Electric Circuits: Current, Resistance, Power, and Kirchhoff's Rules

Current and Resistance

Current in Conductors

Electric current is the flow of electric charge through a conductor, typically measured in amperes (A). The amount of current produced for a given applied electric field or potential difference depends on the properties of the conductor, which determine its resistance.

• Conventional current flows from high potential to low potential, along the direction of the electric field.

• The current is proportional to the drift velocity of the charge carriers in the conductor.

• For many materials, especially metals at room temperature, current is proportional to voltage (Ohmic behavior).

Example: In a metal wire, increasing the applied voltage increases the current linearly.

Ohm's Law

Ohm's Law is an empirical relationship that describes how current flows in many conductors:

• Ohm's Law: $V = IR$

• Where V is the potential difference, I is the current, and R is the resistance (measured in Ohms, $\Omega$).

• Resistance is a constant for Ohmic materials over a range of temperatures.

Example: If a resistor has $R = 10\,\Omega$ and $V = 5\,\text{V}$, then $I = 0.5\,\text{A}$.

Resistance of a Wire

The resistance of a cylindrical wire depends on its length, cross-sectional area, and the material's resistivity:

• Resistance increases with wire length ($L$).

• Resistance decreases with increasing cross-sectional area ($A$).

• Resistance depends on the material's resistivity ($\rho$), a property at a given temperature.

Formula:

$R = \rho \frac{L}{A}$

Example: Copper has a low resistivity, so copper wires have low resistance compared to steel or tungsten.

Substance	Resistivity ($\Omega\cdot m$)
Copper	1.72 × 10-8
Aluminum	2.82 × 10-8
Iron	1.0 × 10-7
Wood	~1010

Additional info: Metal resistivity generally increases with temperature.

Electromotive Force (emf) and Circuits

Electromotive Force (emf)

To maintain a current in a circuit, a device must move charges from low to high potential, against the direction of the electric field. This is achieved by an electromotive force (emf), provided by batteries, generators, solar cells, etc.

• Ideal emf sources maintain a constant potential difference between terminals.

• The device uses a force other than the electric field to move charges.

Example: A battery provides emf to drive current through a resistor.

emf Provides Constant Potential, Not Constant Current

In a circuit, the emf source maintains a constant voltage. The current depends on the total resistance in the circuit.

• Increasing resistance decreases current.

• Opening a switch increases resistance, decreasing current and the brightness of bulbs.

Example: In a series circuit, opening a switch does not change the current through a bulb if the path remains unchanged.

Measurement: Voltmeters and Ammeters

Measurement tools are used to quantify voltage and current in circuits:

• Ideal voltmeter: Infinite resistance, draws no current.

• Ideal ammeter: Zero resistance, causes no potential change.

Formulas:

$I = \frac{V}{R}$

$V = IR$

Power in Electrical Circuits

Power Dissipation

The power transferred into a circuit element is:

$P = IV$

• For a resistor: $V = IR$ so $P = I^2R = \frac{V^2}{R}$

• For an ideal emf source: $V = -\mathcal{E}$, $P = -\mathcal{E}I$ (negative indicates power is transferred out)

Example: If a resistor dissipates 10 W, replacing it with a resistor of $2R$ in the same circuit halves the power to 5 W (if voltage is constant).

Resistors in Series and Parallel

Series Connection

Resistors in series have the same current through each, and the total resistance is the sum of individual resistances:

• Current: $I_a = I_b = I_c = I_d$

• Potential: $V_a > V_b > V_c > V_d$

• Total resistance: $R = R_1 + R_2 + R_3$

Formulas:

$V_1 = IR_1$, $V_2 = IR_2$, $V_3 = IR_3$

$V_a - V_d = IR_1 + IR_2 + IR_3 = IR$

Parallel Connection

Resistors in parallel have the same potential difference across each, and the total current is the sum of the branch currents:

• Potential: $V_1 = V_2 = V_3$

• Current: $I = I_1 + I_2 + I_3$

• Total resistance: $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$

Example: Adding more resistors in parallel decreases the total resistance.

Bulbs as Resistors

Bulb Brightness and Power

Bulbs act as resistors, converting electrical energy into heat and light. Their brightness is proportional to the power loss:

• $P = \frac{V^2}{R}$

• Greater current or potential drop increases brightness.

Example: In a series circuit, all identical bulbs have the same brightness; in parallel, the brightness may differ depending on the arrangement.

Special Circuit Scenarios

Shorting Out a Bulb

If a wire is connected across a bulb, the current prefers the path of least resistance, and all charge flows through the wire, bypassing the bulb.

Big Resistance, Small Resistance

In parallel, the resistor with the smallest resistance carries the largest current:

$I_1 = \frac{V}{R_1}$, $I_2 = \frac{V}{R_2}$

If $R_2 \ll R_1$, then $I_2 \gg I_1$.

Kirchhoff's Rules

Junction Rule

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

$\sum I = 0$

Example: If $I_1$ and $I_2$ enter a junction and $I_3$ leaves, $I_1 + I_2 - I_3 = 0$.

Loop Rule

The algebraic sum of the potential differences around any closed loop is zero:

$\sum V = 0$

Sign conventions: traversing a resistor in the direction of current is $-IR$; traversing an emf source from negative to positive is $+\mathcal{E}$.

Kirchhoff's Rules Example

To solve a circuit:

• Choose a direction for current (arbitrary; negative result means opposite direction).

• Apply junction and loop rules to set up equations.

• Solve for unknown currents and voltages.

Example: In a circuit with two batteries and three resistors, applying the loop rule yields $I = -\frac{2}{3}\,\text{A}$, indicating the guessed direction was wrong.