Resistance

Resistance is the opposition to the movement and flow of charge within a material. Some materials let current flow freely; others impede it heavily. Based on how much they resist current, materials fall into three categories:

• Conductors: offer almost no resistance to charge flow. Metals like copper and silver are excellent conductors, which is why they're used extensively in wiring.

• Insulators: offer very high resistance, effectively blocking charge flow. Rubber and glass are common examples.

• Resistors: fall between the two extremes — components intentionally designed to provide measurable, controlled resistance in a circuit.

Key Takeaways

• Resistance opposes charge flow; materials are classified as conductors (low resistance), insulators (high resistance), or resistors (intentional, controlled resistance).

• R = ρL/A: resistance increases with resistivity and length, decreases with cross-sectional area; resistance also increases with temperature in most conductors.

• Ohm's Law: V = IR (and its rearrangements I = V/R, R = V/I); real batteries have internal resistance (V = Ecell − i rint).

• Power dissipated by a resistor: P = IV = I²R = V²/R.

• Series resistors: same current, voltage divides, Rs = R1 + R2 + … (resistance adds).

• Parallel resistors: same voltage, current divides, 1/Rp = 1/R1 + 1/R2 + … (resistance shrinks).

The Resistance Equation: R = ρL/A

Resistance depends on both the material and the geometry of the resistor:

R = ρL/A

• Resistivity (ρ): a material-specific property describing how strongly a material opposes current flow, measured in ohm-meters (Ω·m). Metals have low resistivity (conduct well); materials with high resistivity conduct poorly.

• Length (L): resistance increases linearly with length — double the length of a resistor, and its resistance doubles.

• Cross-sectional area (A): resistance decreases as area increases — a larger area gives charge more pathways to flow through. Double the cross-sectional area, and resistance is cut in half.

Temperature also affects resistance. In most conductors, raising the temperature makes atoms vibrate more, making it harder for electrons to pass through — so resistance increases at higher temperatures. This is also why electrical components can overheat and fail under prolonged high current: as they heat up, their increased resistance further disrupts current flow.

Ohm's Law and Power Dissipation

Ohm's Law relates voltage, current, and resistance:

V = IR

The voltage drop across a resistor reflects energy converted into other forms — typically heat — as current passes through. Ohm's Law rearranges easily to solve for any of its three variables:

I = V/R R = V/I

MCAT Callout — Ohm's Law Triangle: Arranging V (top), I and R (bottom row) into a triangle is a standard memory aid: cover the variable you're solving for, and the triangle shows you the remaining relationship (V over I gives R; V over R gives I; I times R gives V).

Even voltage sources like batteries aren't perfect — they have internal resistance, denoted rint, which opposes current flow within the source itself. As current flows through this internal resistance, it generates heat and causes a voltage drop:

V = Ecell − i rint

Here, Ecell is the battery's ideal emf (maximum potential difference), i is the current, and V is the actual output voltage once internal resistance is accounted for. This is why batteries become less effective as they age or under high current loads — internal resistance increases, reducing output voltage.

Since resistors convert electrical energy into heat, it's useful to quantify power — the rate at which energy is transferred:

P = W/t = ΔE/t

For resistors, power dissipated can be calculated three equivalent ways, depending on which quantities are known:

P = IV = I²R = V²/R

If you know current and resistance, use P = I²R. If you know voltage and resistance, use P = V²/R. Each version answers the same question — how much energy is the resistor converting to heat per second — using whatever variables you have.

Resistors in Series

In a series circuit, resistors are connected end-to-end along a single path, so current has no alternate route — the same current flows through every resistor. As current passes through each one, there's a voltage drop across it.

The total voltage across all resistors is the sum of the individual drops:

Vs = V1 + V2 + V3 + …

The total resistance is likewise the sum of the individual resistances:

Rs = R1 + R2 + R3 + …

Each additional resistor increases total resistance, which — for a fixed voltage source — reduces overall current. This makes series circuits useful for limiting current. The downside: if any resistor fails or is removed, the entire circuit opens, and current stops flowing everywhere.

Resistors in Parallel

In a parallel configuration, each resistor has its own direct path to the voltage source, so every resistor experiences the same voltage:

Vp = V1 = V2 = V3 = …

The total current supplied by the source splits among the parallel branches, according to each branch's resistance — more current flows through lower-resistance paths. The total (equivalent) resistance follows a different rule than series: the reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances:

1/Rp = 1/R1 + 1/R2 + 1/R3 + …

Adding more resistors in parallel decreases total resistance, because each additional branch gives current another path to flow through. This is why household wiring uses parallel circuits: if one appliance stops working, the others on separate branches keep functioning.

Series vs. Parallel: Key Differences

Common MCAT Mistakes

• Adding resistances the wrong way for the circuit type. Series resistances add directly (Rs = R1+R2+…), but parallel resistances add as reciprocals (1/Rp = 1/R1+1/R2+…) — mixing these up gives an equivalent resistance that's off by orders of magnitude, not just a small error.

• Assuming a larger cross-sectional area increases resistance. It's the opposite: resistance is inversely proportional to area (R = ρL/A) — a wider conductor gives charge more pathways, lowering resistance, not raising it.

• Forgetting real batteries have internal resistance. Treating a battery's emf as identical to its output voltage ignores V = Ecell − i rint — the actual voltage delivered to a circuit is always somewhat less than the ideal emf, especially under high current.

• Picking the wrong power formula for the given variables. P = I²R and P = V²/R aren't interchangeable shortcuts — using P = V²/R when only current and resistance are known (or vice versa) requires first solving for the missing variable via Ohm's Law, which wastes time and invites arithmetic errors.

MCAT-Style Concept Check

Question: A 10 Ω resistor and a 40 Ω resistor are connected in parallel across a 20 V battery. What is the equivalent resistance of the combination?

• A) 8 Ω

• B) 12.5 Ω

• C) 25 Ω

• D) 50 Ω

Answer: A

Explanation: For parallel resistors, 1/Rp = 1/R1 + 1/R2 = 1/10 + 1/40 = 4/40 + 1/40 = 5/40 = 1/8. So Rp = 8 Ω. Note the equivalent resistance (8 Ω) is smaller than either individual resistor, as expected for a parallel combination.

FAQ

What's the difference between a conductor, an insulator, and a resistor?

Conductors (like copper) offer almost no resistance to charge flow. Insulators (like rubber) offer very high resistance, effectively blocking charge flow. Resistors fall between the two extremes — components intentionally designed to provide measurable, controlled resistance in a circuit.

How does temperature affect resistance?

In most conductors, raising the temperature makes atoms vibrate more, making it harder for electrons to pass through — so resistance increases at higher temperatures. This is also why electrical components can overheat and fail under prolonged high current: as they heat up, their increased resistance further disrupts current flow.

Why do resistors add differently in series versus parallel?

In series, the same current must pass through every resistor along a single path, so total resistance is the sum of the individual resistances (Rs = R1+R2+…). In parallel, each resistor offers its own separate path, so adding more paths makes it easier for current to flow overall — total resistance follows 1/Rp = 1/R1+1/R2+… and always ends up smaller than the smallest individual resistor.

What's the difference between a battery's emf and its actual output voltage?

Emf (Ecell) is a battery's ideal, maximum potential difference. Because real batteries have internal resistance (rint), some voltage is lost as current flows through the battery itself: V = Ecell − i rint. This is why output voltage drops under high current loads or as a battery ages.