Topic 11.3: Resistance, Resistivity, and Ohm's Law

What Topic 11.3 requires

Topic 11.3 carries two learning objectives, three essential knowledge statements, and six sub-statements. Everything else here unpacks them.

11.3.A: describe the resistance of an object using physical properties of that object.

• 11.3.A.1 Resistance is a measure of the degree to which an object opposes the movement of electric charge.
• 11.3.A.2 The resistance of a resistor with uniform geometry is proportional to its resistivity and length and is inversely proportional to its cross-sectional area. The CED gives $R = \frac{\rho \ell}{A}$ as the relevant equation.
• 11.3.A.2.i Resistivity 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.
• 11.3.A.2.ii The resistivity of a conductor typically increases with temperature.

11.3.B: describe the electrical characteristics of elements of a circuit.

• 11.3.B.1 Ohm's law relates current, resistance, and potential difference across a conductive element of a circuit. The CED gives $I = \frac{\Delta V}{R}$ as the relevant equation.
• 11.3.B.1.i Materials that obey Ohm's law have constant resistance for all currents and are called ohmic materials.
• 11.3.B.1.ii The resistivity of an ohmic material is constant regardless of temperature.
• 11.3.B.1.iii Resistors can also convert electrical energy to thermal energy, which may change the temperature of both the resistor and the resistor's environment.
• 11.3.B.1.iv The resistance of an ohmic circuit element can be determined from the slope of a graph of the current in the element as a function of the potential difference across the element.

Topic 11.3 prints no boundary statement of its own. The nearest one in the unit belongs to Topic 11.2 and says that unless otherwise specified, all circuit schematic diagrams will be drawn using conventional current.

The CED lists five suggested skills here: 1.B, create quantitative graphs with appropriate scales and units, including plotting data; 2.B, calculate or estimate an unknown quantity with units from known quantities, by selecting and following a logical computational pathway; 2.D, predict new values or factors of change of physical quantities using functional dependence between variables; 3.A, create experimental procedures that are appropriate for a given scientific question; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim. Two of the five, 1.B and 3.A, are about producing and reading data rather than about algebra. Unit 11 carries 15 to 18 percent of the exam and a suggested 12 to 20 class periods.

Resistance belongs to the object, resistivity to the material

The Unit 11 opener makes an unusual request. Under Preparing for the AP Exam, the CED says students should know the differences in meaning between current, potential difference, resistance, resistivity, and capacitance. Topic 11.3 is where two of those five get mixed up, so start with the split.

Resistivity ($\rho$) is a property of a substance. Graphite has one, copper has another, and a block of copper the size of a house has the same resistivity as a copper shaving. Essential knowledge 11.3.A.2.i says it depends on the material's atomic and molecular structure and quantifies how strongly the material opposes the motion of electric charge.

Resistance ($R$) is a property of one particular object: this wire, of this length and this thickness, made of that substance. Essential knowledge 11.3.A.1 defines it as a measure of the degree to which an object opposes the movement of electric charge.

So two objects made of the same material can have very different resistances, and two objects with the same resistance can be made of different materials. Resistivity is what the material contributes, and the geometry supplies the rest.

Neither unit needs memorizing. Ohm's law gives the ohm: $1 \, \Omega = 1 \, \mathrm{V/A}$. Rearranging $R = \rho \ell / A$ gives $\rho = RA/\ell$, so resistivity comes out in $\Omega \cdot \mathrm{m}^2 / \mathrm{m}$, which is $\Omega \cdot \mathrm{m}$. Deriving a unit is faster than recalling it, and it checks that you wrote the equation the right way up.

Geometry sets the resistance (11.3.A.2)

This is one of the 20 equations in the Electricity group of the AP Physics 2 equation sheet, so you do not have to recall it. What you do have to recall is what each symbol points at.

• $\rho$ is the resistivity of the material, in $\Omega \cdot \mathrm{m}$.
• $\ell$ is the length along the direction the charge travels.
• $A$ is the cross-sectional area perpendicular to that direction.

Swap those two and the answer inverts. For a cylindrical wire, $\ell$ is the length of the wire and $A = \pi r^2$ is the area of the circular end face. A problem that hands you a diameter is asking you to halve it first, and that is where factor-of-four errors are born, because the area goes as the square of the radius.

