Resistance, Resistivity, and Ohm's Law
▶︎ Watch it animatedinteractive step-through · ~3 min · optional ⚙︎ Open the appletOhm Lab · build a resistor from a metal, a length and a diameter, then read V over I and dV over dI off the graphResistance is defined as $R = V/I$ for any device, and measured in ohms. Ohm's law is the separate empirical claim that this ratio stays constant as the voltage varies, which holds for metals at fixed temperature and fails for filaments, diodes and thermistors. Geometry enters through $R = \rho L/A$, derived from $J = \sigma E$: resistivity $\rho$ is a property of the material alone, resistance is a property of the object, and because $A = \pi r^2$ the radius enters squared. On a graph, $V$ versus $I$ has slope $R$ while $I$ versus $V$ has slope $1/R$.
Three errors dominate. Treating $R = V/I$ as a proportionality, so doubling the applied voltage is said to double the resistance, when for an ohmic device the current doubles and the ratio holds still. Scrambling the geometry: halving $R$ for a doubled length, forgetting that doubling a radius quarters $R$ rather than halving it, or quoting a material's $\rho$ in ohms as though it were an object's $R$. And reading resistance off the wrong slope, taking the steeper line on an $I$-versus-$V$ plot as the larger resistance when it is the smaller, or confusing the chord $V/I$ with the tangent $dV/dI$ on a curved characteristic.
The work
3 ways in · any order
Lesson
Resistance, Resistivity, and Ohm's Law
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Pulls apart the definition of resistance, the empirical law that some materials obey, and the geometry factor, then drills the radius-squared scaling and the axis check that decides whether a slope is R or 1/R.
Diagnostic
10-item topic check
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Ten items spanning the failure modes of this topic: reading R = V/I as a proportionality so resistance tracks voltage, scrambling how length and radius scale R, confusing a material's resistivity with an object's resistance, and taking the steeper line on an I-versus-V plot as the larger resistance. Take it cold to find which one is yours, or after the lesson to confirm it is not.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.