Ohm's Law
Learning Objectives
- State Ohm's Law and identify voltage, current, and resistance in a circuit diagram.
- Rearrange V = IR to solve for any one quantity given the other two.
- Combine Ohm's Law with the power formula to find power dissipation in a component.
- Distinguish ohmic (linear) components from non-ohmic (non-linear) components using their V-I graphs.
- Apply Ohm's Law inside series and parallel circuits to find branch currents and voltage drops.
- Recognize the practical limits of Ohm's Law (temperature, frequency, non-linear devices).
Quick Answer
Ohm's Law says the current through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance: V = I × R. Discovered experimentally by Georg Simon Ohm in 1827, it is the single most-used equation in electronics because it links the three quantities you measure or design around in every circuit — voltage, current, and resistance. Every resistor calculation, every power budget, and every circuit analysis technique (Kirchhoff's Laws, Thevenin's theorem, AC impedance) ultimately rests on this one relationship, which is why it's the first thing taught in any circuits course.
The Formula
V = I × R
- V = voltage (volts, V) — the electrical "push" between two points
- I = current (amperes, A) — charge flow per second
- R = resistance (ohms, Ω) — opposition to that flow
| To find | Formula | Use when you know |
|---|---|---|
| Voltage | V = I × R | current and resistance |
| Current | I = V / R | voltage and resistance |
| Resistance | R = V / I | voltage and current |
Memory aid: draw a triangle with V on top, I and R on the bottom. Cover the letter you want to find — the remaining two show you the operation (side by side means multiply, one over the other means divide).
Worked Example 1: Finding Current
A 9V battery is connected across a 3Ω resistor. What current flows?
I = V / R = 9 V / 3 Ω = 3 A
Worked Example 2: Finding Resistance
An LED driver circuit draws 0.5A from a 12V supply. What is the effective resistance of the load?
R = V / I = 12 V / 0.5 A = 24 Ω
Worked Example 3: Finding Voltage Drop
A current of 2A flows through a 100Ω resistor. Find the voltage dropped across it.
V = I × R = 2 A × 100 Ω = 200 V
This is a big voltage drop — a reminder that Ohm's Law doesn't care about "small" components; a 100Ω resistor at 2A needs a supply capable of at least 200V just for that one part.
Power and Ohm's Law
Power is the rate of energy conversion: P = V × I. Substituting Ohm's Law gives two more useful forms:
- P = I² × R — use when you know current and resistance
- P = V² / R — use when you know voltage and resistance
| Given | Power formula |
|---|---|
| V and I | P = V × I |
| I and R | P = I² × R |
| V and R | P = V² / R |
Worked Example 4: A 60Ω resistor carries 2A. Find the power it dissipates, and check whether a standard 0.25W resistor would survive.
P = I² × R = (2)² × 60 = 4 × 60 = 240 W
240W is nowhere near a 0.25W-rated resistor's limit — it would burn out almost instantly. This is exactly why datasheets specify a power rating: exceeding it causes overheating and failure, even though the resistance value itself is "correct."
Ohmic vs. Non-Ohmic Behavior
Ohm's Law applies exactly to ohmic (linear) resistors, where R stays constant no matter what V or I is. Plot V against I for an ideal resistor and you get a straight line through the origin — the slope is R.
Many real components are non-ohmic: their resistance changes with operating conditions.
| Component | V-I behavior | Why |
|---|---|---|
| Ideal resistor | Straight line through origin | R is constant |
| Diode | Sharp exponential curve | Forward and reverse conduction differ physically |
| Filament bulb | Resistance rises as it heats up | Tungsten resistivity increases with temperature |
| NTC thermistor | Resistance falls as temperature rises | More charge carriers become available |
| Varistor (MOV) | Resistance collapses above a threshold voltage | Designed for surge protection |
For a non-ohmic device you can still apply Ohm's Law locally, using the slope of the V-I curve at one operating point (the "small-signal" or "dynamic" resistance) — but a single R value won't describe the whole curve.
Ohm's Law in Series and Parallel Circuits
Series circuit — same current everywhere, voltages add:
- R_total = R1 + R2 + R3 + ...
