5.8: Current and Resistance (Summary)

Key Terms

alternating current (AC)	the flow of electric charge that periodically reverses direction
AC voltage	voltage that fluctuates sinusoidally with time, expressed as \(V = V_0 \sin(2\pi f t) \), where \(V_0\) is the peak voltage and \(f\) is frequency in hertz
AC current	current that fluctuates sinusoidally with time, expressed as \(I = I_0 \sin(2\pi f t) \), where \(I_0\) is the peak voltage and \(f\) is frequency in hertz
ampere (amp)	SI unit for current; \(\displaystyle 1 \, \mathrm{A}=1 \, \mathrm{C}/\mathrm{s}\)
circuit	complete path that an electrical current travels along
conventional current	current that flows through a circuit from the positive terminal of a battery through the circuit to the negative terminal of the battery
current density	flow of charge through a cross-sectional area divided by the area
direct current (DC)	the flow of electric charge in only one direction
diode	nonohmic circuit device that allows current flow in only one direction
drift velocity	velocity of a charge as it moves nearly randomly through a conductor, experiencing multiple collisions, averaged over a length of a conductor, whose magnitude is the length of conductor traveled divided by the time it takes for the charges to travel the length
electrical conductivity	measure of a material’s ability to conduct or transmit electricity
electrical current	rate at which charge flows, \(\displaystyle I=\frac{dQ}{dt}\)
electrical power	time rate of change of energy in an electric circuit
nonohmic	type of a material for which Ohm’s law is not valid
ohm	(\(\displaystyle Ω\)) unit of electrical resistance, \(\displaystyle 1\, \Omega=1\, \mathrm{V}/\mathrm{A}\)
ohmic	type of a material for which Ohm’s law is valid, that is, the voltage drop across the device is equal to the current times the resistance
Ohm’s law	empirical relation stating that the current I is proportional to the potential difference \(\Delta V\); it is often written as \(\displaystyle \Delta V=IR\), where \(R\) is the resistance
resistance	electric property that impedes current; for ohmic materials, it is the ratio of voltage to current, \(\displaystyle R=\Delta V/I\)
resistivity	intrinsic property of a material, independent of its shape or size, directly proportional to the resistance, denoted by \(\displaystyle \rho\)
rms current	the root mean square of the current, \(I_{\mathrm{rms}}/\sqrt{2}\), where \(I_0\) is the peak currrent, in an AC system
rms voltage	the root mean square of the voltage, \(V_{\mathrm{rms}}/\sqrt{2}\), where \(V_0\) is the peak voltage, in an AC system
schematic	graphical representation of a circuit using standardized symbols for components and solid lines for the wire connecting the components

Key Equations

Average electrical current	\(\displaystyle I_{ave}=\frac{ΔQ}{Δt}\)
Definition of an ampere	\(\displaystyle 1A=1C/s\)
Electrical current	\(\displaystyle I=\frac{dQ}{dt}\)
Drift velocity	\(\displaystyle v_d=\frac{I}{nqA}\)
Current density	\(\displaystyle I=∬_{area}\vec{J}⋅d\vec{A}\)
Resistivity	\(\displaystyle ρ=\frac{E}{J}\)
Common expression of Ohm’s law	\(\displaystyle V=IR\)
Resistivity as a function of temperature	\(\displaystyle ρ=ρ_0[1+α(T−T_0)]\)
Definition of resistance	\(\displaystyle R≡\frac{V}{I}\)
Resistance of a cylinder of material	\(\displaystyle R=ρ\frac{L}{A}\)
Temperature dependence of resistance	\(\displaystyle R=R_0(1+αΔT)\)
Electric power	\(\displaystyle P=IV\)
Power dissipated by a resistor	\(\displaystyle P=I^2R=\frac{V^2}{R}\)
Average AC power	\(\displaystyle P_{\mathrm{ave}}=\frac{1}{2}I_0 V_0 = I_{\mathrm{rms}} V_{\mathrm{rms}}\)

Summary

Electrical Current

• The average electrical current \(\displaystyle I_{\mathrm{ave}}\) is the rate at which charge flows, given by \(\displaystyle I_{\mathrm{ave}}=\frac{\Delta Q}{\Delta t}\), where \(\displaystyle \Delta Q\) is the amount of charge passing through an area in time \(\displaystyle \Delta t\).
• The instantaneous electrical current, or simply the current I, is the rate at which charge flows. Taking the limit as the change in time approaches zero, we have \(\displaystyle I=\frac{dQ}{dt}\), where \(\displaystyle \frac{dQ}{dt}\) is the time derivative of the charge.
• The direction of conventional current is taken as the direction in which positive charge moves. In a simple direct-current (DC) circuit, this will be from the positive terminal of the battery to the negative terminal.
• The SI unit for current is the ampere, or simply the amp (A), where \(\displaystyle 1\, \mathrm{A}=1\, \mathrm{C}/\mathrm{s}\).
• Current consists of the flow of free charges, such as electrons, protons, and ions.

