Current, Resistance, Voltage, and Power
Current
Current is a measure of the flow of electric charge through a material. A material that can carry a flow of charge is called a conductor. Current is defined as the amount of charge that flows through a conductor in a certain amount of time. The unit of current is the Ampere (A), which is equal to one Coulomb per second (Coulomb is the unit of charge),
![](/notes/ap_physics/images/current_1.png)
The symbol I is used to represent current (though J is often used in engineering sources). The current I through a conductor depends on its area A, the concentration n of charge carriers, the magnitude of the charge q of each carrier, and the magnitude of their average (or "drift") velocity vd,
![](/notes/ap_physics/images/current_2.png)
Current density is the amount of current flowing through a conductor divided by its area,
![](/notes/ap_physics/images/current_3.png)
The direction of the flow of current is defined in terms of a flow of positive charges (even if the actual charge carriers are negative). The unit of current density is Amperes per meter squared (A/m2).
Resistivity
Some conductors carry charge more easily than others. The resistivity of a material describes how easily charge can flow. Good conductors have a small resistivity, and good insulators have a large resistivity. The resistivity ρ (the Greek letter "rho") is equal to magnitude of the electric field in the material divided by the current density,
![](/notes/ap_physics/images/resistance_1.png)
The unit for the magnitude of an electric field is a Volt per meter (V/m), and the unit of current density is an Ampere per meter squared (A/m2), and so the unit for resistivity is a Volt-meter per Ampere,
![](/notes/ap_physics/images/resistance_2.png)
Many conductors follow Ohm's Law. Materials that follow Ohm's law have a constant resistivity no matter what the values of the electric field E and current density J are. The formulas related to circuits are true for "Ohmic" materials, and "non-Ohmic" materials are not discussed in this course.
The resistivity of an Ohmic conductor depends on the temperature of the material. The temperature-dependent resistivity ρ(T) can be found using the formula,
![](/notes/ap_physics/images/resistance_3.png)
This formula requires ρ0, the resistivity at a reference temperature T0. The temperature coefficient of resistivity α is different for every material. For temperatures in degrees Celsius (℃), the temperature coefficient of resistivity has units 1/℃ = (℃)(-1)
Resistance
Resistivity is a property of a type of material, but resistance is a property of a certain piece of that material. The resistance of a piece of conductor depends on its length L, area A, and resistivity ρ,
![](/notes/ap_physics/images/resistance_4.png)
The unit of resistance is the Ohm, which is represented with the Greek letter Ω ("omega"). One Ohm is equal to one Volt per Ampere,
![](/notes/ap_physics/images/resistance_5.png)
Resistance depends on temperature in the same way as resistivity,
![](/notes/ap_physics/images/resistance_6.png)
This formula requires R0, the resistance at a reference temperature T0. The temperature coefficient α is different for every material, as described in the Resistance section.
A resistor is a device that is used in electric circuits, and has a certain fixed resistance. Resistors are made by choosing a piece of material with a certain resistivity, length, and area, and wrapping it in an insulator with wires leading out of each end. In circuit diagrams, it is represented with the symbol,
![](/notes/ap_physics/images/resistance_7.png)
Voltage
Voltage is a difference in electric potential between two points. If an electric field is uniform through a conductor, the potential difference is,
![](/notes/ap_physics/images/voltage_1.png)
Using equations in the Current, Resistivity, and Resistance sections, another equation for the potential difference can be found,![](/notes/ap_physics/images/voltage_1.png)
![](/notes/ap_physics/images/voltage_2.png)
![](/notes/ap_physics/images/voltage_3.png)
![](/notes/ap_physics/images/voltage_4.png)
![](/notes/ap_physics/images/voltage_5.png)
The equation V = IR means that the potential difference, or voltage, across a resistor can be found by multiplying its resistance by the current flowing through it. The unit of potential difference is the Volt (V), which is equal to a Joule per Coulomb (J/C).
A voltage source is a device used in electric circuits that has a fixed potential difference between its ends. A voltage source can be a battery, or another source of direct current with a fixed potential difference. In circuit diagrams, it is represented with the symbol,
![](/notes/ap_physics/images/voltage_13.png)
If the ends of a voltage source are connected through a circuit with any number of resistors or other components, a complete circuit is formed, and current can flow from one terminal to the other. If current is flowing, it will be the same on both terminals of the voltage source.
