Difficulty: Introductory–Intermediate | Prerequisites: Basic understanding of voltage, current, and electric fields from PHY 211.
This material forms the foundation for analysing any electric circuit. You need Ohm's Law to relate voltage, current, and resistance at the component level, and Kirchhoff's Rules to solve for unknowns in circuits with multiple loops and junctions. Internal resistance explains why real batteries behave differently from ideal ones. If you cannot confidently apply these tools, every later topic (RC circuits, AC circuits, electromagnetic induction) will be harder than it needs to be.
Ohm's Law connects current density to the electric field inside a conductor, and at the macroscopic level gives V = IR. Kirchhoff's Voltage Rule (loop law) says voltage gains and drops around any closed loop sum to zero. Kirchhoff's Current Rule (junction law) says current in equals current out at every node. Real batteries have internal resistance that causes their terminal voltage to sag under load.
Current density (J)
The current per unit cross-sectional area of a conductor: J = I/A. It points in the direction of conventional current flow. In simple terms, it tells you how "concentrated" the current is across the wire's cross-section.
Conductivity (σ)
A material property describing how easily current flows. High σ means a good conductor. Ohm's Law in microscopic form is J = σE.
Resistivity (ρ)
The inverse of conductivity: ρ = 1/σ. Higher resistivity means the material resists current flow more strongly. Think of it as the material's built-in opposition to electron movement.
Resistance (R)
For a uniform conductor of length L and cross-sectional area A: R = L/(σA) = ρL/A. This is the macroscopic quantity you plug into V = IR.
Kirchhoff's Voltage Rule (KVR, loop law)
The sum of all voltage changes around any closed loop in a circuit is zero: ΣΔVᵢ = 0. In simple terms, what goes up must come down, energy-wise, around a complete path.
Kirchhoff's Current Rule (KCR, junction law)
At any junction (node) in a circuit, the total current flowing in equals the total current flowing out: ΣI_in = ΣI_out. Charge is conserved, so nothing piles up at a junction.
Internal resistance (r)
The resistance inside a real battery or power source. It causes the terminal voltage to drop below the ideal EMF when current flows.
Conventional current
The direction positive charges would flow (opposite to actual electron flow). This is the standard convention used in circuit analysis.
The microscopic form is J = σE, linking current density to the electric field inside the conductor.
For a conductor of length L, the voltage across it is V = EL, so E = V/L.
Substituting: J = σ(V/L), and since J = I/A, you get I/A = σV/L.
Rearranging gives I = V/(L/σA), which is I = V/R, the familiar macroscopic Ohm's Law.
Resistance depends on geometry and material: R = L/(σA) = ρL/A.
Longer conductor → more resistance.
Larger cross-section → less resistance.
Higher resistivity material → more resistance.
KVR (loop law): Choose any closed loop in a circuit. Add up every voltage rise (e.g. crossing a battery from − to +) and every voltage drop (e.g. crossing a resistor in the direction of current). The total is zero.
KCR (junction law): At any node where wires meet, the sum of currents entering equals the sum leaving.
Label all currents. Assign a direction to each branch current. If you guess wrong, the algebra will return a negative value, which simply means the current flows the other way.
Label polarities on every element. For resistors, current enters the + side. For batteries, the longer line is +.
Choose loops and a traversal direction. You must start on a wire, not on a component. The first sign you encounter on each element determines whether you write a positive or negative term.
Write the KVR equation for each loop: sum of voltage changes = 0.
Write the KCR equation at each junction: current in = current out.
Solve the system of equations for the unknowns.
A real battery with EMF V₀ and internal resistance r delivers a terminal voltage V_L = V₀ − Ir to the external load.
The current drawn is I = V₀/(R + r), where R is the external load resistance.
As R → 0 (short circuit), I → V₀/r, which is the maximum current the battery can supply.
As R → ∞ (open circuit), I → 0 and V_L → V₀. This is why you measure the full EMF with a high-impedance voltmeter.