Basic Model of Conduction in Metals

• The current through a conductor depends mainly on the motion of free electrons.
• When an electrical field is applied to a conductor, the free electrons in a conductor do not move through a conductor at a constant speed and direction; instead, the motion is almost random due to collisions with atoms and other free electrons.
• Even though the electrons move in a nearly random fashion, when an electrical field is applied to the conductor, the overall velocity of the electrons can be defined in terms of a drift velocity.
• The current density is a vector quantity defined as the current through an infinitesimal area divided by the area.
• The current can be found from the current density, \(\displaystyle I=∬_{area}\vec{J}⋅d\vec{A}\).
• An incandescent light bulb is a filament of wire enclosed in a glass bulb that is partially evacuated. Current runs through the filament, where the electrical energy is converted to light and heat.

Resistivity and Resistance

• Resistance has units of ohms (\(\displaystyle Ω\)), related to volts and amperes by \(\displaystyle 1 \, \Omega =1\, \mathrm{V}/\mathrm{A}\).
• The resistance \(R\) of a cylinder of length \(L\) and cross-sectional area \(A\) is \(\displaystyle R=\frac{\rho L}{A}\), where \(\displaystyle \rho \) is the resistivity of the material.
• Values of \(\displaystyle \rho \) in the table in this section show that materials fall into three groups—conductors, semiconductors, and insulators.
• Temperature affects resistivity; for relatively small temperature changes \(\displaystyle \Delta T\), resistivity is \(\displaystyle \rho=\rho_0(1+\alpha\Delta T)\), where \(\displaystyle \rho_0\) is the original resistivity and \(\displaystyle \alpha \) is the temperature coefficient of resistivity.
• The resistance \(R\) of an object also varies with temperature: \(\displaystyle R=R_0(1+\alpha \Delta T)\), where \(\displaystyle R_0\) is the original resistance, and \(R\) is the resistance after the temperature change.

Ohm's Law

• Ohm’s law is an empirical relationship for current, voltage, and resistance for some common types of circuit elements, including resistors. It does not apply to other devices, such as diodes.
• One statement of Ohm’s law gives the relationship among current \(I\), voltage \(\Delta V\), and resistance \(R\) in a simple circuit as \(\displaystyle \Delta V=IR\).
• Another statement of Ohm’s law, on a microscopic level, is \(\displaystyle J=\sigma E\).

Electrical Energy and Power

• Electric power is the rate at which electric energy is supplied to a circuit or consumed by a load.
• Power dissipated by a resistor depends on the square of the current through the resistor and is equal to \(\displaystyle P=I^2R=\frac{(\Delta V)^2}{R}\).
• The SI unit for electric power is the watt and the SI unit for electric energy is the joule. Another common unit for electric energy, used by power companies, is the kilowatt-hour (\(\mathrm{kW} \cdot  \mathrm{h}\)).
• The total energy used over a time interval can be found by \(\displaystyle E=\int P \, dt\).

Alternating Current versus Direct Current

• Direct current (DC) is the flow of electric current in only one direction. It refers to systems where the source voltage is constant.
• The voltage source of an alternating current (AC) system puts out \(\Delta V= V_{0} \sin({2\pi} ft)\), where \(\Delta V\) is the voltage at time \(t\), \(V_{0}\) is the peak voltage, and \(f\) is the frequency in hertz.
• In a simple circuit, \(I = \Delta V/R\) and AC current is \(I = I_{0} \sin({2\pi}ft)\), where \(I\) is the current at time \(t\), and \(I_{0} = V_{0}/R\) is the peak current.
• The average AC power is \(P_{\mathrm{ave}} = \frac{1}{2} I_{0}V_{0}\).
• Average (rms) current \(I_{\mathrm{rms}}\) and average (rms) voltage \(V_{\mathrm{rms}}\) and \(I_{\mathrm{rms}} = \frac{I_{0}}{\sqrt{2}}\) and \(V_{\mathrm{rms}} = \frac{V_{0}}{\sqrt{2}}\), where rms stands for root mean square.
• Thus, \(P_{\mathrm{ave}} = I_{\mathrm{rms}}V_{\mathrm{rms}}\).
• Ohm's law for AC is \(I_{\mathrm{rms}} = \frac{V_{\mathrm{rms}}}{R}\).
• Expressions for the average power of an AC circuit are \(P_{\mathrm{ave}} = I_{\mathrm{rms}}V_{\mathrm{rms}}\), \(P_{\mathrm{ave}} = \frac{V_{\mathrm{rms}}^{2}}{R}\), and \(P_{\mathrm{ave}} = I_{\mathrm{rms}}^{2}R\), analogous to the expressions for DC circuits.

Contributors and Attributions

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).