Resistance, Resistivity, and Ohm’s Law
Separate object resistance from material resistivity and test a controlled-temperature ohmic model.
AP Physics 2 Topic 11.3 · College Board CED printed p. 64, 65 / PDF p. 71, 72 · See this topic in the course outline
Resistance depends on both material and geometry. A graph can test whether treating it as constant is justified.
For a supporting original reading, see Current and Resistance. Compare material resistivity with object resistance. Interpret constant-resistivity calculations as ideal models; real metallic resistivity usually rises with temperature.
Resistance is an object property; resistivity is a material property
Resistance measures how strongly an object opposes current: for a given potential difference, higher resistance means smaller current. For a uniform resistor of length ℓ and cross-sectional area A, . R is in ohms (Ω = V/A); resistivity ρ is in Ω·m. A longer object gives carriers a longer resistive path; a wider cross section provides more parallel flow paths.
Resistivity reflects the material’s atomic and molecular structure and scattering of carriers. Changing the material can change ρ without changing geometry. Real metallic conductor resistivity typically increases with temperature, so a heated wire can have greater resistance even when its dimensions are nearly unchanged. “Typically” allows exceptions and other materials with different temperature behavior.
| Change | R factor |
|---|---|
| Length doubles, area fixed | 2 |
| Radius doubles for circular cross section, length fixed | 1/4, because A = πr² |
| ρ and length both double, area triples | 4/3 |
An ohmic model has a fixed slope
Ohm’s relation is . An ohmic element has constant resistance over the currents being modeled, giving an I-versus-ΔV line through the origin. Its slope is , so R is the reciprocal slope. Swapping axes gives a ΔV-versus-I slope R. Check axis labels before interpreting any slope.
The ideal ohmic model used here holds resistivity and geometry fixed; the framework also treats its resistivity as temperature independent. This is a modeling idealization, not a universal consequence of the word “ohmic.” A real metal can be approximately ohmic at each fixed temperature and still have larger resistivity at a higher temperature. Therefore use controlled temperature or negligible heating for the constant-R graph. The real-metal trend in the previous passage and the ideal fixed-ρ model describe different assumptions.
Resistors convert electrical energy to thermal energy, which can raise their own temperature and the surroundings’ temperature. A bulb’s filament often heats strongly; its changing resistance can curve an I–V graph and invalidate a single constant-R description over a broad range.
| ΔV (V) | I (A) |
|---|---|
| 0 | 0 |
| 2 | 0.10 |
| 4 | 0.20 |
| 6 | 0.30 |
The slope is 0.050 A/V and resistance 20 Ω. A line with a nonzero intercept may reflect a sensor offset or a different device model; do not silently treat it as an ideal resistor through the origin.
For a 20 Ω resistor, the table values (ΔV in V, I in A) are (0,0), (1,0.050), (2,0.100), and (3,0.150). Plot them with quantitative scales: the slope is 0.050 A/V, so R = 1/slope = 20 Ω. This is a tested range at fixed temperature; it does not prove that the real material ignores heating.
Material and geometry, then an I–V check
A uniform wire has ρ = 1.0 × 10⁻⁶ Ω·m, length 0.80 m, and circular radius 0.20 mm. Find R and the ideal current at 4.0 V. Assume its temperature remains fixed.
Convert radius to 2.0 × 10⁻⁴ m; A = πr² = 1.26 × 10⁻⁷ m².
Why this step? The equation uses area, not radius or diameter.
R = ρℓ/A = 6.37 Ω. I = ΔV/R = 0.628 A.
Why this step? Uniform geometry and fixed resistivity permit the object-resistance and Ohm relations.
The expected I-versus-ΔV slope is 1/R = 0.157 A/V. Doubling radius at the same length would quadruple area and reduce R to 1.59 Ω.
Why this step? A graph and a geometry scaling provide independent representations of the same model.
Answer and check
R = 6.37 Ω and I = 0.628 A in the stated fixed-temperature model. Heating in a real trial could change these predictions.
Structure sets resistivity; geometry sets the object resistance
In a metal, an electric field supports drift, but carrier interactions and scattering transfer energy to the material. Its atomic and molecular structure determines how strongly it opposes charge motion. Resistivity ρ quantifies that intrinsic material response, in Ω·m. Composition, arrangement and defects can change carrier behavior, so equal-shaped objects made from different materials need not have the same ρ.
For uniform geometry, R = ρL/A. Doubling length increases the resistive path and R; doubling area provides more parallel paths and lowers R. Neither change alone alters ρ at the same material state. Higher ρ gives smaller current at the same field in the stipulated linear-response comparison and larger R at the same geometry. A resistance or I–V graph alone does not uniquely identify microscopic structure.