Essential knowledge 11.3.A.2 states the dependence as a proportionality rather than as a calculation, which makes the equation a scaling rule and is what suggested skill 2.D asks you to use.

The fourth row is the one that catches people. Stretching conserves the volume $A\ell$, so doubling $\ell$ also halves $A$, and the two effects multiply instead of cancelling.

The phrase "uniform geometry" in 11.3.A.2 matters: the equation describes an object whose cross-section does not change along its length, which is why AP problems are written around cylinders and rectangular blocks.

The same geometry turns up in Topic 9.5, where the sheet prints thermal conduction as $\frac{Q}{\Delta t} = \frac{k A \Delta T}{L}$. There $A$ is on top because that equation describes a conductance: a short, fat slab passes more heat. Turn it over and the thermal resistance goes as $L/(kA)$, the same skeleton as $\rho \ell / A$.

Temperature, and the two CED statements that look opposed

Put 11.3.A.2.ii and 11.3.B.1.ii next to each other and they read like a contradiction.

• 11.3.A.2.ii: the resistivity of a conductor typically increases with temperature.
• 11.3.B.1.ii: the resistivity of an ohmic material is constant regardless of temperature.

They sit on facing pages of the CED and both are true, because they describe different things. The first is about real conductors. The second is part of the definition of the ohmic model: 11.3.B.1.i defines ohmic materials as those with constant resistance for all currents, and a resistance that does not move with current does not move as the element warms. Being ohmic is what rules the temperature dependence out.

Essential knowledge 11.3.B.1.iii is the bridge between the two: resistors can also convert electrical energy to thermal energy, which may change the temperature of both the resistor and the resistor's environment. Push current through a real resistor and it warms. Warm a real conductor and its resistivity climbs. So a real resistor's resistance drifts upward as it works, and calling something ohmic is the decision to ignore that drift.

An incandescent filament is the standard case where you cannot ignore the drift. It runs hot enough that a graph of its current against potential difference bends over instead of staying straight. That is not a faulty bulb, it is a conductor doing what 11.3.A.2.ii says conductors typically do. Which statement applies on an exam question is not left to you to guess, and the next section is where that gets settled.

Ohm's law, exactly as the sheet prints it

The current in a conductive element equals the potential difference across it divided by its resistance. That is essential knowledge 11.3.B.1, and the same arrangement is printed in the Electricity group of the equation sheet.

Keep the delta. Most textbooks write $V = IR$. The AP Physics 2 sheet does not, and the reason is physical rather than cosmetic. Electric potential $V$ is a property of a single point in space, which is Topic 10.5. A resistor does not have a potential. It has two ends, each at its own potential, and what drives current through it is the difference between them. Writing $\Delta V$ keeps that two-point nature visible, and on a free-response item that asks you to justify a claim the notation is doing part of the justifying.

The algebra is unchanged: $\Delta V = IR$ and $R = \Delta V / I$ are the same statement rearranged. The Ohm's law guide works all three arrangements with a unit check for each, and the Ohm's law calculator will solve for whichever one you are missing.

What this topic asks for sits one level above that arithmetic: the discipline of pairing a potential difference with the element it was measured across. The $\Delta V$ and the $R$ have to describe the same piece of circuit. Use a battery's potential difference with one resistor's resistance and you get a current that flows nowhere in the actual circuit. Topic 11.5 is where multi-element circuits get reduced so that the pairing becomes legal, and the series and parallel guide walks that reduction.

One piece of vocabulary is worth care. Ohm's law is a relationship that some materials obey and others do not, which makes it unlike conservation of charge or conservation of energy. The CED's wording is careful in the same way: 11.3.B.1 says Ohm's law relates current, resistance, and potential difference across a conductive element, and 11.3.B.1.i then names the materials that obey it. Nothing in physics obliges a device to.

What the ohmic convention buys you

The AP Physics 2 Table of Information prints a short list of assumptions, introduced as the conventions used in this exam unless otherwise stated. One line reads: "Resistors and lightbulbs are ohmic."

