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CED objectives

Resistance, Resistivity, and Ohm's Law

▶︎  Watch it animatedinteractive step-through · ~3 min · optional ⚙︎  Open the appletResistance Bench · cut a wire from a metal, a length and a diameter, lock it as a reference, then swap the graph axes

Resistivity is a property of a material and resistance is a property of an object, joined by $R = \rho L/A$. Length scales the resistance in direct proportion while the cross-sectional area divides it, and for a round wire the radius or diameter reaches $R$ through a square, so doubling the diameter quarters the resistance. Ohm's law is the claim that $R$ holds constant as the current varies, true of ohmic materials and not of a lamp filament, whose resistance rises with temperature so its characteristic curve bends. On a plot of current against potential difference the slope is $1/R$, so the steeper line belongs to the smaller resistance.

Four errors dominate. Using a table value of resistivity as the resistance of a particular wire, or arguing that two wires of the same metal must have equal resistance. Scaling resistance linearly with diameter or radius when the area carries the square. Carrying a lamp's resistance from one operating point to another as though it were constant. And ranking resistances off the slope of a graph without checking which quantity sits on which axis.

same metal, same ρ, three different resistances L, area A R = 8 Ω 2L, area A R = 16 Ω L, DOUBLE the diameter R = 2 Ω length is LINEAR: twice as long, twice the resistance diameter is SQUARED: twice as thick, one QUARTER the resistance, not one half
The material constant is identical in all three rows. Everything separating the three numbers on the right is geometry.
I against ΔV: slope is 1/R, so steeper means SMALLER R I ΔV steep: 4 Ω shallow: 12 Ω lamp: R grows as it heats more amps per volt = less opposition swap the axes and the slope becomes R an ohmic device is a straight line through the origin; the bending curve is the whole content of the word ohmic
Saying what one unit of rise over one unit of run means physically settles the ranking on either arrangement of the axes.

The work

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Lesson
Resistance, Resistivity, and Ohm's Law ›

Splits the material constant from the object's resistance, squares the diameter dependence, restricts Ohm's law to ohmic devices, and reads a characteristic graph off its axes rather than off habit.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items spanning the failure modes of this topic: using resistivity as a resistance, scaling resistance linearly with diameter, carrying a lamp's resistance to a different operating point, and ranking resistances backward off a graph. Take it cold to find which one is yours, or after the lesson to confirm it is not.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

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.

Take the diagnostic to identify your misconceptions
Diagnostic first