DC Circuits: Principles, Components, and Measurement

DC Circuits

EMF and Terminal Voltage

Direct current (DC) circuits require a source of electromotive force (EMF), such as a battery or generator, to produce current. The EMF is the energy per unit charge supplied by the source. Real batteries have internal resistance, which reduces the actual terminal voltage compared to the ideal EMF.

• EMF (Electromotive Force): The maximum potential difference a source can provide when no current is flowing.

• Terminal Voltage: The voltage measured across the terminals of a battery when it is supplying current; it is less than the EMF due to internal resistance.

• Internal Resistance: Acts as if it is in series with the EMF, causing a voltage drop inside the battery.

Formula:

$V = \text{EMF} - Ir$

where $V$ is terminal voltage, $I$ is current, and $r$ is internal resistance.

Resistors in Series and in Parallel

Resistors can be connected in series or parallel, affecting the total resistance and current distribution in a circuit.

• Series Connection: All resistors are connected end-to-end, forming a single path for current.

• Current: Same through each resistor.

• Voltage: Divided among resistors; sum of voltage drops equals battery voltage.

• Equivalent Resistance (Series):

$R_{\text{eq}} = R_1 + R_2 + R_3 + \ldots$

• Parallel Connection: Resistors are connected across the same two points, splitting the current.

• Voltage: Same across each resistor.

• Current: Total current is the sum of currents through each resistor.

• Equivalent Resistance (Parallel):

$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$

• Analogy: Water flow analogy helps visualize parallel circuits, where multiple paths allow more flow.

Kirchhoff’s Rules

For complex circuits that cannot be simplified into series or parallel, Kirchhoff’s rules are used to analyze current and voltage.

• Junction Rule: The sum of currents entering a junction equals the sum of currents leaving it.

• Loop Rule: The sum of the changes in potential around a closed loop is zero.

• Problem Solving Steps:

  1. Label each current.

  2. Identify unknowns.

  3. Apply junction and loop rules to form equations.

  4. Solve equations, paying attention to sign conventions.

EMFs in Series and Parallel; Charging a Battery

Multiple sources of EMF can be combined in circuits to increase voltage or current.

• EMFs in Series (Same Direction): Total voltage is the sum of individual voltages.

• EMFs in Series (Opposite Direction): Total voltage is the difference; the lower-voltage battery may be charged.

• EMFs in Parallel: Only valid if voltages are equal; increases available current.

Circuits Containing Capacitors in Series and in Parallel

Capacitors store electrical energy and can be arranged in series or parallel, affecting total capacitance.

• Parallel Connection: Same voltage across each capacitor.

• Total Capacitance (Parallel):

$C_{\text{eq}} = C_1 + C_2 + C_3 + \ldots$

• Series Connection: Same charge on each capacitor.

• Total Capacitance (Series):

$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots$

RC Circuits—Resistor and Capacitor in Series

RC circuits consist of a resistor and capacitor in series. When the circuit is closed, the capacitor charges or discharges exponentially.

• Charging: Voltage across the capacitor increases with time.

• Charge: Follows an exponential curve.

• Time Constant ($\tau$): Characteristic time for charging/discharging, given by:

$\tau = RC$

• Discharging: An isolated charged capacitor connected across a resistor will discharge exponentially.

Formulas:

Charging voltage: $V_C(t) = V_0 \left(1 - e^{-t/RC}\right)$

Charge: $Q(t) = Q_0 \left(1 - e^{-t/RC}\right)$

Discharging voltage: $V_C(t) = V_0 e^{-t/RC}$

Electric Hazards

Electric currents as low as 10–100 mA can be dangerous, potentially disrupting the nervous system or causing burns. Household voltages can be lethal, especially if the body is wet and in good contact with the ground.

• Shock: Occurs when a person becomes part of a complete circuit.

• Safety: Faulty wiring and improper grounding increase risk; three-prong plugs provide a separate ground line.

• Household Wiring: Color codes vary; always identify the hot wire before working.

Ammeters and Voltmeters—Measurement Affects the Quantity Being Measured

Measurement devices can affect the circuit quantities they are designed to measure.

• Ammeter: Measures current; should have low resistance to minimize impact on circuit.

• Voltmeter: Measures voltage; should have high resistance to avoid drawing current.

• Ohmmeter: Measures resistance; uses a battery to supply current.

• Measurement Effects: Incorrect resistance in meters can alter the measurement.

Summary Table: Series vs. Parallel Connections

Property	Series	Parallel
Resistors	Same current, voltage divides $R_{\text{eq}} = R_1 + R_2 + \ldots$	Same voltage, current divides $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \ldots$
Capacitors	Same charge, voltage divides $\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \ldots$	Same voltage, charge divides $C_{\text{eq}} = C_1 + C_2 + \ldots$

Summary Table: Measurement Devices

Device	Measures	Resistance Requirement
Ammeter	Current	Low resistance
Voltmeter	Voltage	High resistance
Ohmmeter	Resistance	Uses internal battery

Examples and Applications

• Example: Calculating the terminal voltage of a battery with internal resistance when supplying a known current.

• Application: Household wiring safety, use of three-prong plugs, and proper measurement techniques to avoid hazards.

• Example: Using Kirchhoff’s rules to solve for unknown currents in a multi-loop circuit.

Additional info: Academic context and formulas have been expanded for clarity and completeness.