- I = V_supply / R_total
- V1 = I × R1, V2 = I × R2, ...
Parallel circuit — same voltage everywhere, currents add:
- 1/R_total = 1/R1 + 1/R2 + 1/R3 + ...
- Two resistors only: R_total = (R1 × R2) / (R1 + R2)
- I1 = V / R1, I2 = V / R2, ...
Worked Example 5: A 10V supply feeds two resistors in parallel: R1 = 20Ω, R2 = 5Ω. Find the total resistance and the current from the supply.
R_total = (R1 × R2) / (R1 + R2) = (20 × 5) / (20 + 5) = 100 / 25 = 4 Ω
I_total = V / R_total = 10 / 4 = 2.5 A
Notice the parallel combination (4Ω) is smaller than either individual resistor — that's always true for resistors in parallel, and it's a quick sanity check on your arithmetic.
Practical Limitations of Ohm's Law
- Temperature dependence — resistance of most metals rises with temperature, so heavy currents that self-heat a conductor change R as they flow, making the "constant" resistance drift.
- Frequency — at AC frequencies, capacitance and inductance contribute reactance. The AC version of Ohm's Law replaces R with impedance Z: V = I × Z, where Z is a complex number combining resistance and reactance.
- Non-ohmic devices — diodes, transistors, and MOSFETs operate on curved V-I relationships, so a single resistance value doesn't describe them.
- Superconductors — below their critical temperature, R = 0. Ohm's Law still holds (V = I × 0 = 0) but it's a degenerate, special case.
Key Terms
| Term | Definition |
|---|---|
| Voltage (V) | Electrical potential difference between two points, measured in volts; the driving force for current |
| Current (I) | Rate of charge flow, measured in amperes |
| Resistance (R) | Opposition to current flow, measured in ohms |
| Ohmic component | A device whose resistance stays constant regardless of voltage or current |
| Power (P) | Rate of energy conversion, measured in watts (P = VI) |
| Impedance (Z) | The AC generalization of resistance, including reactive effects from capacitors and inductors |
| Small-signal resistance | The local slope of a non-linear device's V-I curve at one operating point |
Common Mistakes
-
Misconception: "Ohm's Law works for every electrical component." Why it's wrong: Diodes, LEDs, transistors, and bulbs have curved V-I relationships — a single R doesn't hold across their whole operating range. Correct: Ohm's Law strictly applies to ohmic (linear) resistors; non-ohmic devices need their own V-I characteristic curve or a small-signal resistance at a specific point.
-
Misconception: "Resistors in parallel add up like resistors in series." Why it's wrong: Adding a second path for current to flow through can only make it easier for current to flow, so total resistance in parallel must go down, not up. Correct: Parallel resistances combine via 1/R_total = 1/R1 + 1/R2 + ..., and R_total is always less than the smallest individual resistor.
-
Misconception: "If I know a resistor's rated wattage, I don't need to check current." Why it's wrong: Power rating tells you the maximum P a resistor can safely dissipate — you still must calculate the actual I² × R (or V²/R) for your specific circuit to know if it's within that limit. Correct: Always compute the expected power dissipation with your actual circuit values and compare it to the component's rated power before choosing it.
Comparison and Connections
| Concept | Ohm's Law | Kirchhoff's Laws | AC Impedance Law |
|---|---|---|---|
| Scope | Single component (V, I, R) | Whole circuit (current/voltage conservation) | Single component under AC (V, I, Z) |
| Formula | V = I × R | ΣI = 0 (node), ΣV = 0 (loop) | V = I × Z (complex numbers) |
| Applies to | Resistors (ohmic) | Any circuit topology | Resistors, capacitors, inductors |
| Typical use | Finding one unknown quantity | Solving multi-loop/multi-node circuits | AC circuit analysis, filters |
Practice Questions
Recall 1: State Ohm's Law in words and as a formula. Answer guidance: Voltage across a conductor is directly proportional to current through it; V = I × R, with R as the proportionality constant.
Recall 2: What are the three forms of the power formula derived using Ohm's Law? Answer guidance: P = VI, P = I²R, P = V²/R.
