11.3 Resistance, Resistivity, and Ohm's Law

Resistance measures how much an object opposes the flow of electric charge, and it depends on the material's resistivity along with the object's length and cross sectional area through $R = \frac{\rho \ell}{A}$. Ohm's law, $I = \frac{\Delta V}{R}$, ties current, potential difference, and resistance together for ohmic elements, where resistance stays constant and an $I$ versus $\Delta V$ graph is a straight line. Topic 11.3, Resistance, Resistivity, and Ohm's Law is part of AP Physics 2 in Unit 11 - Electric Circuits.

Why This Matters for the AP Physics 2 Exam

This topic gives you the tools to predict and explain how circuit elements behave, which supports both calculation and explanation tasks on the exam. You will use $R = \frac{\rho \ell}{A}$ to reason about how a wire's material and shape change its resistance, and Ohm's law to connect current and potential difference. The free-response section often asks you to analyze data and graphs, so being able to read the slope of an $I$-versus-$\Delta V$ graph and decide whether a material is ohmic is directly useful. You also need precise vocabulary here, since "resistance" and "resistivity" mean different things and mixing them up costs points when you justify claims.

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Key Takeaways

• Resistance ($R$, in ohms) measures opposition to charge flow; resistivity ($\rho$) is a material property set by atomic and molecular structure.
• Use $R = \frac{\rho \ell}{A}$: longer length raises resistance, larger cross-sectional area lowers it.
• Ohm's law $I = \frac{\Delta V}{R}$ applies to ohmic materials, which have constant resistance regardless of current.
• For an ohmic element, an $I$-versus-$\Delta V$ graph is a straight line whose slope equals $\frac{1}{R}$.
• Resistivity of a real conductor typically increases with temperature, while an ohmic material is modeled with constant resistivity.
• Resistors convert electrical energy to thermal energy, which can change the temperature of the resistor and its surroundings.

Resistance and Resistivity

Resistance is a measure of how strongly an object opposes the movement of electric charge. Resistivity is a fundamental property of a material that depends on its atomic and molecular structure and describes how strongly that material opposes the motion of charge. As charges move through a material, they interact with the atoms in it, and that interaction limits how easily charge flows.

• Resistance ($R$) is measured in ohms ($\Omega$).
• Higher resistance means less current flows for a given potential difference.
• Resistance depends on both the material and the object's physical dimensions.

For a resistor with uniform geometry, resistance follows this relationship:

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

Where:

• $\rho$ (rho) is the resistivity, an intrinsic material property set by the material's atomic and molecular structure.
• $\ell$ is the length of the material.
• $A$ is the cross-sectional area.

This relationship shows that:

• Doubling the length doubles the resistance.
• Doubling the cross-sectional area halves the resistance.
• Different materials have different resistivities, so they oppose charge motion by different amounts.

Temperature affects resistivity in most materials. For a typical conductor, resistivity increases as temperature rises, because stronger thermal vibrations of the atoms make it harder for charge to move through.

Ohm's Law and Electrical Characteristics

Ohm's law relates current, resistance, and potential difference across a conductive element of a circuit.

$I = \frac{\Delta V}{R}$

Where:

• $\Delta V$ is the potential difference measured in volts (V).
• $I$ is the current measured in amperes (A).
• $R$ is the resistance measured in ohms ($\Omega$).

You can rearrange this relationship to solve for any of the three variables:

• $\Delta V = IR$
• $R = \frac{\Delta V}{I}$

Materials that obey Ohm's law are called ohmic:

• Ohmic materials have constant resistance for all currents.
• For an ohmic element with fixed dimensions, current is directly proportional to potential difference, so the current-voltage relationship is linear.
• In this course model, the resistivity of an ohmic material is taken to be constant regardless of temperature.
• For an ohmic element, a graph of current $I$ as a function of potential difference $\Delta V$ is a straight line.
• The resistance of an ohmic circuit element can be found from the slope of an $I$-versus-$\Delta V$ graph. Since $I = \frac{\Delta V}{R}$, the slope is $\frac{1}{R}$, so $R = \frac{1}{\text{slope}}$.
• A steeper $I$-versus-$\Delta V$ line means smaller resistance, and a less-steep line means larger resistance.
• On a $\Delta V$-versus-$I$ graph, the slope is $\frac{\Delta V}{I} = R$.