Real metal resistivity typically rises as temperature rises because carrier interactions change. An approximately ohmic metal at each controlled temperature need not have temperature-independent resistivity. The AP ideal ohmic model explicitly holds ρ fixed, including its temperature-independent idealization; apply that model only when stated rather than treating it as a universal property of real materials.
A graph compares two material coefficients
Two uniform wires have L = 0.40 m and A = 1.0 × 10⁻⁷ m² at the same controlled temperature. Material A has ρ_A = 1.0 × 10⁻⁶ Ω·m and B has ρ_B = 2.0 × 10⁻⁶ Ω·m. Both are ohmic over 0–4 V. In the stipulated material model, B’s different structure produces stronger opposition to carrier drift.
- R_A = ρ_AL/A = 4.0 Ω; R_B = 8.0 Ω. Equal geometry isolates the material coefficient.
- At 4.0 V, I_A = 1.0 A and I_B = 0.50 A. On I versus V axes the slopes are 0.25 A/V and 0.125 A/V. The higher-resistivity material has the shallower current slope.
- The coefficient difference represents atomic/molecular structure and carrier interactions, not a longer path. Doubling A-wire length doubles its R to 8.0 Ω but leaves ρ_A = 1.0 × 10⁻⁶ Ω·m.
- These straight-line predictions assume maintained temperature and constant geometry; heating in an uncontrolled real trial could shift slopes.
Answer and check
Wire resistances 4.0 and 8.0 Ω, with a 2:1 current ratio at any common nonzero voltage in the tested linear range. Material coefficient and geometry can produce the same R by different causes.
Make the model respond
Focused Physics 2 model
Two-resistor DC circuit
Predict first. Predict how total current changes when the same resistors are rewired in parallel.
Rₑq = R₁ + R₂; I₁ = I₂ = I; Vbattery = V₁ + V₂; P = I²R
On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.
Series: I₁ = 0.2 A, I₂ = 0.2 A, and total current = 0.2 A.
Read the model data and comparisons
| Quantity | Value | Unit | Meaning or limit |
|---|---|---|---|
| Battery voltage | 12 | V | |
| Equivalent resistance | 60 | Ω | |
| Battery current | 0.2 | A | |
| R₁ current | 0.2 | A | |
| R₂ current | 0.2 | A | |
| R₁ voltage drop | 4 | V | |
| R₂ voltage drop | 8 | V | |
| R₁ power | 0.8 | W | |
| R₂ power | 1.6 | W | |
| Battery power | 2.4 | W | |
| Power residual Pbattery − P₁ − P₂ | -2.220 × 10^−16 | W |
In series, every resistor carries the same current, and the voltage drops sum to the battery voltage. In parallel, every branch has the full battery voltage, and the branch currents sum at the junction. In either topology, the ideal battery power equals the sum of resistor powers. The battery moves charge around a closed circuit; resistors transfer electric energy to thermal energy.
Assumptions and limits. Ideal constant-voltage battery, ideal wires, two ohmic resistors with fixed resistance and no temperature dependence. Steady DC state; conventional-current direction from the positive terminal. No internal battery resistance or transient behavior.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
A wire is twice as long and has twice the radius as another of the same material at the same temperature. Compare R. Would heating an actual metallic wire generally preserve ρ?
Worked answer and reason
R is half as large: length factor 2 divided by area factor 4. Actual metallic resistivity typically increases when heated; a fixed-ρ calculation needs its model assumption.
Area depends on radius squared, and real temperature behavior differs from the ideal constant-resistivity model.
Question 2
An I-versus-ΔV graph has slope 0.025 A/V. Find R. What if ΔV is instead the vertical axis? How could resistor heating alter the plot?
Worked answer and reason
R = 40 Ω. For ΔV vertically versus I horizontally, slope is 40 V/A. Heating can change resistivity and resistance, producing curvature or a changed slope.
Axis choice determines slope interpretation; thermal change can violate the constant-R assumption.
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Catch the tempting mistake
- Every ohmic metal has temperature-independent resistivity.
- Constant-R ohmic behavior is usually specified at a controlled temperature. Real metallic ρ can rise with temperature; the temperature-independent idealization must be named.
- I-versus-V slope equals resistance.
- It equals 1/R; a V-versus-I slope equals R.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 5 · symbolic geometry and temperature model
- Problem 6 · I–V experiment and graph slope
- Problem 17 · resistance experiment, uncertainty, and power
Explain and apply the model
Try these problems before opening the solutions. Show the physical model, your calculation or sketch, and the reason your conclusion follows.
Can two equal wires resist differently?
At a common controlled temperature, uniform wires A and B each have L = 0.50 m and cross-sectional area 2.0 × 10⁻⁷ m². Data are A: (V,I) = (0,0), (1.0,0.25), (2.0,0.50), (3.0,0.75), with V in volts and I in amperes. B has (0,0), (1.0,0.125), (2.0,0.250), (3.0,0.375). Both are approximately ohmic over this tested range.
