DC Circuits: Current and Resistance (AAMC 4C)

Current electricity deals with the steady flow of charge through conductors. The AAMC MCAT content outline zeroes in on Ohm's law, resistance and resistivity, the internal resistance of a cell, the condition for maximum power transfer, and the meaning of a steady current. This is one of the most heavily tested physics topics on the MCAT.

MCAT Chem/Phys focus. Circuits are part of content category 4C. High-yield: current and Ohm's law (V = IR), resistance and resistivity, series/parallel networks, power (P = IV = I²R = V²/R), EMF and internal resistance, Kirchhoff's laws, and capacitors in circuits (the RC time constant is covered in the Electrostatics notes).

Steady Current

An electric current is the rate of flow of charge: I = dq/dt. Its SI unit is the ampere (1 A = 1 C s⁻¹). A steady current is one whose magnitude does not change with time, so I = q/t directly.

Conventional vs electronic current

Conventional current flows from + to − outside the source (the direction in which positive charge would move). In metals the actual carriers are electrons, drifting in the opposite direction.

Drift velocity

Free electrons in a conductor move randomly at very high thermal speeds (~10⁵ m s⁻¹) but, when an electric field is applied, they acquire a small drift velocity v_d superimposed on this motion.

I = nAev_d ⇒ v_d = I/(nAe)

where n = number density of free electrons, A = cross-sectional area, e = electronic charge. Drift velocity is typically only a fraction of a millimetre per second, yet the electrical signal travels at near light speed because the field sets the entire electron sea in motion almost instantly.

Ohm's Law

For a metallic conductor at constant temperature, the current through it is directly proportional to the potential difference across its ends.

V = IR

The constant of proportionality is the resistance R, measured in ohms (Ω). Conductors that obey this law are called ohmic; semiconductors, gas discharges, and diodes are non-ohmic.

Common trap. Ohm's law applies only to ohmic conductors at constant temperature. A filament bulb, a diode, or an electrolyte does not obey V = IR linearly because R changes with current/voltage.

Electrical Power

The rate at which a circuit element converts electrical energy is

P = IV = I²R = V²/R

measured in watts (1 W = 1 J s⁻¹). For a fixed resistor all three forms agree; choose the version that matches the quantities you are given. Energy delivered over time t is W = Pt (the kilowatt-hour is the everyday billing unit, 1 kWh = 3.6 × 10⁶ J).

Resistance and Resistivity

Resistance depends on a conductor's geometry and material:

R = ρL/A

where ρ is the resistivity (units: Ω m), L is length, A is cross-sectional area. Resistivity is a material property; resistance is a sample property.

Temperature dependence

For most metals, resistance rises with temperature: R_T = R₀(1 + αΔT), where α is the temperature coefficient of resistance. For semiconductors and electrolytes, R decreases with temperature because more charge carriers are released.

Series and parallel combinations

Series vs Parallel, resistors and capacitors
Quantity	Resistors in series	Resistors in parallel	Capacitors in series	Capacitors in parallel
Combined value	R_s = R₁ + R₂ + …	1/R_p = 1/R₁ + 1/R₂ + …	1/C_s = 1/C₁ + 1/C₂ + …	C_p = C₁ + C₂ + …
Current / charge	Same I through each	I splits; sum at junction	Same Q on each	Q splits
Voltage	V splits; V = V₁ + V₂	Same V across each	V splits; V = V₁ + V₂	Same V across each
Result vs largest element	Larger than any individual R	Smaller than any individual R	Smaller than smallest C	Larger than any individual C
Application	Voltage divider, festive lights	House wiring (each device same V)	Increases voltage rating	Increases storage capacity

Quick rule: resistors and capacitors behave oppositely, resistors add in series, capacitors add in parallel.

Kirchhoff's laws (steady current networks)

Junction rule: ΣI_in = ΣI_out at any junction (charge conservation).
Loop rule: ΣV = 0 around any closed loop (energy conservation).

Internal Resistance

Every real cell has some internal resistance r, opposition to current within the cell itself. If the EMF is ε and a current I flows, the terminal potential difference (the voltage measured at the cell's terminals) is

V = ε − Ir

This means V < ε whenever current flows. On open circuit (I = 0) the terminal pd equals the EMF.

Definitions

EMF (ε): The energy supplied by the source per unit charge that passes through it. SI unit: volt.
Terminal potential difference (V): The pd between the cell's terminals when current flows. V = ε − Ir.
Internal resistance (r): Resistance of the electrolyte and electrodes inside the cell. Causes a "lost-volts" drop Ir.

Maximum Power Output

For a cell of EMF ε and internal resistance r connected to a load R, the current is I = ε/(R + r) and the power dissipated in the load is

P = I²R = ε²R / (R + r)².

Differentiating dP/dR = 0 gives the maximum power transfer condition:

R = r

At this matched load the maximum power delivered is P_max = ε²/(4r), and the efficiency is exactly 50% (half the power goes into heating the internal resistance).

Memory aid. "Match for max", load resistance must match internal resistance for maximum power. But efficiency is only 50%, so for power-hungry distribution networks (mains) we use very low source resistance instead, not impedance matching.

Worked MCQs

Five MCQs that capture the high-yield testing patterns for this chapter. Read the explanation even when you get the answer right, it's where the deeper concept lives.

Q1. A wire of resistance R is stretched to twice its original length without losing material. The new resistance is:

Volume is constant: A × L = constant, so doubling L halves A. R = ρL/A → new R = ρ(2L)/(A/2) = 4ρL/A = 4R. Resistance is proportional to L² when volume is fixed.

Q2. A cell of EMF 6 V and internal resistance 1 Ω delivers maximum power to an external resistance of:

0 Ω
1 Ω
2 Ω
6 Ω

Maximum power transfer occurs when R = r. So R = 1 Ω. Maximum power = ε²/(4r) = 36/4 = 9 W; efficiency 50%.

Q3. The terminal potential difference of a cell of EMF ε and internal resistance r drawing current I is:

ε + Ir
ε − Ir
Ir − ε
ε

When current flows, the cell loses Ir volts inside itself, so the pd at the terminals is V = ε − Ir. On open circuit (I = 0) we measure the EMF; under load we measure less.

Q4. Three resistors of 6 Ω each are connected in parallel. The equivalent resistance is:

18 Ω
6 Ω
3 Ω
2 Ω

For n equal resistors in parallel: R_p = R/n = 6/3 = 2 Ω. The parallel combination is always less than the smallest single resistance.

Q5. The drift velocity of free electrons in a copper wire carrying a steady current is typically of the order of:

3 × 10⁸ m s⁻¹
10⁵ m s⁻¹
10² m s⁻¹
10⁻⁴ m s⁻¹

Drift velocity is surprisingly slow, under a millimetre per second, because n is huge. The electrical signal itself, however, propagates close to the speed of light because the entire electron sea responds to the field almost instantly.

Quick Recap

I = q/t for steady current; SI unit ampere.
I = nAev_d; drift velocity ~ 10⁻⁴ m/s in metals.
Ohm's law V = IR (only ohmic conductors at constant T).
R = ρL/A; resistivity ρ is a material property.
Power: P = IV = I²R = V²/R (watts).
Series: R_s = ΣR; Parallel: 1/R_p = Σ1/R.
Internal resistance: V = ε − Ir.
Maximum power transfer when R_load = r; efficiency = 50%.
Kirchhoff: junction rule (charge), loop rule (energy).

Test yourself. Take a timed Current Electricity quiz or browse all Physics MCQs to lock these concepts in.