The "unless otherwise stated" clause is not decoration. The convention is a default, and questions are free to override it. While it holds, it buys you a great deal:

• One number $R$ describes the element at every operating point, so a resistance measured at 2 V is still the resistance at 8 V.
• The equivalent-resistance rules of Topic 11.5 are legitimate, because they assume each element has one fixed resistance, and the power forms in Topic 11.4 that carry an $R$ mean something, because there is a single $R$ to carry.
• A lightbulb, genuinely non-ohmic on a laboratory bench, can be treated as a plain resistor. That is the convention doing its heaviest lifting, and it is why a bulb in an AP problem behaves nothing like the filament in the previous section.

The default comes off in two situations. The first is when a question says so: a line telling you the filament's resistance rises with temperature cancels the convention for that item. The second is when a question asks you to determine whether an element is ohmic, which the CED's own sample free-response Question 3 on experimental design does, giving a student ohmic resistors of known resistance and ideal meters and asking what to measure and graph. Its scoring guidelines list learning objectives 11.2.A, 11.3.B and 11.5.B. You cannot assume the answer to the question you were asked.

The same list also states that strings, springs, batteries, wires, and meters are ideal, and that current is conventional current. The circuit topics fill in what ideal means: ideal batteries and wires have negligible resistance (11.5.B.1), ideal ammeters have zero resistance (11.5.C.1.ii), and ideal voltmeters have infinite resistance (11.5.C.2.ii). Those defaults are what let a Topic 11.3 problem put all of a circuit's resistance into the one element you are studying.

Reading a current-versus-voltage graph (11.3.B.1.iv)

Essential knowledge 11.3.B.1.iv is specific about the axes, and the specificity is the point: the resistance of an ohmic circuit element can be determined from the slope of a graph of the current in the element as a function of the potential difference across the element.

Current as a function of potential difference means current on the vertical axis and potential difference on the horizontal axis. On those axes the slope is a change in current over a change in potential difference, which by Ohm's law is $1/R$. The resistance is the reciprocal of the slope, not the slope.

Both graphs are legitimate and both show up on exams; the CED's wording picks the first. Read the axis labels before you touch the slope, and carry a unit: a slope in $\mathrm{A/V}$ inverts to $\mathrm{V/A}$, which is an ohm.

The shape of the plot carries the physics.

• A straight line through the origin means the element is ohmic over the range you tested. It certifies nothing outside that range.
• A curve that flattens as $\Delta V$ grows means the current is rising more slowly than the potential difference, so the resistance is increasing. Self-heating, through 11.3.B.1.iii and then 11.3.A.2.ii, is the usual cause.
• A straight line that misses the origin is a warning about the measurement rather than about the material. An ohmic element carries no current at zero potential difference.

Suggested skills 1.B and 3.A both live here. 1.B asks for quantitative graphs with appropriate scales and units, so label each axis with a symbol and a unit, pick a scale that spreads the points across the grid, and draw a best-fit line rather than joining the dots. 3.A asks for a procedure that answers a stated question, and the CED's Unit 11 sample activities include exactly that shape of task: measuring the potential difference across and the current in a length of mechanical pencil lead to find the resistivity of graphite.

How Topic 11.3 is tested, and where it leads

The five suggested skills map onto five recognizable question shapes.

1. Compute a resistance or a resistivity from dimensions (2.B). Halve the diameter before you square it.
2. Predict a factor of change (2.D). Often with no numbers at all: a wire is stretched, lengthened, or swapped for a thicker one, and you report the multiplier on $R$.
3. Apply Ohm's law to one element (3.B), and justify the pairing of the potential difference with that element.
4. Plot and interpret data (1.B) to decide whether a device is ohmic and to pull $R$ out of a slope.
5. Design the experiment that produces those data (3.A), naming the meters, where each one goes, and what you would repeat to reduce uncertainty.

The unit's opening pages are worth reading with that list in hand. The CED says Unit 11 requires more than calculating currents, resistances and potential differences in a simple circuit, and it names testing whether a light bulb is ohmic as an example of the experimental design work students should practice.

From here the unit runs two ways. Topic 11.4 takes the same $I$, $\Delta V$ and $R$ and asks how fast energy moves, which is where the thermal conversion in 11.3.B.1.iii gets an equation. Topic 11.5 puts several resistors in one circuit and supplies the equivalent-resistance rules. Underneath both sit Topic 11.1, which defines the current in $I = \Delta V / R$, and Topic 11.2, which defines the circuit it flows around. The Unit 11 overview lays out all eight topics in order.