Resistance
High-Yield Summary
- R = ρL/A — resistance increases with resistivity and length, decreases with cross-sectional area; also increases with temperature in most conductors.
- Ohm's Law: V = IR (rearranges to 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 — pick the form matching known variables.
- Series resistors: same current through all, voltage divides, Rs = R1+R2+... (resistance adds).
- Parallel resistors: same voltage across all, current divides, 1/Rp = 1/R1+1/R2+... (resistance shrinks).
Resistance, Ohm's Law, and Power
R = ρL/A | V = IR | P = IV = I²R = V²/R
- ρ = Resistivity (Ω·m), material-specific
- L, A = Length and cross-sectional area of the resistor
- V, I, R = Voltage, current, resistance
- P = Power dissipated (rate of energy transfer)
- Internal resistance: V = Ecell − i·rint, where Ecell is ideal emf and i is current through the source.
Series vs. Parallel Resistors
| Series | Parallel |
|---|---|
| Same current through every resistor | Same voltage across every resistor |
| Voltage divides among resistors | Current divides among branches |
| Rs = R1+R2+... (resistance adds, always increases) | 1/Rp = 1/R1+1/R2+... (resistance always decreases) |
| One resistor fails → entire circuit opens | One branch fails → others keep working |
Common MCAT Trap
- Adding a resistor in series always increases total resistance; adding one in parallel always decreases it — a common sign-flip error under time pressure.
- Resistance increases with temperature in most conductors — don't assume heating a wire makes current flow more easily.
- Choose the power formula that matches your known variables (P=I²R if you know I and R; P=V²/R if you know V and R) rather than solving for the missing variable first.
Quick Recall
Two identical resistors are added in parallel vs. in series. Which arrangement has lower total resistance?
Why does a battery's output voltage drop under high current?