The term "voltage sagging" refers to the terminal voltage dropping as the current increases, because more voltage is lost across the internal resistance.
Quantity | Formula |
|---|---|
Ohm's Law (microscopic) | J = σE |
Ohm's Law (macroscopic) | V = IR |
Resistance from geometry | R = ρL/A = L/(σA) |
Current density | J = I/A |
Terminal voltage (real battery) | V_L = V₀ − Ir |
Current with internal resistance | I = V₀/(R + r) |
KVR | ΣΔV = 0 around any loop |
KCR | ΣI_in = ΣI_out at any node |
Internal resistance is why car batteries can start an engine (low R, high current) but the headlights dim slightly while cranking. Engineers designing power supplies must account for internal resistance to ensure the output voltage stays within specification under varying loads.
Students often assume current is "used up" as it passes through a resistor. It is not. Current is the same entering and leaving a resistor; what changes is the voltage (energy per charge).
Conventional current flows from + to −, but electrons actually move from − to +. When applying Kirchhoff's rules, you use conventional current direction.
A negative result for a current variable does not mean you made an error. It means the actual current flows opposite to the direction you initially assumed.
Students sometimes apply KVR starting on a component rather than on a wire. The sign convention becomes ambiguous if you do this. Always start on a wire.
⚠️ You will almost certainly need to apply Kirchhoff's rules to a multi-loop circuit. Practise setting up the equations quickly and consistently.
⚠️ Know the derivation from J = σE to V = IR. This is a conceptual favourite.
⚠️ Internal resistance problems are common: expect a question asking for terminal voltage or current under load.
⚠️ Sign errors in KVR are the most frequent source of lost marks. Be meticulous about traversal direction and polarity labels.
1. True or false: Current density J has units of A/m².
A: True.
2. Fill in the blank: Kirchhoff's Voltage Rule states that the sum of all voltage changes around a closed loop equals ______.
A: Zero.
3. True or false: If you assume a current direction and solve to get a negative value, you must redo the problem.
A: False. The negative sign simply indicates the current flows in the opposite direction to what you assumed.
4. Fill in the blank: A battery with EMF 12 V and internal resistance 2 Ω connected to a 4 Ω load delivers a current of ______ A.
A: 2 A (I = 12/(4+2) = 2).
Q: A copper wire has resistivity ρ, length L, and diameter d. If you double the length and halve the diameter, by what factor does the resistance change?
A: R = ρL/A. Doubling L multiplies R by 2. Halving the diameter quarters the cross-sectional area (A = πd²/4), which multiplies R by 4. Combined factor: 2 × 4 = 8. Resistance increases by a factor of 8.
Q: A 9 V battery with internal resistance 0.5 Ω is connected to a 4 Ω resistor. What is the terminal voltage across the external resistor?
A: I = 9/(4 + 0.5) = 2 A. Terminal voltage = 9 − 2(0.5) = 8 V.
Q: In a circuit with two loops sharing a middle branch, you write KVR for each loop and KCR at the shared junction. How many independent equations do you need if there are three unknown currents?
A: Three. Two KVR loop equations and one KCR junction equation.
Q: Why does a battery's terminal voltage decrease when you connect a low-resistance load?
A: More current flows, increasing the voltage drop across the internal resistance (V_drop = Ir). The terminal voltage is V₀ − Ir, so it falls.
This material connects directly to RC Circuits: you need KVR to write the differential equation governing capacitor charging and discharging. Kirchhoff's rules will reappear when you study AC circuits and RLC circuits later in the course. The microscopic form of Ohm's Law (J = σE) ties back to electric fields and will connect forward to how current density appears in Ampere's Law.
Ohm's law, current density, conductivity, resistivity, resistance, Kirchhoff's voltage rule, Kirchhoff's current rule, loop law, junction law, node law, KVR, KCR, internal resistance, terminal voltage, EMF, voltage sag, series circuits, parallel circuits, circuit analysis, PHY 212, university physics, electricity and magnetism