11.3 Resistance, Resistivity, and Ohm's Law

Resistance measures how strongly an object opposes the flow of electric charge, and it depends on the material's resistivity, the object's length, and its cross sectional area through $R = \frac{\rho \ell}{A}$. Ohm's law, $I = \frac{\Delta V}{R}$, connects current, potential difference, and resistance for circuit elements, and ohmic materials keep a constant resistance shown by a straight line through the origin on an I-V graph.

Why This Matters for the AP Physics C: E&M Exam

This topic gives you the tools to predict how current responds to potential difference and to connect a resistor's physical shape to its electrical behavior. Both the resistance formula and Ohm's law show up across the circuit unit, which carries a large share of the multiple-choice section, so being fast with them helps everywhere from simple circuits to compound circuits and RC circuits.

The free-response section includes an experimental design and analysis question. The data-analysis skills here, plotting current versus potential difference, finding resistance from a slope, and linearizing data, are exactly the kind of reasoning that question rewards. You may be asked to design a procedure to test whether a material is ohmic, collect and graph data, and justify a claim from your graph.

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Key Takeaways

• Resistance comes from material and shape: $R = \frac{\rho \ell}{A}$. Longer means more resistance, wider cross-section means less.
• Resistivity $\rho$ is a material property measured in ohm-meters, and for conductors it usually increases as temperature rises.
• Ohm's law is $I = \frac{\Delta V}{R}$. Ohmic materials have constant resistance and a linear I vs V graph through the origin.
• On an I vs V graph, the slope equals $\frac{1}{R}$, so a steeper line means lower resistance. On a V vs I graph, the slope equals $R$.
• When resistivity varies along the length, integrate: $R = \int \frac{\rho(\ell)\, d\ell}{A}$.
• Resistors convert electrical energy to thermal energy, and this heating can change the temperature of the resistor and its surroundings.

Resistance from Physical Properties

For an object with uniform geometry, resistance depends on both the material and the object's dimensions:

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

Resistance increases when the resistivity $\rho$ increases, increases when the length $\ell$ increases, and decreases when the cross-sectional area $A$ increases. A longer path gives charges more material to move through, while a larger area provides more room for charge to flow.

• Resistivity ($\rho$) is a fundamental property of a material that depends on its atomic and molecular structure and quantifies how strongly the material opposes the motion of electric charge.
  • Resistivity is measured in ohm-meters (Ω·m).
  • For conductors, resistivity typically increases as temperature increases, so the resistance of a metal wire usually increases when it gets hotter.
• If the resistor has uniform cross-sectional area $A$ but the resistivity varies along its length, the total resistance is found by summing small pieces:

$R = \int \frac{\rho(\ell)\, d\ell}{A}$

Electrical Characteristics of Circuit Elements

Ohm's Law in Circuits

Ohm's law relates current, resistance, and potential difference across a conductive circuit element. The current through the element is directly proportional to the potential difference across it, with resistance as the proportionality constant.

• Ohm's law is written as $I=\frac{\Delta V}{R}$, where:
  • $\Delta V$ is the potential difference (voltage) measured in volts (V)
  • $I$ is the current measured in amperes (A)
  • $R$ is the resistance measured in ohms (Ω)
• This is equivalent to the familiar form $\Delta V = IR$.
• Ohmic materials maintain a constant resistance regardless of the current passing through them.
  • For an ohmic material, current is proportional to potential difference, so the material has constant resistance and a constant resistivity. In the model used here, the resistivity of an ohmic material is treated as constant regardless of temperature.
  • Their current-voltage ($I$-$V$) graph is a straight line passing through the origin.
  • Examples include most metals over limited temperature ranges.
• Non-ohmic materials have resistance that varies with current or voltage.
  • Their $I$-$V$ graph is non-linear.
  • Examples include diodes, transistors, and light bulbs when their temperature changes.
• The resistance of an ohmic circuit element can be determined from the slope of a graph of current as a function of potential difference:
  • On a graph of current $I$ as a function of potential difference $\Delta V$ for an ohmic element, the slope is $\frac{\Delta I}{\Delta V} = \frac{1}{R}$. The resistance is the reciprocal of the slope: $R = \frac{1}{(\Delta I / \Delta V)}$.
  • A steeper slope on an $I$ vs. $\Delta V$ graph means lower resistance.
  • If voltage is graphed as a function of current (a $V$ vs. $I$ graph), the slope is $\frac{\Delta V}{\Delta I} = R$.

Experimental Determination of Resistance

To determine whether a circuit element is ohmic and to find its resistance, vary the potential difference across the element, measure the current for several trials, and plot current $I$ (in amperes) versus potential difference $\Delta V$ (in volts).

• If the graph is linear and passes through the origin, the element is ohmic under those conditions.
• The slope of an $I$ versus $\Delta V$ graph is $\frac{1}{R}$, so the resistance is the reciprocal of the slope.
• To make the conclusion reliable, keep physical conditions such as temperature as constant as possible while collecting the data.
• Resistors convert electrical energy to thermal energy.
  • This power follows $P = I^2R = \frac{\Delta V^2}{R} = I\Delta V$.
  • This heating can change the temperature of the resistor and its surroundings.
  • For example, a light bulb filament heats up and glows when current flows through it.

