Resistance, Resistivity, and Ohm’s Law
Resistance describes how strongly a particular object opposes charge flow; resistivity describes the material itself. For a uniform straight resistor, R=ρℓ/A. When resistivity varies along a uniform cross section, add differential series slices: R=∫ρ(ℓ)dℓ/A. Ohm’s law, I=ΔV/R, is a model for an ohmic element whose resistance is constant under the stated conditions—not a universal law for every device.
Topic sequence · 3 of 8
Learning objectives
Objective 1
Separate resistance R (an object-and-geometry property) from resistivity ρ (a material property), and predict how length, area, material, and temperature affect resistance.
Objective 2
Use R=ρℓ/A and the calculus form R=∫ρ(ℓ)dℓ/A for a constant-area resistor whose resistivity varies along its length.
Objective 3
Interpret I–ΔV data correctly for ohmic and nonohmic elements, including the graph-slope convention and the conditions under which Ohm’s law is valid.
- Topic 11.1 electric current and the local relation E⃗=ρJ⃗.
- Topic 11.2 circuit topology, current direction, and potential difference across an element.
- Calculus fluency with definite integrals and graph slopes.
Definitions first
Representations you must read and create
Representation 1
Two wires made from the same material but with different lengths and diameters; label which changes geometry and which would require a different material to change ρ.
Representation 2
A chain of differential resistor slices dR in series along a wire whose resistivity varies with position; the integral sums those local contributions.
Representation 3
I-versus-ΔV graph for an ohmic element: a straight line through the origin with slope 1/R, not R, when I is on the vertical axis.
Representation 4
ΔV-versus-I graph for the same element: a straight line through the origin whose slope is R.
Representation 5
Nonlinear I–ΔV curve showing why a single constant R does not describe every operating point of a nonohmic device.
Core relationships and calculus
Variables: R Ω; ρ Ω·m; ℓ m; A m²
Use: uniform material, uniform cross section, steady conduction model
Variables: dR Ω
Use: differential series slice for constant A when resistivity varies with position
Variables: R Ω
Use: total resistance for a constant-area resistor with position-dependent resistivity
Variables: I A; ΔV V; R Ω
Use: Ohm’s law for an ohmic circuit element
Variables: ΔV V
Use: same ohmic model written as a potential drop for a known current
Variables: slope A/V = S
Use: if current is vertical and potential difference horizontal; inverse slope gives R
Variables: slope V/A = Ω
Use: if potential difference is vertical and current horizontal
The integral form is not a different physical law from R=ρℓ/A; it is the continuous version of adding many short resistive pieces in series. Divide the conductor into slices of length dℓ. Each slice has dR=ρ(ℓ)dℓ/A. Because the same steady current passes through all slices, their potential drops add, so their resistances add. Taking the limit of infinitely thin slices gives R=∫ρ(ℓ)dℓ/A. Graph calculus also matters: from I=ΔV/R, dI/d(ΔV)=1/R for a constant-R element.
Start locally with E=ρJ for a uniform straight conductor. With uniform current density J=I/A and approximately uniform axial field E=ΔV/ℓ, substitute to get ΔV/ℓ=ρI/A. Rearranging gives ΔV=Iρℓ/A, so R≡ΔV/I=ρℓ/A. For a position-dependent ρ but constant A, write dV=E dℓ=ρ(ℓ)(I/A)dℓ. Integrate ΔV=I∫ρ(ℓ)dℓ/A and divide by I to obtain the required calculus form.
Do not treat Ohm’s law as a universal definition of all circuit elements. The current framework distinguishes the general geometry/material dependence of resistance from the special ohmic model. It also notes that conductor resistivity typically increases with temperature, while an ideal ohmic material is modeled with constant resistance/resistivity over its stated conditions. For an I-versus-ΔV graph, the slope is 1/R; for a ΔV-versus-I graph, the slope is R.
Physical examples
Example 1
Doubling the length of a uniform wire while keeping ρ and A fixed doubles its resistance because the same current must pass through twice as much resistive material.
Example 2
Doubling a wire’s radius quadruples its cross-sectional area and therefore reduces its resistance by a factor of four if length and resistivity are unchanged.
Example 3
Two geometrically identical wires can have very different resistances if their materials have different resistivities.
