Current, Resistance, and Resistivity: Study Notes for PHY 131

Current and Current Density

Definition and Measurement of Electric Current

Electric current is a fundamental concept in physics, describing the flow of electric charge through a conductor. It is measured as the amount of charge passing through a cross-sectional area per unit time.

• Current (I): Defined mathematically as $I = \frac{dq}{dt}$, where $dq$ is the differential amount of charge and $dt$ is the differential time interval.

• Units: The SI unit of current is the ampere (A), where $1~\text{A} = 1~\text{C}/\text{s}$.

• Direction: By convention, current flows in the direction opposite to the movement of electrons in a wire.

Example: If a current of 80.0 mA exists in a wire for 10.0 minutes, the number of electrons passing a cross-section is:

• $\Delta q = I \Delta t = (80.0 \times 10^{-3}~\text{A})(600~\text{s}) = 48.0~\text{C}$

• Number of electrons $= \frac{48.0~\text{C}}{1.60 \times 10^{-19}~\text{C/electron}} = 3.00 \times 10^{20}$ electrons

Current Density

Current density describes how much current flows per unit area of a conductor.

• Current Density (J): $J = \frac{I}{A}$, where $A$ is the cross-sectional area.

• In terms of charge carriers: $J = nqv_d$, where $n$ is the number density of charge carriers, $q$ is the charge of each carrier, and $v_d$ is the drift velocity.

Drift Speed of Charge Carriers

Microscopic View of Current

Electrons in a metal move randomly at high speeds, but without an electric field, there is no net flow (drift velocity is zero). When a potential difference is applied, an electric field causes electrons to drift, resulting in a net current.

• Drift Velocity ($v_d$): The average velocity of charge carriers due to an electric field, typically much less than their random thermal velocity.

• Formula: $I = nqAv_d$

Example: For a copper wire with $A = 1.00~\text{mm}^2$, $I = 2.0~\text{A}$, and $n = 1.10 \times 10^{29}~\text{electrons/m}^3$:

• $v_d = \frac{I}{neA} = \frac{2.0~\text{A}}{(1.10 \times 10^{29}~\text{m}^{-3})(1.60 \times 10^{-19}~\text{C})(1.00 \times 10^{-6}~\text{m}^2)} = 1.1 \times 10^{-4}~\text{m/s}$

Resistance and Resistivity

Ohm's Law and Ohmic Materials

Ohm's Law relates the voltage across a conductor to the current flowing through it, provided the material is ohmic (linear relationship).

• Ohm's Law: $\Delta V = IR$

• Resistance (R): The proportionality constant, measured in ohms ($\Omega$), where $1~\Omega = 1~\text{V}/\text{A}$.

• Ohmic Material: A material that obeys Ohm's Law, i.e., $\Delta V \propto I$.

Resistivity and Conductivity

Resistivity is a material property that quantifies how strongly a material opposes the flow of electric current.

• Resistivity ($\rho$): $\rho$ is the intrinsic property of a material, measured in $\Omega\cdot\text{m}$.

• Conductivity ($\sigma$): $\sigma = \frac{1}{\rho}$, measured in $\text{S/m}$ (siemens per meter).

• Current Density and Electric Field: $J = \sigma E = \frac{1}{\rho}E$

Resistance of a Conductor

The resistance of a uniform conductor depends on its geometry and material properties.

• Formula: $R = \rho \frac{L}{A}$, where $L$ is the length and $A$ is the cross-sectional area.

• Materials with low $\rho$ are conductors; those with high $\rho$ are insulators.

Resistor Identification and Circuit Symbols

Resistors are identified by color codes and represented by standard symbols in circuit diagrams.

Color	Digit	Multiplier
Black	0	$10^0$
Brown	1	$10^1$
Red	2	$10^2$
Orange	3	$10^3$
Yellow	4	$10^4$
Green	5	$10^5$
Blue	6	$10^6$
Violet	7	$10^7$
Gray	8	$10^8$
White	9	$10^9$

ANSI and IEC symbols are used for resistors in circuit diagrams.

Change of Resistivity with Temperature

Temperature Dependence

The resistivity of most materials changes with temperature, typically increasing for metals.

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

• $\rho_0$ is the resistivity at reference temperature $T_0$; $\alpha$ is the temperature coefficient of resistivity.

Example: If $R = 19.8~\Omega$ at $15.0^\circ$C and $R = 25.0~\Omega$ at $85.0^\circ$C, then:

• $\alpha = \frac{R - R_0}{R_0 \Delta T} = \frac{25.0~\Omega - 19.8~\Omega}{19.8~\Omega \times (85.0 - 15.0)^\circ\text{C}} = 3.75 \times 10^{-3}~\text{C}^{-1}$

Superconductivity

Some materials exhibit zero resistivity below a critical temperature, becoming superconductors.

• Superconductor: A material with $\rho = 0$ below its critical temperature $T_c$.

• Example: Mercury is a superconductor below $T_c \approx 4.2~\text{K}$.

Superconductors can exclude magnetic fields, leading to phenomena such as magnetic levitation.

Power and Resistive Dissipation

Energy Dissipation in Circuits

When current flows through a resistor, electrical energy is converted into thermal energy (Joule heating).

• Power (P): $P = I \Delta V$

• For a resistor: $P = I^2 R = \frac{(\Delta V)^2}{R}$

Example: A resistor with $\Delta V = 15~\text{V}$ and $P = 327~\text{W}$:

• $R = \frac{(\Delta V)^2}{P} = \frac{225}{327} = 0.688~\Omega$

• $I = \sqrt{\frac{P}{R}} = \sqrt{\frac{327}{0.688}} = 21.8~\text{A}$

Measurement Devices: Ammeters and Voltmeters

Ammeters

An ammeter measures current and is connected in series with the circuit element. It has low internal resistance to minimize its effect on the circuit.

• Placement: Series connection

• Function: Measures current through a component or total current drawn from EMF source

Voltmeters

A voltmeter measures the potential difference across a circuit element and is connected in parallel. It has high internal resistance to minimize current draw.

• Placement: Parallel connection

• Function: Measures voltage drop across a component

Ohmic and Non-Ohmic Devices

Experimental Determination

To determine if a device is ohmic, measure the current and voltage and plot $V$ vs. $I$. A linear relationship indicates ohmic behavior.

• Ohmic Device: $V$ vs. $I$ is linear (e.g., most metal resistors)

• Non-Ohmic Device: $V$ vs. $I$ is nonlinear (e.g., Thyrite)

Example: Thyrite does not follow Ohm's Law; its $V$ vs. $I$ plot is nonlinear, and resistance varies with current.

Summary Table: Key Equations and Concepts

Concept	Equation
Current	$I = \frac{dq}{dt}$
Current Density	$J = nqv_d$; $J = \frac{I}{A}$
Ohm's Law	$J = \sigma E = \frac{1}{\rho}E$; $\Delta V = IR$
Resistance	$R = \rho \frac{L}{A}$
Power Dissipation	$P = I \Delta V$; $P = I^2 R$; $P = \frac{(\Delta V)^2}{R}$
Resistivity vs. Temperature	$\rho(T) = \rho_0 [1 + \alpha (T - T_0)]$

Additional info:

• Superconductors are used in applications requiring zero electrical resistance, such as MRI machines and maglev trains.

• Resistor color codes are standardized for easy identification in electronic circuits.