A voltage source that is part of a complete circuit can produce an electromotive force, which is represented with the symbol ε ("script e"). The unit of electromotive force is the Volt (V), which is equal to a Joule per Coulomb (J/C). For an ideal source, the electromotive force is equal to the voltage difference,
![](/notes/ap_physics/images/voltage_7.png)
Real sources like batteries are not ideal, and so there is some amount of internal resistance. If the internal resistance of a battery is r, then the voltage difference across the battery is,
![](/notes/ap_physics/images/voltage_8.png)
This is also called the terminal voltage of the battery. If a complete circuit is made using a resistor with a resistance R, the current flowing through the circuit can be found by using the equation V = IR,
![](/notes/ap_physics/images/voltage_6.png)
![](/notes/ap_physics/images/voltage_9.png)
![](/notes/ap_physics/images/voltage_10.png)
![](/notes/ap_physics/images/voltage_11.png)
![](/notes/ap_physics/images/voltage_12.png)
The current is equal to the electromotive force of the source divided by the total circuit resistance.
Power
Power (P) is a measure of the rate at which energy is delivered or used by a circuit element. Voltage sources deliver power, while resistors use power (by dissipating it as heat). Power is equal to the voltage across a circuit element multiplied by the current flowing through it,
![](/notes/ap_physics/images/power_1.png)
The unit for power is the Watt (W), which is equal to a Joule per second,
![](/notes/ap_physics/images/power_2.png)
This relation can be found from the formula for power,
![](/notes/ap_physics/images/power_1.png)
![](/notes/ap_physics/images/power_3.png)
![](/notes/ap_physics/images/power_4.png)
![](/notes/ap_physics/images/power_5.png)
![](/notes/ap_physics/images/power_6.png)
The power used or dissipated by a resistor can be found using the formula V = IR. This formula can be used to substitute for the voltage or for the current in the power formula,
![](/notes/ap_physics/images/power_1.png)
,
and,
![](/notes/ap_physics/images/power_1.png)
![](/notes/ap_physics/images/power_8.png)
The power output of a battery that has internal resistance can be found using the formula V = ε-Ir and the formula for power,
![](/notes/ap_physics/images/power_1.png)
![](/notes/ap_physics/images/power_9.png)
![](/notes/ap_physics/images/power_10.png)
Current is a measure of the flow of electric charge through a material. A material that can carry a flow of charge is called a conductor. Current is defined as the amount of charge that flows through a conductor in a certain amount of time. The unit of current is the Ampere (A), which is equal to one Coulomb per second (Coulomb is the unit of charge),
![](/notes/ap_physics/images/current_1.png)
The symbol I is used to represent current (though J is often used in engineering sources). The current I through a conductor depends on its area A, the concentration n of charge carriers, the magnitude of the charge q of each carrier, and the magnitude of their average (or "drift") velocity vd,
![](/notes/ap_physics/images/current_2.png)
Current density is the amount of current flowing through a conductor divided by its area,
![](/notes/ap_physics/images/current_3.png)
The direction of the flow of current is defined in terms of a flow of positive charges (even if the actual charge carriers are negative). The unit of current density is Amperes per meter squared (A/m2).
Resistivity
Some conductors carry charge more easily than others. The resistivity of a material describes how easily charge can flow. Good conductors have a small resistivity, and good insulators have a large resistivity. The resistivity ρ (the Greek letter "rho") is equal to magnitude of the electric field in the material divided by the current density,
![](/notes/ap_physics/images/resistance_1.png)
The unit for the magnitude of an electric field is a Volt per meter (V/m), and the unit of current density is an Ampere per meter squared (A/m2), and so the unit for resistivity is a Volt-meter per Ampere,
![](/notes/ap_physics/images/resistance_2.png)
Many conductors follow Ohm's Law. Materials that follow Ohm's law have a constant resistivity no matter what the values of the electric field E and current density J are. The formulas related to circuits are true for "Ohmic" materials, and "non-Ohmic" materials are not discussed in this course.
