Resistance and Ohm's Law

If current is the flow of charge, resistance is what opposes that flow. Every material resists current to some degree. Copper barely fights it (great conductor — that’s why your wires are copper). Rubber fights it almost completely (great insulator — that’s why the wires are coated with rubber).

Understanding resistance and its relationship to voltage and current — captured by Ohm’s law — is the single most important skill for MCAT circuit problems. Ohm’s law shows up in 90% of circuit questions, often in disguise.

Resistance: The Opposition to Flow

Think of resistance as a narrow section of pipe in our water-park analogy. A narrow, long, twisty pipe is hard to push water through. A short, wide, straight pipe is easy. Same idea for charges in a wire.

When charges fight their way through a resistor, they lose energy — that energy becomes heat. This is exactly why phone chargers, laptops, light bulbs, and toaster elements all warm up: the resistance of the circuit is dissipating electrical energy as thermal energy.

This formula gets tested heavily — the MCAT loves to change one variable and ask what happens:

• Double the length → resistance doubles ($R \propto L$).
• Double the cross-sectional area → resistance halves ($R \propto 1/A$).
• Switch to a higher-resistivity material → resistance goes up ($R \propto \rho$).

Ohm’s Law

This is the single most-used equation in circuit physics.

Ohm’s law tells you three different things depending on which form you use:

• $V = IR$ — The voltage drop across a resistor equals the current times the resistance.
• $I = V/R$ — More voltage drives more current; more resistance reduces it.
• $R = V/I$ — Resistance is the ratio of voltage drop to current.

Temperature Dependence of Resistance

For metals (conductors), resistance increases as temperature increases. Higher temperature → metal atoms vibrate more → more collisions with the drifting electrons. It’s like trying to walk through a crowd where everyone is dancing chaotically — the more they move, the harder it is to get through.

For semiconductors (silicon, germanium), the opposite is true: resistance decreases as temperature increases. The thermal energy frees up extra charge carriers that more than make up for the increased atomic vibration.

Ohmic vs. Non-Ohmic Materials

An ohmic material obeys Ohm’s law: its resistance stays constant regardless of the voltage applied. On a $V$ vs. $I$ graph, an ohmic material produces a straight line through the origin — and the slope of that line is $R$.

A non-ohmic material has resistance that changes with voltage or current. Examples:

• Diodes — only conduct in one direction; almost no current in the other direction.
• Light bulb filaments — heat up sharply when current flows, raising R as they warm.
• Batteries under heavy load — internal resistance changes as the battery drains.

The MCAT defaults to assuming resistors are ohmic unless the passage explicitly says otherwise.

Worked Example

A copper wire has resistance 5 Ω. You replace it with a piece of the same copper but twice as long and half the cross-sectional area. What’s the new resistance?

• $R \propto L/A$.
• Length doubles → factor of 2.
• Area halves → factor of 2 (since R is inversely proportional to A).
• Combined: $R_{new} = 5 \times 2 \times 2 = 20$ Ω.