Understanding 1: Why doesn't Ohm's Law apply to a diode in the same way it applies to a resistor? Answer guidance: A diode's current depends exponentially on voltage, so its "resistance" (V/I) changes drastically at different operating points — there's no single constant R.
Understanding 2: Explain why total resistance decreases when resistors are added in parallel. Answer guidance: Each additional parallel branch gives current another path, increasing total current for the same voltage, which by R = V/I means total resistance drops.
Application 1: A heating element needs to dissipate 500W when connected to a 100V supply. What resistance should it have, and what current will flow? Answer guidance: R = V²/P = 100²/500 = 20 Ω; I = V/R = 100/20 = 5 A.
Application 2: Two resistors, 6Ω and 3Ω, are connected in parallel across a 12V battery. Find the current through each and the total current. Answer guidance: I1 = 12/6 = 2A, I2 = 12/3 = 4A, I_total = 6A (check: R_total = 2Ω, I = 12/2 = 6A ✓).
Analysis 1: A resistor is rated 1W and 100Ω. What is the maximum current it can safely carry, and what happens if a fault raises the current to 200mA? Answer guidance: I_max = √(P/R) = √(1/100) = 0.1 A (100mA). At 200mA, P = I²R = 0.04 × 100 = 4W — four times the rating, so the resistor will overheat and likely fail.
Analysis 2: Compare how Ohm's Law changes when moving from a DC resistive circuit to an AC circuit containing a capacitor. What replaces R, and why can't the same single number be used at all frequencies? Answer guidance: R is replaced by impedance Z, which for a capacitor is Z = 1/(jωC) — it depends on angular frequency ω, so Z (and hence the V/I relationship) changes with frequency, unlike a resistor's constant R.
FAQ
Q1: Does Ohm's Law apply to AC circuits? Yes, but resistance R is replaced by impedance Z, a complex quantity that accounts for the phase-shifting effects of capacitors and inductors: V = I × Z.
Q2: Why do resistors have power ratings if Ohm's Law only involves V, I, and R? Because current flowing through resistance always generates heat (P = I²R). The power rating tells you how much heat the resistor can dissipate before it's damaged — a separate physical limit from its resistance value.
Q3: Is a light bulb filament ohmic? Not really. Its resistance is low when cold and rises sharply once the filament heats up, so the V-I graph curves rather than forming a straight line.
Q4: Can resistance ever be negative? Not for a passive component like a resistor — but some active devices (like tunnel diodes in part of their curve) exhibit a "negative differential resistance" region, where increasing voltage causes current to decrease.
Q5: Why is the parallel resistance formula "product over sum" only for two resistors? Because 1/R_total = 1/R1 + 1/R2 simplifies algebraically to R_total = (R1R2)/(R1+R2) only when there are exactly two terms. For three or more resistors you must add the reciprocals and then invert the sum.
Quick Revision
- Ohm's Law: V = I × R (voltage = current × resistance).
- Rearranged: I = V/R, R = V/I.
- Power: P = VI = I²R = V²/R.
- Ohmic components have constant R (straight-line V-I graph through the origin).
- Non-ohmic components (diodes, bulbs, thermistors) have curved V-I graphs.
- Series circuits: same current, resistances add (R_total = R1 + R2 + ...).
- Parallel circuits: same voltage, reciprocal resistances add (1/R_total = 1/R1 + 1/R2 + ...).
- Two-resistor parallel shortcut: R_total = (R1×R2)/(R1+R2).
- Parallel R_total is always smaller than the smallest branch resistance.
- At AC frequencies, R generalizes to impedance Z (a complex number).
- Component power ratings must exceed the calculated I²R or V²/R, not just match resistance value.
- Superconductors (R=0) are a valid but degenerate case of Ohm's Law.
Related Topics
Prerequisites: Basic electric charge, voltage, and current concepts; SI units for electrical quantities.
Related Topics: Series and Parallel Circuits, Power Calculations, Kirchhoff's Laws.
Next Topics: Kirchhoff's Laws, Thevenin's and Norton's Theorems.