Resistance is how much an object opposes motion of electric charge—think of it as “electrical friction.” For a uniform resistor, R = ρℓ/A: it increases with the material’s resistivity ρ and length ℓ and decreases with cross-sectional area A. Resistivity is a material property (usually rises with temperature for conductors); if ρ varies along the length use R = ∫ρ(ℓ) dℓ / A. Ohm’s law links current, resistance, and potential difference: I = ΔV / R. Materials with constant R (for all ΔV and I) are ohmic; nonohmic devices (diodes, filament lamps) don’t follow that linear relation. On the AP exam you may be asked to find R from the slope of an I vs ΔV graph (slope = 1/R, conductance). Resistors convert electric energy to thermal energy (Joule heating), so power P = I²R = ΔV·I = ΔV²/R. For a focused review see the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa); more unit content and practice problems are at the Unit 11 overview (https://library.fiveable.me/ap-physics-2-revised/unit-11) and the practice bank (https://library.fiveable.me/practice/ap-physics-2-revised).

A longer wire has more resistance because charge carriers have a longer path and therefore more chances to scatter off atoms—so the wire opposes current more. For a uniform wire the CED gives the exact relation: R = ρℓ/A. That means R is directly proportional to length ℓ (so doubling ℓ doubles R) and inversely proportional to cross-sectional area A (thicker wire → less R). ρ (resistivity) is a material property that can change with temperature (for most conductors ρ rises as T rises). Ohm’s law (I = ΔV/R) then tells you a longer wire draws less current for the same voltage. If you want practice applying this or seeing IV-graphs for ohmic materials, check the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and Unit 11 review (https://library.fiveable.me/ap-physics-2-revised/unit-11).

Resistance and resistivity are related but different. Resistivity (ρ) is a material property that quantifies how strongly a material opposes charge flow—it depends on the material’s atomic structure and usually changes with temperature (CED 11.3.A.2.i–ii). Resistance (R) is a property of a particular object or resistor and depends on both the material and its shape: R = ρℓ/A for a uniform resistor (CED 11.3.A.2). So for two wires made of the same metal (same ρ), the longer or thinner one has larger R. If ρ varies along the length, use R = ∫ρ(ℓ) dℓ / A (CED 11.3.A.2.iii). Ohm’s law (I = ΔV/R) connects R to circuit behavior; materials with constant ρ (over the tested range) are “ohmic” (CED 11.3.B.1.i–ii). For a quick review, see the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and more practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Ohm’s law says I = ΔV / R for a conductive element. To solve circuit problems, follow these steps: 1. Identify the element(s) and whether they’re ohmic (constant R). Use R = ρℓ/A if you need resistance from geometry (CED 11.3.A.2). 2. Redraw the circuit, label ΔV across each resistor, and mark currents with directions. For series, Rtot = R1+R2+…; for parallel, 1/Rtot = Σ(1/Ri) (CED 11.5). 3. Use Ohm’s law: for a known ΔV find I = ΔV/R; for a known I find ΔV = I·R. For circuits with a battery+internal resistance use ε = I(R + r). 4. Check power: P = IΔV = I^2R = (ΔV)^2 / R (CED 11.4). 5. If asked for R from IV data, plot I vs ΔV; slope = 1/R (conductance) or plot ΔV vs I; slope = R (CED 11.3.B.1.iv). That graph/derivation style appears on the AP FRQ rubric. Practice problems and the Topic 11.3 study guide can help you apply these steps (Fiveable study guide: https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa; unit overview: https://library.fiveable.me/ap-physics-2-revised/unit-11).

R = ρℓ/A just tells you what determines how hard a material makes it for charges to flow. - ρ (resistivity) is a material property (Ω·m). Metals have low ρ, insulators high ρ; ρ usually increases with temperature. - ℓ (length)—resistance is proportional to length. Double the length → double R (more collisions/opposition). - A (cross-sectional area)—resistance is inversely proportional to area. Double A → half R (more paths for charge to flow). So R = ρℓ/A means long, thin wires have high R; short, fat wires have low R. For nonuniform ρ(ℓ) use R = ∫ ρ(ℓ) dℓ / A. Combine with Ohm’s law I = ΔV/R to find currents. On the AP CED this is 11.3.A.2 (use functional dependence to predict changes on the exam). For a quick review and practice problems, see the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and Fiveable practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