Many materials are non-ohmic. Their resistance changes with current or potential difference, so their current-voltage graphs are curved rather than straight.

When current flows through a resistor, electrical energy is converted to thermal energy. This can change the temperature of both the resistor and the resistor's environment. The rate of that energy conversion is given by $P = I\Delta V = I^2 R = \frac{(\Delta V)^2}{R}$.

How to Use This on the AP Physics 2 Exam

Problem Solving

• When a problem changes a wire's dimensions, go straight to $R = \frac{\rho \ell}{A}$ and track how each factor scales. Length and resistance scale together; area and resistance scale in opposite directions.
• For Ohm's law problems, identify which two of $I$, $\Delta V$, and $R$ you know, then solve for the third. Keep units consistent (volts, amperes, ohms).
• If a resistor's temperature changes in a real conductor, expect resistivity and resistance to rise, which lowers current for a fixed potential difference.

Free Response

• To find resistance from data, plot current versus potential difference and use $R = \frac{1}{\text{slope}}$. Plotting $\Delta V$ versus $I$ instead gives slope equal to $R$ directly; state which axes you used.
• To decide whether a material is ohmic, check whether the $I$-versus-$\Delta V$ data form a straight line through the origin. A curved graph signals non-ohmic behavior.
• When you justify a claim, use the correct term. Say "resistance" when you mean opposition for a specific object, and "resistivity" when you mean the material property.

Common Trap

• Do not confuse resistance and resistivity. Resistivity ($\rho$) belongs to the material; resistance ($R$) depends on resistivity and the object's geometry.

Practice Problem 1: Calculating Resistance

Solution

Use the resistance equation that relates resistivity, length, and cross-sectional area:

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

Substituting the given values:

• Resistivity ($\rho$) = $1.7 \times 10^{-8}\ \Omega \cdot \text{m}$
• Length ($\ell$) = 10 m
• Cross-sectional area ($A$) = $2.0 \times 10^{-6}\ \text{m}^2$

$R = (1.7 \times 10^{-8}\ \Omega \cdot \text{m}) \times \frac{10\ \text{m}}{2.0 \times 10^{-6}\ \text{m}^2}$

$R = (1.7 \times 10^{-8}\ \Omega \cdot \text{m}) \times (5.0 \times 10^{6}\ \text{m}^{-1})$

$R = 0.085\ \Omega$

The resistance of the copper wire is $0.085\ \Omega$.

Practice Problem 2: Applying Ohm's Law

Solution

First, find the resistance using Ohm's law, rearranged as $R = \frac{\Delta V}{I}$.

Given:

• Potential difference ($\Delta V$) = 12 V
• Current ($I$) = 0.5 A

$R = \frac{12\ \text{V}}{0.5\ \text{A}} = 24\ \Omega$

For the second part, switch from the ohmic course-model simplification to the behavior of a real metal resistor. For most conductors, resistivity increases with temperature, so resistance increases.

If resistance increases while the potential difference stays at 12 V, then by $I = \frac{\Delta V}{R}$ the current must decrease, because current is inversely proportional to resistance when potential difference is constant.

So as the resistor heats up, you would expect the current in the circuit to decrease.

Common Misconceptions

• Resistance and resistivity are not the same. Resistivity is a property of the material; resistance also depends on the object's length and cross-sectional area.
• A bigger cross-sectional area lowers resistance, not raises it. More room for charge to move means easier flow.
• Ohm's law does not apply to every material. Only ohmic materials have a constant resistance and a straight-line current-voltage graph.
• The slope of an $I$-versus-$\Delta V$ graph is $\frac{1}{R}$, not $R$. The slope equals $R$ only when you plot $\Delta V$ versus $I$.
• Current is not "used up" in a resistor. The resistor converts some electrical energy to thermal energy, but charge is not consumed.
• Real conductors change resistance with temperature, even though the idealized ohmic model treats resistivity as constant.

Related AP Physics 2 Guides

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