DC Circuits: EMF, Resistors, Kirchhoff's Rules, and RC Circuits

DC Circuits

Electromotive Force (EMF)

Electromotive Force (EMF) is a fundamental concept in electric circuits, representing the energy provided by a source (such as a battery) to move charge through a circuit.

• Definition: EMF is the influence that makes current flow from low potential to high potential inside a battery, opposite to the direction of current in the external circuit.

• Symbol: The circuit symbol for a battery (EMF source) is shown with the longer line at high potential and the shorter line at low potential.

• Ideal Battery: Maintains a constant potential difference, called the battery's EMF ($\varepsilon$).

• Work Done: The work done by an ideal battery in moving a charge $q$ is $W = q\varepsilon$.

• Energy Conversion: Batteries convert chemical energy into electrical energy. When chemical reactions cease, the battery can no longer do work.

Example: In a circuit, the battery provides the energy needed to move charges, maintaining a potential difference across its terminals.

Resistors in Series and Parallel

Resistors are circuit elements that impede the flow of electric current, and their arrangement affects the overall resistance and current distribution in a circuit.

Resistors in Series

• Current: The same current flows through all resistors in series.

• Equivalent Resistance: The total resistance is the sum of individual resistances:

$R_{eq} = R_1 + R_2 + \ldots + R_n = \sum_{i=1}^n R_i$

Resistors in Parallel

• Voltage: The voltage drop across each resistor in parallel is the same.

• Equivalent Resistance: The reciprocal of the total resistance is the sum of reciprocals of individual resistances:

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} = \sum_{i=1}^n \frac{1}{R_i}$

Example: For three resistors $R_1 = 15\,\Omega$, $R_2 = 12\,\Omega$, $R_3 = 24\,\Omega$ arranged as shown, the total resistance between points A and B is $R_{eq} = 23\,\Omega$.

Kirchhoff's Rules

Kirchhoff's rules are essential for analyzing complex circuits with multiple loops and junctions.

• Junction Rule: The current that flows into a junction equals the current that flows out (conservation of charge).

• Loop Rule: The sum of the voltage drops around any closed loop is zero (conservation of energy).

Application: Assign polarities to EMF sources, assign currents to each branch, apply the rules, and solve for unknowns.

Solving Multi-Loop Circuit Problems

• Assign polarity (+/–) to all EMF sources.

• Assign currents to each branch of the circuit.

• Apply Kirchhoff's rules to write equations for each loop and junction.

• Solve the resulting system of equations for the unknown currents.

Example: For a circuit with three resistors and two batteries, set up equations using the loop and junction rules, then solve for $I_1$, $I_2$, and $I_3$.

RC Circuits: Charging and Discharging a Capacitor

RC circuits consist of a resistor and capacitor in series, and are used to study the time-dependent behavior of charge and current.

Charging a Capacitor

• When the switch is closed at $t=0$, the capacitor begins to charge.

• Applying Kirchhoff's loop rule:

$\varepsilon - IR - \frac{Q}{C} = 0$

• This leads to a differential equation for $Q$:

$\frac{dQ}{dt} = \frac{\varepsilon}{R} - \frac{Q}{RC}$

• Solving gives:

$Q(t) = Q_0 (1 - e^{-t/\tau})$ Where $\tau = RC$ is the time constant and $Q_0 = \varepsilon C$.

• The current in the circuit:

$I(t) = \frac{dQ}{dt} = I_0 e^{-t/\tau}$

• Voltage across the capacitor:

$V_C(t) = \frac{Q(t)}{C} = \varepsilon (1 - e^{-t/\tau})$

• Voltage across the resistor:

$V_R(t) = I(t) R$

Discharging a Capacitor

• After the capacitor is fully charged, the battery is removed and the capacitor discharges through the resistor.

• Applying Kirchhoff's loop rule:

$-IR + \frac{Q}{C} = 0$

• Solving gives:

$Q(t) = Q_0 e^{-t/\tau}$

$I(t) = I_0 e^{-t/\tau}$

$V_C(t) = \varepsilon e^{-t/\tau}$

RC Time Constant

• Definition: The time constant $\tau = RC$ characterizes the rate at which the capacitor charges or discharges.

• After a time $t = \tau$, the charge (or voltage) has changed by about 63% of its total change.

Examples and Applications

• Calculating Resistance: Use series and parallel formulas to find total resistance in complex circuits.

• Current and Power: Use Ohm's law and power formulas to determine current through and power dissipated by resistors.

• RC Circuit Analysis: Use time constant and exponential equations to analyze charging/discharging behavior.

Configuration	Current	Voltage	Equivalent Resistance
Series	Same through all resistors	Sum of voltage drops	$R_{eq} = R_1 + R_2 + \ldots + R_n$
Parallel	Splits among branches	Same across all resistors	$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}$

Summary of Key Equations

• Series Resistance: $R_{eq} = \sum_{i=1}^n R_i$

• Parallel Resistance: $\frac{1}{R_{eq}} = \sum_{i=1}^n \frac{1}{R_i}$

• Kirchhoff's Rules: Junction rule (current conservation), Loop rule (energy conservation)

• Charging Capacitor: $Q(t) = Q_0 (1 - e^{-t/\tau})$

• Discharging Capacitor: $Q(t) = Q_0 e^{-t/\tau}$

• RC Time Constant: $\tau = RC$

Example Application: In an RC circuit with $C = 1.00\,\mu F$, $R = 2.00 \times 10^6\,\Omega$, and $\varepsilon = 10.0\,V$, the time constant is $\tau = 2.00\,s$, and the charge on the capacitor at $t = 10.8\,s$ is $Q(t) = 9.95 \times 10^{-6}\,C$.

Additional info: These notes cover the essential concepts and calculations for DC circuits, including EMF, resistor networks, Kirchhoff's rules, and the analysis of RC circuits, as required for a college-level physics course.