How to Use This on the AP Physics C: E&M Exam

Problem Solving

• Pick the right form of the equation for what you are given. Use $R = \frac{\rho \ell}{A}$ when you have material and geometry, and use $I = \frac{\Delta V}{R}$ when you have circuit quantities.
• For "factor of change" questions, write a ratio. If length doubles with the same material and area, $R$ doubles, so for fixed $\Delta V$ the current halves.
• Carry units through every step. Resistivity in ohm-meters, length in meters, and area in square meters give resistance in ohms.
• When resistivity is given as a function of position, set up the integral $R = \int \frac{\rho(\ell)\, d\ell}{A}$ instead of plugging into the simple formula.

Free Response

• For the experimental design and analysis question, describe a clear procedure: vary $\Delta V$, measure $I$ at several settings, and plot the data.
• Justify whether a material is ohmic using your graph. A straight line through the origin supports an ohmic claim; curvature supports a non-ohmic claim.
• Find resistance from a slope and state it correctly. On an I vs V graph the slope is $\frac{1}{R}$, so take the reciprocal.
• Mention controlling temperature as a way to reduce error, since heating changes resistivity for many materials.

Common Trap

• Reading resistance directly as the slope of an I vs V graph. The slope is $\frac{1}{R}$, not $R$.

Practice Problem 1: Ohm's Law Application

Solution

To find the resistance, apply Ohm's law: $\Delta V = IR$

Rearranging for resistance: $R = \frac{\Delta V}{I} = \frac{12\text{ V}}{2\text{ A}} = 6\text{ Ω}$

When the length is doubled while keeping the same material and cross-sectional area, use $R = \frac{\rho \ell}{A}$:

• The original resistance is $R_1 = \frac{\rho \ell_1}{A}$
• The new resistance is $R_2 = \frac{\rho (2\ell_1)}{A} = 2 \times \frac{\rho \ell_1}{A} = 2R_1$

So the resistance doubles to 12 Ω.

Using Ohm's law again with the new resistance: $I_2 = \frac{\Delta V}{R_2} = \frac{12\text{ V}}{12\text{ Ω}} = 1\text{ A}$

When the length doubles, the resistance doubles and the current is reduced by half.

Practice Problem 2: I-V Graph Analysis

Solution

To determine if this is an ohmic material, check whether the relationship between potential difference and current is linear.

Calculate the resistance at each data point using $R = \frac{\Delta V}{I}$:

• At (2V, 0.4A): $R = \frac{2\text{ V}}{0.4\text{ A}} = 5\text{ Ω}$
• At (4V, 0.8A): $R = \frac{4\text{ V}}{0.8\text{ A}} = 5\text{ Ω}$
• At (6V, 1.2A): $R = \frac{6\text{ V}}{1.2\text{ A}} = 5\text{ Ω}$
• At (8V, 1.6A): $R = \frac{8\text{ V}}{1.6\text{ A}} = 5\text{ Ω}$

Since the resistance is constant (5 Ω) at all values, and the I-V relationship is linear (current doubles when potential difference doubles), this is an ohmic material.

The resistance is 5 Ω. On a graph of $I$ vs. $\Delta V$, the slope is $\frac{\Delta I}{\Delta V} = \frac{1.6\text{ A} - 0.4\text{ A}}{8\text{ V} - 2\text{ V}} = \frac{1.2\text{ A}}{6\text{ V}} = 0.2\text{ A/V}$. Since the slope equals $\frac{1}{R}$, the resistance is $R = \frac{1}{0.2\text{ A/V}} = 5\text{ Ω}$.

Common Misconceptions

• "Resistance and resistivity are the same thing." Resistivity $\rho$ is a property of the material itself, while resistance $R$ also depends on the shape and size of the object through $R = \frac{\rho \ell}{A}$.
• "The slope of an I vs V graph is the resistance." The slope is $\frac{1}{R}$. You have to take the reciprocal to get resistance.
• "Every material obeys Ohm's law." Only ohmic materials have constant resistance. Diodes, transistors, and heated filaments are non-ohmic and have curved I-V graphs.
• "Resistivity never changes." For most conductors, resistivity increases as temperature rises. The constant-resistivity idea applies to the idealized ohmic model.
• "A wider resistor has more resistance because there is more material." Increasing the cross-sectional area lowers resistance because there is more room for charge to flow.
• "Ohm's law lets current flow without any energy cost." Resistors convert electrical energy to thermal energy, and that heating can raise the temperature of the resistor and its surroundings.

Related AP Physics C: E&M Guides

• 11.1 Electric Current
• 11.2 Electric Circuits
• 11.5 Compound Direct Current Circuits
• 11.8 Resistor-Capacitor (RC) Circuits
• 11.4 Electric Power
• 11.6 Kirchhoff's Loop Rule