Example 4
For many ordinary conductors, increasing temperature raises resistivity. That statement describes a real-material trend; an idealized ohmic-element calculation assumes the operating conditions keep R effectively constant.
Example 5
A curved I–ΔV graph signals that the ratio ΔV/I changes with operating point, so blindly using one fixed resistance across the full curve is not justified.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Functional dependence: if a wire is stretched or its radius changes, translate the geometry change into ℓ and A before predicting R.
Reasoning pattern 2
Material versus object: changing ρ changes the material response; changing ℓ or A changes the resistance of the object without changing the material’s resistivity.
Reasoning pattern 3
Integral setup: when ρ varies along a constant-area wire, integrate ρ(ℓ)dℓ/A rather than averaging endpoints without justification.
Reasoning pattern 4
Graph slope: read axis labels first. I on y and ΔV on x gives slope 1/R; ΔV on y and I on x gives slope R.
Reasoning pattern 5
Model validity: a nonlinear I–ΔV curve means a single constant-R Ohm’s-law model does not describe the entire operating range.
Deeper calculus and limiting-case check
For a conductor whose cross-sectional area also varies, the same differential-series idea becomes dR=ρ(x)dx/A(x), so R=∫ρ(x)dx/A(x). This is the natural generalization of the course’s constant-area variable-resistivity integral. A useful limiting check is that if both ρ and A become constant, the integral collapses immediately to R=ρL/A.
Model-selection workflow: decide what the problem is before choosing an equation
For Resistance, Resistivity, and Ohm’s Law, the fastest reliable route is not equation hunting. Start by identifying the nodes, branches, and circuit state, encode it with the circuit schematic/graph, choose the Kirchhoff/constitutive or transient equation, and then test the ideal-element and topology assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat resistance R and resistivity ρ as physical quantities first and symbols second.
Choose a representation
Build Two wires made from the same material but with different lengths and diameters; label which changes geometry and which would require a different material to change ρ.. If the prompt supplies A chain of differential resistor slices dR in series along a wire whose resistivity varies with position; the integral sums those local contributions., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is R = ρℓ/A. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Do not treat Ohm’s law as a universal definition of all circuit elements. The current framework distinguishes the general geometry/material dependence of resistance from the special ohmic model. It also notes that conductor resistivity typically increases with temperature, while an ideal ohmic material is modeled with constant resistance/resistivity over its stated conditions. For an I-versus-ΔV graph, the slope is 1/R; for a ΔV-versus-I graph, the slope is R. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Two wires made from the same material but with different lengths and diameters; label which changes geometry and which would require a different material to change ρ. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | R = ρℓ/A | uniform material, uniform cross section, steady conduction model. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | dR = ρ(ℓ)dℓ/A | differential series slice for constant A when resistivity varies with position. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | The integral form is not a different physical law from R=ρℓ/A; it is the continuous version of adding many short resistive pieces in series. Divide the conductor into slices of length dℓ. Each slice has dR=ρ(ℓ)dℓ/A. Because the same steady current passes through all slices, their potential drops add, so their resistances add. Taking the limit of infinitely thin slices gives R=∫ρ(ℓ)dℓ/A. Graph calculus also matters: from I=ΔV/R, dI/d(ΔV)=1/R for a constant-R element. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A wire has length L=0.80 m, cross-sectional area A=0.50 mm², and resistivity that varies as ρ(x)=ρ₀(1+x/L), with ρ₀=1.6×10⁻⁸ Ω·m. Treat each slice dx as a series element: dR=ρ(x)dx/A. Then R=(ρ₀/A)∫₀ᴸ(1+x/L)dx=(ρ₀L/A)(1+1/2)=1.5ρ₀L/A. With A=5.0×10⁻⁷ m², R≈0.0384 Ω. If ΔV=0.120 V and the element is treated ohmically at these conditions, I=ΔV/R≈3.13 A. Using only ρ₀L/A would miss the higher-resistivity portion of the wire and underpredict R. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 11.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship R = ρℓ/A should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Two wires made from the same material but with different lengths and diameters; label which changes geometry and which would require a different material to change ρ. and A chain of differential resistor slices dR in series along a wire whose resistivity varies with position; the integral sums those local contributions.