A material is called ohmic if the current through it is directly proportional to the potential difference across it—that means a constant resistance R = ΔV/I for all applied voltages. On an I vs. V graph an ohmic element makes a straight line; the slope is the conductance (1/R) and R is constant. The CED also notes an ohmic material has resistivity that doesn’t change with temperature. Non-ohmic materials don’t follow that linear relationship: their resistance changes with voltage, current, or temperature. Examples: diodes, incandescent lamp filaments (resistance rises as they heat), and thermistors. For AP style questions, you’ll often identify ohmic behavior by plotting I vs. V and checking linearity—slope gives R (CED 11.3.B.1.iv). For a quick review, see the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and practice lots of IV-graph problems (https://library.fiveable.me/practice/ap-physics-2-revised).

As temperature rises, the atoms in a conductor vibrate more. Those stronger lattice vibrations scatter the free electrons that carry current, so electrons have a shorter average time (and distance) between collisions. That makes it harder for charge to flow—i.e., the material’s resistivity ρ increases—and since R = ρℓ/A, the resistance R goes up (for fixed length ℓ and area A). This is why most metallic conductors show increasing resistance with temperature (CED: 11.3.A.2.ii). Note: on the AP exam you may need to cite R = ρℓ/A or explain scattering qualitatively; some problems also treat “ohmic” behavior as having constant resistance over the measured range (CED: 11.3.B.1.i–ii). For a quick review of these ideas, see the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and try more practice problems at (https://library.fiveable.me/practice/ap-physics-2-revised).

If your graph plots current (I) on the vertical axis and potential difference (ΔV) on the horizontal axis, use the slope. For an ohmic device (CED 11.3.B.1.i–iv) the I vs V plot is a straight line through the origin: - slope = ΔI / ΔV = 1/R (that slope is the conductance, in siemens S) - so R = 1 / (slope) = ΔV / ΔI (units: ohms, Ω) If the I–V curve is nonlinear (nonohmic), the resistance isn't constant. You can still find: - average resistance over a range: Ravg = ΔV / ΔI for that interval - instantaneous (dynamic) resistance: Rinst = 1 / (dI/dV) = dV/dI (reciprocal of the derivative at the point) Practice reading slopes and distinguishing ohmic vs nonohmic for the AP exam (see CED 11.3.B.1.iv). For extra practice and examples, check the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and more problems at (https://library.fiveable.me/practice/ap-physics-2-revised).

Use R = ρℓ/A (CED 11.3.A.2). If you make the wire thicker but keep the same length and material, its cross-sectional area A increases, so R decreases in inverse proportion to A. Example: if you double the wire’s diameter, A goes up by 4 (because A ∝ diameter²), so the resistance drops to one quarter of the original value. Resistivity ρ is a property of the material (and can change with temperature), so if you change material or heat the wire the result can differ. For AP work, apply R = ρℓ/A to predict numerical factors of change (skill 2.D) and show reasoning on exams. For a quick review, see the Topic 11.3 study guide on Fiveable (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and more unit resources at (https://library.fiveable.me/ap-physics-2-revised/unit-11).

Resistors turn electrical energy into heat because moving charges lose kinetic energy to the material. Microscopically, free electrons drifting under a potential difference collide with ions/atoms in the resistor’s lattice; those collisions transfer energy to atomic vibrations (thermal energy). Macroscopically you can use Ohm’s law and power formulas: I = ΔV/R and P = IV = I^2R = V^2/R. That P is the rate electrical energy is converted to thermal (Joule) heating. Resistivity ρ (R = ρℓ/A) sets how strongly a material opposes charge motion, so materials with higher ρ produce more heating for the same current. Resistivity usually rises with temperature, so heating can change R (non-ohmic behavior if large). This concept appears in CED 11.3 (Resistance, resistivity, Ohm’s law) and in problems on electric power—practice these ideas on the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and unit review (https://library.fiveable.me/ap-physics-2-revised/unit-11). For extra practice, try the AP practice set (https://library.fiveable.me/practice/ap-physics-2-revised).