The resistivity of an Ohmic conductor depends on the temperature of the material. The temperature-dependent resistivity ρ(T) can be found using the formula,
![](/notes/ap_physics/images/resistance_3.png)
This formula requires ρ0, the resistivity at a reference temperature T0. The temperature coefficient of resistivity α is different for every material. For temperatures in degrees Celsius (℃), the temperature coefficient of resistivity has units 1/℃ = (℃)(-1)
Resistance
Resistivity is a property of a type of material, but resistance is a property of a certain piece of that material. The resistance of a piece of conductor depends on its length L, area A, and resistivity ρ,
![](/notes/ap_physics/images/resistance_4.png)
The unit of resistance is the Ohm, which is represented with the Greek letter Ω ("omega"). One Ohm is equal to one Volt per Ampere,
![](/notes/ap_physics/images/resistance_5.png)
Resistance depends on temperature in the same way as resistivity,
![](/notes/ap_physics/images/resistance_6.png)
This formula requires R0, the resistance at a reference temperature T0. The temperature coefficient α is different for every material, as described in the Resistance section.
A resistor is a device that is used in electric circuits, and has a certain fixed resistance. Resistors are made by choosing a piece of material with a certain resistivity, length, and area, and wrapping it in an insulator with wires leading out of each end. In circuit diagrams, it is represented with the symbol,
![](/notes/ap_physics/images/resistance_7.png)
Voltage
Voltage is a difference in electric potential between two points. If an electric field is uniform through a conductor, the potential difference is,
![](/notes/ap_physics/images/voltage_1.png)
Using equations in the Current, Resistivity, and Resistance sections, another equation for the potential difference can be found,
![](/notes/ap_physics/images/voltage_1.png)
![](/notes/ap_physics/images/voltage_2.png)
![](/notes/ap_physics/images/voltage_3.png)
![](/notes/ap_physics/images/voltage_4.png)
![](/notes/ap_physics/images/voltage_5.png)
The equation V = IR means that the potential difference, or voltage, across a resistor can be found by multiplying its resistance by the current flowing through it. The unit of potential difference is the Volt (V), which is equal to a Joule per Coulomb (J/C).
A voltage source is a device used in electric circuits that has a fixed potential difference between its ends. A voltage source can be a battery, or another source of direct current with a fixed potential difference. In circuit diagrams, it is represented with the symbol,
![](/notes/ap_physics/images/voltage_13.png)
If the ends of a voltage source are connected through a circuit with any number of resistors or other components, a complete circuit is formed, and current can flow from one terminal to the other. If current is flowing, it will be the same on both terminals of the voltage source.
A voltage source that is part of a complete circuit can produce an electromotive force, which is represented with the symbol ε ("script e"). The unit of electromotive force is the Volt (V), which is equal to a Joule per Coulomb (J/C). For an ideal source, the electromotive force is equal to the voltage difference,
![](/notes/ap_physics/images/voltage_7.png)
Real sources like batteries are not ideal, and so there is some amount of internal resistance. If the internal resistance of a battery is r, then the voltage difference across the battery is,
![](/notes/ap_physics/images/voltage_8.png)
This is also called the terminal voltage of the battery. If a complete circuit is made using a resistor with a resistance R, the current flowing through the circuit can be found by using the equation V = IR,
![](/notes/ap_physics/images/voltage_6.png)
![](/notes/ap_physics/images/voltage_9.png)
![](/notes/ap_physics/images/voltage_10.png)
![](/notes/ap_physics/images/voltage_11.png)
![](/notes/ap_physics/images/voltage_12.png)
The current is equal to the electromotive force of the source divided by the total circuit resistance.
Power
Power (P) is a measure of the rate at which energy is delivered or used by a circuit element. Voltage sources deliver power, while resistors use power (by dissipating it as heat). Power is equal to the voltage across a circuit element multiplied by the current flowing through it,
![](/notes/ap_physics/images/power_1.png)
The unit for power is the Watt (W), which is equal to a Joule per second,
![](/notes/ap_physics/images/power_2.png)
This relation can be found from the formula for power,
![](/notes/ap_physics/images/power_1.png)
![](/notes/ap_physics/images/power_3.png)
![](/notes/ap_physics/images/power_4.png)
![](/notes/ap_physics/images/power_5.png)
![](/notes/ap_physics/images/power_6.png)
The power used or dissipated by a resistor can be found using the formula V = IR. This formula can be used to substitute for the voltage or for the current in the power formula,
![](/notes/ap_physics/images/power_1.png)
![](/notes/ap_physics/images/power_7.png)
and,
![](/notes/ap_physics/images/power_1.png)
![](/notes/ap_physics/images/power_8.png)
The power output of a battery that has internal resistance can be found using the formula V = ε-Ir and the formula for power,
![](/notes/ap_physics/images/power_1.png)
![](/notes/ap_physics/images/power_9.png)
![](/notes/ap_physics/images/power_10.png)
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