(a) Construct both quantitative I-versus-V graphs on common axes, find their slopes and resistances, and infer both resistivities. Explain what higher resistivity says about opposition to charge motion and how atomic/molecular structure can cause a difference at equal geometry. State what the graph cannot uniquely tell you microscopically.
(b) A third wire uses material A at the same temperature but has twice the length and twice the area. Predict its R and ρ, and explain why changing the object’s dimensions does not force a change in the material coefficient.
(c) The actual A wire is then heated while geometry changes negligibly. Predict the typical metallic trend of ρ, R and the I-versus-V slope. Contrast it explicitly with the stated temperature-independent ideal ohmic model, and explain whether one linear I–V graph at fixed temperature proves that idealization across temperatures.
Written working workspace
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For the teacher
- Read both axes aloud before extracting R.
- Have students identify which assumptions make constant ρ legitimate.
- Use low-current readings and cooldowns in resistance measurements.
Source objectives and scope
- 11.3.A — Describe the resistance of an object using physical properties of that object.
- 11.3.B — Describe the electrical characteristics of elements of a circuit.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 Course and Exam Description, Unit 11 (PDF pages 68–81) — Official topic order, learning objectives, all essential-knowledge clauses and boundaries. Equation pages 68,71,73,75,77–80 and relevant continuation pages visually inspected; independent reviewed-source-equation overlay consulted.
- OpenStax University Physics Volume 2, 9.3 Resistivity and Resistance — Independent verification of real material temperature dependence and the fixed-temperature qualification for linear metallic response. Teaching and assessment prose are original.
- OpenStax University Physics Volume 2, 10.3 Kirchhoff’s Rules — Independent verification of conservation-based branch and loop analysis; no calculus is needed in these authored tasks.
- Corrected original reading: Current and Resistance — Follow charge-flow rate and conventional-current direction; keep electron drift opposite the arrows.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Introduction to Circuits — Practice node connectivity, switches and common symbols. Check that any short-circuit idealization retains a finite current-limiting resistance.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Resistors in Circuits — Connect resistor voltage and current to energy transfer per second. Brightness comparisons require a stated bulb model.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Kirchhoff’s Laws — Trace actual closed paths and keep signed voltage changes. Use independent equations rather than combining disconnected branches into one loop.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: Capacitors in Circuits — Compare equal-charge series and equal-voltage parallel capacitor arrangements.Supporting context; original AP teaching and practice above remain independently authored.
- Corrected original reading: RC Circuits — Use initial and long-time states, τ and qualitative curves. Source exponential and logarithmic transient calculations extend beyond required AP Physics 2 modeling.Supporting context; original AP teaching and practice above remain independently authored.
Glossary and deeper reading · 10 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Resistivity
Resistivity is a material property describing resistance to electrical current under specified conditions. It is distinct from the resistance of a particular object and is measured in ohm-meters. For example, changing a wire’s length changes its resistance without changing the material’s resistivity when temperature and composition stay fixed.
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Electrical resistance
Electrical resistance relates the potential difference across an element to the current through it. It is measured in ohms and depends on the element’s material, geometry, and conditions. For example, a longer uniform wire has greater resistance when cross-sectional area and material remain fixed.
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Electric current
Electric current is the net rate of charge flow through a specified cross section. It is measured in amperes, or coulombs per second, with direction defined by positive-charge motion. For example, the same steady current passes through each element of a simple unbranched series loop.
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Potential difference
Potential difference is the change in electric potential between two locations. It represents electric potential-energy change per unit charge and is measured in volts. For example, the potential difference across a resistor describes the energy transferred per coulomb passing through it.
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Electrical conductor
An electrical conductor contains charge carriers that can move through the material. In electrostatic equilibrium, mobile charges redistribute until the electric field within the conducting material is zero. For example, excess charge on an isolated metal object resides on its surface after the charges settle.
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Temperature
Temperature is a state quantity that determines the direction of net energy transfer through thermal contact. For a classical ideal gas it is proportional to the average translational kinetic energy per particle, rather than to the total energy of the sample. For example, a small hot metal sample can have a higher temperature but less total internal energy than a large cooler sample.
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Heat
Heat is energy transferred between systems because of a temperature difference. It describes a transfer rather than a substance or energy store contained in an object. For example, when a warmer mug transfers energy to a cooler hand, the mug loses internal energy and the hand gains it.
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Electric field
An electric field assigns the electric force per unit positive test charge to each location. The field is produced by the sources and is a vector, while the force on a negative charge points opposite the field. For example, the electric field between ideal parallel capacitor plates can accelerate an electron opposite the field arrows.
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Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
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Force
A force describes an interaction that can change an object’s momentum. It is a vector measured in newtons, so both size and direction matter when forces are added. For example, a rope can pull a cart while the floor exerts friction in another direction.