Think of the resistor as lots of tiny pieces in series. For a small slice of length dl the resistance is dR = ρ(l)·dl / A (from R = ρℓ/A applied to a differential piece). If ρ varies with position, you add (integrate) all those tiny dR’s along the whole length to get the total: R = ∫[0 to L] ρ(l) dl / A. That’s it—the integral just replaces the simple ρ·L/A when ρ isn’t constant. If the cross-sectional area also changes, use A(l) in the denominator: R = ∫ ρ(l) dl / A(l). Ohm’s law still uses the total R (I = ΔV/R)—the integral gives the effective resistance you plug into I = ΔV/R (CED: 11.3.A.2.iii and 11.3.B.1). Quick example: if ρ(l)=ρ0(1+αl) and A is constant, R = (ρ0/A)∫0^L(1+αl)dl = (ρ0L/A)(1 + αL/2). For more practice, check the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and AP practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Look at V and I data and check whether V and I are proportional. Practically: - Compute R = V/I for several data points. If R is (approximately) the same for all V, the material is ohmic. - Or better, plot I vs. V (or V vs. I). If the points lie on a straight line through the origin, the element is ohmic. The slope of I vs. V is the conductance (1/R); the slope of V vs. I is the resistance R (this is exactly what the CED says: “Resistance … can be determined from the slope of a graph of current as a function of potential difference.”). - Watch out for temperature effects: heating can change resistivity, producing curved I–V behavior even for a metal. Also diode-like or filament devices show nonlinear (nonohmic) I–V curves. If you want practice identifying ohmic vs. nonohmic behavior, check the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and more problems at Fiveable practice (https://library.fiveable.me/practice/ap-physics-2-revised).

Resistance (R) tells you how a particular object opposes current—it depends on both the material and the shape. Resistivity (ρ) is the material’s intrinsic property: R = ρℓ/A. That matters because two wires with the same R could be different materials (one long/thin, one short/thick). ρ lets you compare materials (metals vs. insulators), predict how R changes with length or area, and include temperature effects (ρ usually increases with T for conductors). For nonuniform materials you need the integral R = ∫ρ(ℓ) dℓ / A to get the total resistance. On the AP exam you might extract R from an I vs ΔV slope (Ohm’s law) but you’ll use ρ when the question asks about material properties, geometry changes, or temperature dependence (CED 11.3.A.2, 11.3.B.1). For a quick review check the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and try practice problems at (https://library.fiveable.me/practice/ap-physics-2-revised).

Think of an electric circuit like water in pipes: voltage (ΔV) is the “push” that makes charge flow, current (I) is how much charge flows per second, and resistance (R) is how much the material or wire resists that flow. Ohm’s law puts it simply: I = ΔV / R—more push or less resistance gives more current. For a physical resistor, R depends on the material and shape: R = ρℓ/A (ρ = resistivity, ℓ = length, A = cross-sectional area). Metals usually get more resistive as they heat up (ρ rises with temperature). Materials with constant R over different voltages are called ohmic; you find R from the slope of an I vs. ΔV graph. This is exactly what AP Topic 11.3 covers—review the CED points and the Topic 11.3 study guide on Fiveable (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised) to prep for exam-style questions.

You can measure a resistor’s resistance two easy ways in the lab. 1) Direct V/I method (good for AP labs): Put the ammeter in series with the resistor and the battery; connect the voltmeter across the resistor (voltmeter parallel). Vary the battery voltage (or use different known series resistors), measure V and I for several settings. For each run compute R = V/I (Ohm’s law). To test ohmic behavior, plot I vs V—a straight line through the origin means the element is ohmic; slope = 1/R (or plot V vs I and slope = R). Repeat measurements and average to reduce random error. Tips to reduce uncertainty: use multiple voltage points, avoid large currents (heating changes resistivity), use high-resistance voltmeter and low-resistance ammeter, record uncertainties, and correct for instrument internal resistance if needed. For more detail and AP-style lab guidance, see the Topic 11.3 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-3/3-resistance-resistivity-and-ohms-law/study-guide/y0ZmKqhOPqeLWZFa) and the unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-11). For extra practice, try problems at (https://library.fiveable.me/practice/ap-physics-2-revised).