College Physics for AP® Courses

# 20.2Ohm’s Law: Resistance and Simple Circuits

College Physics for AP® Courses20.2 Ohm’s Law: Resistance and Simple Circuits

### Learning Objectives

By the end of this section, you will be able to:

• Explain the origin of Ohm's law.
• Calculate voltages, currents, and resistances with Ohm's law.
• Explain the difference between ohmic and non-ohmic materials.
• Describe a simple circuit.

The information presented in this section supports the following AP® learning objectives and science practices:

• 4.E.4.1 The student is able to make predictions about the properties of resistors and/or capacitors when placed in a simple circuit based on the geometry of the circuit element and supported by scientific theories and mathematical relationships. (S.P. 2.2, 6.4)

What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—which are necessary to maintain a current. All such devices create a potential difference and are loosely referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference $VV size 12{V} {}$ that creates an electric field. The electric field in turn exerts force on charges, causing current.

### Ohm's Law

The current that flows through most substances is directly proportional to the voltage $VV size 12{V} {}$ applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is directly proportional to the voltage applied:

$I∝V.I∝V. size 12{I prop V.} {}$
20.12

This important relationship is known as Ohm's law. It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn't always occur.

### Resistance and Simple Circuits

If voltage drives current, what impedes it? The electric property that impedes current (crudely similar to friction and air resistance) is called resistance $RR size 12{R} {}$. Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is defined as inversely proportional to current, or

$I∝1R.I∝1R. size 12{I prop { {1} over {R} } "."} {}$
20.13

Thus, for example, current is cut in half if resistance doubles. Combining the relationships of current to voltage and current to resistance gives

$I=VR.I=VR. size 12{I = { {V} over {R} } "."} {}$
20.14

This relationship is also called Ohm's law. Ohm's law in this form really defines resistance for certain materials. Ohm's law (like Hooke's law) is not universally valid. The many substances for which Ohm's law holds are called ohmic. These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance $RR size 12{R} {}$ that is independent of voltage $VV size 12{V} {}$ and current $II size 12{I} {}$. An object that has simple resistance is called a resistor, even if its resistance is small. The unit for resistance is an ohm and is given the symbol $ΩΩ size 12{ %OMEGA } {}$ (upper case Greek omega). Rearranging $I=V/RI=V/R size 12{I = ital "V/R"} {}$ gives $R=V/IR=V/I size 12{R= ital "V/I"} {}$, and so the units of resistance are 1 ohm = 1 volt per ampere:

$1 Ω= 1 VA.1 Ω= 1 VA. size 12{"1 " %OMEGA =" 1 " { {V} over {A} } "."} {}$
20.15

Figure 20.8 shows the schematic for a simple circuit. A simple circuit has a single voltage source and a single resistor. The wires connecting the voltage source to the resistor can be assumed to have negligible resistance, or their resistance can be included in $RR size 12{R} {}$.

Figure 20.8 A simple electric circuit in which a closed path for current to flow is supplied by conductors (usually metal wires) connecting a load to the terminals of a battery, represented by the red parallel lines. The zigzag symbol represents the single resistor and includes any resistance in the connections to the voltage source.

### Making Connections: Real World Connections

Ohm's law ($V=IRV=IR$) is a fundamental relationship that could be presented by a linear function with the slope of the line being the resistance. The resistance represents the voltage that needs to be applied to the resistor to create a current of 1 A through the circuit. The graph (in the figure below) shows this representation for two simple circuits with resistors that have different resistances and thus different slopes.

Figure 20.9 The figure illustrates the relationship between current and voltage for two different resistors. The slope of the graph represents the resistance value, which is 2Ω and 4Ω for the two lines shown.

### Making Connections: Real World Connections

The materials which follow Ohm's law by having a linear relationship between voltage and current are known as ohmic materials. On the other hand, some materials exhibit a nonlinear voltage-current relationship and hence are known as non-ohmic materials. The figure below shows current voltage relationships for the two types of materials.

Figure 20.10 The relationship between voltage and current for ohmic and non-ohmic materials are shown.

Clearly the resistance of an ohmic material (shown in (a)) remains constant and can be calculated by finding the slope of the graph but that is not true for a non-ohmic material (shown in (b)).

### Example 20.4

#### Calculating Resistance: An Automobile Headlight

What is the resistance of an automobile headlight through which 2.50 A flows when 12.0 V is applied to it?

#### Strategy

We can rearrange Ohm's law as stated by $I=V/RI=V/R size 12{I = ital "V/R"} {}$ and use it to find the resistance.

#### Solution

Rearranging $I=V/RI=V/R size 12{I = ital "V/R"} {}$ and substituting known values gives

$R=VI=12.0 V2.50 A= 4.80 Ω.R=VI=12.0 V2.50 A= 4.80 Ω. size 12{R = { {V} over {I} } = { {"12" "." "0 V"} over {2 "." "50 A"} } =" 4" "." "80 " %OMEGA "."} {}$
20.16

#### Discussion

This is a relatively small resistance, but it is larger than the cold resistance of the headlight. As we shall see in Resistance and Resistivity, resistance usually increases with temperature, and so the bulb has a lower resistance when it is first switched on and will draw considerably more current during its brief warm-up period.

Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of $1012Ω1012Ω size 12{"10" rSup { size 8{"12"} }  %OMEGA } {}$ or more. A dry person may have a hand-to-foot resistance of $105Ω105Ω size 12{"10" rSup { size 8{5} }  %OMEGA } {}$, whereas the resistance of the human heart is about $103Ω103Ω size 12{"10" rSup { size 8{3} }  %OMEGA } {}$. A meter-long piece of large-diameter copper wire may have a resistance of $10−5Ω10−5Ω size 12{"10" rSup { size 8{ - 5} }  %OMEGA } {}$, and superconductors have no resistance at all (they are non-ohmic). Resistance is related to the shape of an object and the material of which it is composed, as will be seen in Resistance and Resistivity.

Additional insight is gained by solving $I=V/RI=V/R size 12{I = ital "V/R"} {}$ for $V,V, size 12{V} {}$ yielding

$V=IR.V=IR. size 12{V = ital "IR."} {}$
20.17

This expression for $VV size 12{V} {}$ can be interpreted as the voltage drop across a resistor produced by the current $II size 12{I} {}$. The phrase $IRIR size 12{ ital "IR"} {}$ drop is often used for this voltage. For instance, the headlight in Example 20.4 has an $IRIR size 12{ ital "IR"} {}$ drop of 12.0 V. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form (such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since $PE=qΔVPE=qΔV size 12{"PE"=qΔV} {}$, and the same $qq size 12{q} {}$ flows through each. Thus the energy supplied by the voltage source and the energy converted by the resistor are equal. (See Figure 20.11.)

Figure 20.11 The voltage drop across a resistor in a simple circuit equals the voltage output of the battery.

### Making Connections: Conservation of Energy

In a simple electrical circuit, the sole resistor converts energy supplied by the source into another form. Conservation of energy is evidenced here by the fact that all of the energy supplied by the source is converted to another form by the resistor alone. We will find that conservation of energy has other important applications in circuits and is a powerful tool in circuit analysis.

Ohm's Law

See how the equation form of Ohm's law relates to a simple circuit. Adjust the voltage and resistance, and see the current change according to Ohm's law. The sizes of the symbols in the equation change to match the circuit diagram.

Figure 20.12
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