unit volume stored in the electric field of a capacitor.

• $u_E = \frac{U}{\text{Volume}} = \frac{1}{2}CV^2 / \text{Volume}$

• For a parallel plate capacitor: $u_E = \frac{1}{2} \varepsilon_0 E^2$

• Key Point: Electrical energy density is proportional to the square of the electric field.

Dielectrics

Role and Properties of Dielectrics

Dielectrics are insulating materials placed between capacitor plates to increase capacitance and prevent charge leakage.

• Dielectric Constant (κ): Factor by which capacitance increases when a dielectric is inserted.

• Capacitance with Dielectric: $C = \kappa \frac{\varepsilon_0 A}{d}$

• Advantages: Prevents plate contact, increases maximum voltage (prevents breakdown), and increases capacitance.

• Example: Common dielectrics include paper, glass, and Teflon, each with a specific dielectric constant.

Material	Dielectric Constant (κ)
Water	80.4
Paper	3.7
Pyrex Glass	5.6
Teflon	2.1
Vacuum	1.00059

Resistors and DC Circuits

Concept and Definition of Resistance and Current

Resistance is a measure of how much a material opposes the flow of electric current. Current is the rate of flow of electric charge.

• Current (I): $I = \frac{dq}{dt}$, measured in amperes (A).

• Resistance (R): $R = \frac{V}{I}$, measured in ohms (Ω).

• Ohm's Law: $V = IR$

• Example: A resistor of 10 Ω with a current of 2 A has a voltage drop of 20 V.

Resistors in Series and Parallel

Resistors can be combined in series or parallel to achieve desired resistance values in circuits.

• Series: $R_{\text{eff}} = R_1 + R_2 + \cdots$

• Parallel: $\frac{1}{R_{\text{eff}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$

• Application: Used to control current and voltage in electronic devices.

Effective Resistance of a Network

Configuration	Formula	Example Calculation
Series	$R_{\text{eff}} = R_1 + R_2 + R_3$	$R_{\text{eff}} = 250\,k\Omega + 350\,k\Omega = 600\,k\Omega$
Parallel	$\frac{1}{R_{\text{eff}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$	$\frac{1}{R_{\text{eff}}} = \frac{1}{100\,k\Omega} + \frac{1}{200\,k\Omega} + \frac{1}{600\,k\Omega}$

Current Density

Current density describes the flow of electric charge per unit area.

• Current Density (J): $J = nqv_d$, where n is charge carrier density, q is charge, and v_d is drift velocity.

• Relation to Current: $I = JA$, where A is cross-sectional area.

Ohm's Law and Resistivity

Ohm's Law relates voltage, current, and resistance. Resistivity is a material property that quantifies how strongly a material opposes current flow.

• Resistivity (ρ): $\rho = \frac{E}{J}$

• Resistance in terms of resistivity: $R = \rho \frac{L}{A}$, where L is length and A is area.

• Temperature Dependence: $\rho(T) = \rho_0[1 + \alpha(T - T_0)]$

• Example: Metals have low resistivity, insulators have high resistivity.

Kirchhoff's Laws

Kirchhoff's Current Law (KCL)

The sum of currents entering a junction equals the sum of currents leaving the junction.

• Mathematical Statement: $\sum I_{\text{in}} = \sum I_{\text{out}}$

• Application: Used to analyze complex circuits with multiple branches.

Kirchhoff's Voltage Law (KVL)

The sum of the potential differences around any closed loop in a circuit is zero.

• Mathematical Statement: $\sum V = 0$ (for a closed loop)

• Application: Used to solve for unknown voltages and currents in multi-loop circuits.

RC Circuits

Charging and Discharging of Capacitors

RC circuits consist of resistors and capacitors and exhibit time-dependent behavior when charging or discharging.

• Charging: $q(t) = CE(1 - e^{-t/RC})$

• Discharging: $q(t) = Q_0 e^{-t/RC}$

• Current during charging: $I(t) = \frac{E}{R} e^{-t/RC}$

• Current during discharging: $I(t) = -\frac{Q_0}{RC} e^{-t/RC}$

• Time constant: $\tau = RC$

• Application: Used in timing circuits, filters, and signal processing.

Summary Table: Series and Parallel Rules

Component	Series	Parallel
Resistors	$R_{\text{eff}} = R_1 + R_2 + \cdots$	$\frac{1}{R_{\text{eff}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$
Capacitors	$\frac{1}{C_{\text{eff}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$	$C_{\text{eff}} = C_1 + C_2 + \cdots$

Key Equations

• $C = \frac{Q}{V}$

• $C = \frac{\varepsilon_0 A}{d}$

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

• $V = IR$

• $R = \rho \frac{L}{A}$

• $q(t) = CE(1 - e^{-t/RC})$ (charging)

• $q(t) = Q_0 e^{-t/RC}$ (discharging)

• $W = \frac{1}{2}CV^2$

• $u_E = \frac{1}{2} \varepsilon_0 E^2$

Additional info: These notes expand on the provided slides and images, adding definitions, formulas, and context for a self-contained study guide suitable for college-level physics students.