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Measure current through a resistive element at several applied potential differences while keeping the element as close as practical to a constant temperature. Plot both I versus ΔV and ΔV versus I. Fit a line only if the data justify an ohmic model, and show explicitly that the two slope conventions are reciprocals. Repeat with a filament lamp or another element whose temperature changes enough to produce visible nonlinearity. Discuss meter resistance, heating, contact resistance, uncertainty, and why extrapolating far outside the measured range is not justified.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Calling resistivity and resistance the same thing. Resistivity belongs to the material; resistance belongs to the specific object and includes its geometry. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using R=ρℓ/A after the cross-sectional area or resistivity varies substantially without checking whether the uniform-model assumptions still hold. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Reading the slope of I versus ΔV as R. With those axes the slope is 1/R. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming every device obeys a constant-R Ohm’s law. A nonlinear I–ΔV curve is evidence against that model over the full range. | Return to the Kirchhoff/constitutive or transient equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Saying temperature never changes resistivity because an ideal ohmic model holds R constant. Real conductors can have temperature-dependent resistivity; state the model and operating conditions. | Return to the circuit schematic/graph, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 11.1 provides E⃗=ρJ⃗ and current-density ideas from which R=ρℓ/A can be derived for a uniform conductor. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 11.4 combines Ohm’s law with P=IΔV to obtain P=I²R and P=(ΔV)²/R for an ohmic resistor. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 11.5 uses resistor values to reduce series and parallel networks to equivalent resistance. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write R = ρℓ/A and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 11.3 mastery ladder
Interactive variable-resistivity wire
Use ρ(x)=ρ₀[1+βx/L] with constant cross-sectional area. The tool evaluates R=∫ρ(x)dx/A, then applies Ohm’s law at the entered potential difference.
Experimental connection
Measure current through a resistive element at several applied potential differences while keeping the element as close as practical to a constant temperature. Plot both I versus ΔV and ΔV versus I. Fit a line only if the data justify an ohmic model, and show explicitly that the two slope conventions are reciprocals. Repeat with a filament lamp or another element whose temperature changes enough to produce visible nonlinearity. Discuss meter resistance, heating, contact resistance, uncertainty, and why extrapolating far outside the measured range is not justified.
Connections across Physics C
Connection 1
Topic 11.1 provides E⃗=ρJ⃗ and current-density ideas from which R=ρℓ/A can be derived for a uniform conductor.
Connection 2
Topic 11.4 combines Ohm’s law with P=IΔV to obtain P=I²R and P=(ΔV)²/R for an ohmic resistor.
Connection 3
Topic 11.5 uses resistor values to reduce series and parallel networks to equivalent resistance.
Connection 4
Topic 13.5 later uses resistance together with inductance to set LR-circuit time behavior.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Calling resistivity and resistance the same thing. Resistivity belongs to the material; resistance belongs to the specific object and includes its geometry.
Avoid this
Using R=ρℓ/A after the cross-sectional area or resistivity varies substantially without checking whether the uniform-model assumptions still hold.
Avoid this
Reading the slope of I versus ΔV as R. With those axes the slope is 1/R.
Avoid this
Assuming every device obeys a constant-R Ohm’s law. A nonlinear I–ΔV curve is evidence against that model over the full range.
Avoid this
Saying temperature never changes resistivity because an ideal ohmic model holds R constant. Real conductors can have temperature-dependent resistivity; state the model and operating conditions.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
What is the difference between resistance and resistivity?
Resistance R describes a particular object and depends on material plus geometry. Resistivity ρ is a material property measured in Ω·m.
Does Ohm’s law define resistance for every device?
No. ΔV/I can be formed at an operating point, but the ohmic model requires a constant proportionality between current and potential difference over the stated conditions.
Why does a thicker wire have smaller resistance?
A larger cross-sectional area provides more parallel microscopic pathways for current, and R=ρℓ/A decreases as A increases.
What is the slope of an I-versus-V graph?
For an ohmic element with I on the vertical axis and ΔV on the horizontal axis, the slope is 1/R. If the axes are reversed, the slope is R.
When do I integrate resistance?
Use the calculus form when a property such as resistivity varies continuously along the conductor; add differential series slices rather than using one uniform value.