Ohm’s law: definition, formula and examples

What is Ohm’s law? Definition and statement

Ohm’s law states that the current flowing through a conductor is directly proportional to the potential difference across its ends, provided its temperature and other physical conditions remain unchanged. In symbols, V ∝ I, so V = I × R, where the constant of proportionality R is the conductor’s resistance. That is the definition examiners expect, and it is worth learning word for word.

Put a voltage across a piece of metal and a current flows. Double the voltage and the current doubles; halve it and the current halves. The ratio of voltage to current stays the same, and that ratio is the resistance. That is all Ohm’s law says:

The units make it concrete. One volt drives one ampere through one ohm, and the SI defines the ohm exactly that way, as a volt per ampere. Its reciprocal, the siemens, measures conductance, the ease with which current flows:

A 10 Ω resistor with 5 V across it carries 0.5 A. Raise the voltage to 50 V and it carries 5 A, as long as it stays at the same temperature. That last condition matters more than it sounds: Ohm’s law is a description of how many materials behave, most metals among them, not a law of nature like conservation of charge. It also applies to one resistance at a time. A circuit with several parts needs Kirchhoff’s laws as well, though each ohmic part in it still follows V = IR on its own. To watch it happen, open the Ohm’s law circuit in the simulator and change the voltage or the resistance.

The Ohm’s law formula and the power wheel

Solve V = IR for each quantity in turn and you get the three forms everyone learns first:

Add the power relationship, P = V × I (watts equal volts times amps), and substitute Ohm’s law into it. Replacing V with IR gives P = I²R; replacing I with V/R gives P = V²/R:

The two substitutions say different things, and both are worth knowing by heart. P = I²R says the heat in a conductor rises with the square of the current, which is why electrical codes limit the current each conductor size may carry. P = V²/R says a fixed resistance turns out power in proportion to the square of the voltage, which is why an appliance on the wrong supply voltage is badly under- or over-rated.

Rearranging those for each unknown gives twelve formulas, usually drawn as a wheel. Find the quantity you want in the hub, then use the segment in its quarter that contains the two values you already know.

The wheel is a lookup, not a separate set of laws. If you forget it, write down V = IR and P = VI and every segment follows in a line or two of algebra. The triangle some people draw, with V on top and I and R below, is the same three-way rearrangement of V = IR without the power formulas.

The Ohm’s law triangle

The Ohm’s law triangle is a memory aid for the three forms of V = IR. Write V at the top and I and R side by side underneath. Cover the quantity you want with a finger: what is left shows how to get it. I beside R means multiply; V over I or R means divide.

The triangle works only for voltage, current and resistance. For power, use the wheel above, or add P = V × I as a second triangle with P on top and V and I underneath.

Solve any two values

The calculator does the algebra: enter any two of voltage, current, resistance and power and it returns the other two, with each step shown. It reads component shorthand such as 4k7 and 2M2. For worked examples and a resistor wattage check, open the full Ohm’s law calculator.

Worked examples

Each example is computed by the same engine as the calculator, so the working shown here is the working it shows.

The second example is the one that catches people out. On 208 V the element delivers 3,380 W, only 75 % of its rating, because the voltage is 87 % of rated and power goes with its square. The water heater still works; it recovers more slowly. The mistake in the other direction is worse: an element made for 4,500 W at 208 V is 9.614 Ω, and on 240 V it would dissipate 5,991 W, 33 % more than it was built for. The third example is why loop resistors are 250 Ω: 4 mA gives 1 V and 20 mA gives 5 V, and at full scale the resistor dissipates only 100 mW.

Ohm’s law experiment (class 10)

Ohm’s law is verified in the school laboratory by measuring the current through a length of nichrome wire at several voltages and checking that V ÷ I stays the same. The circuit puts the wire in series with a battery, a plug key, an ammeter and a rheostat, with a voltmeter connected across the wire.

Procedure

1. Connect the circuit as shown, with the ammeter in series and the voltmeter in parallel with the wire XY, positive terminals towards the battery’s positive side.
2. Close the key and set the rheostat so a small current flows. Note the ammeter reading I and the voltmeter reading V.
3. Move the rheostat (or add cells) to raise the current in steps, and note V and I each time.
4. Open the key between readings so the wire does not warm up, which would change its resistance.
5. Work out V ÷ I for each pair of readings, then plot a graph of I against V.

Readings and result

For a 5 Ω wire, the readings Ohm’s law predicts are:

Conclusion: V ÷ I is the same for every reading and the graph is a straight line through the origin, so the current through the wire is directly proportional to the potential difference across it. That is Ohm’s law, and the constant ratio is the wire’s resistance. A graph that curves upward or downward at higher currents usually means the wire heated up during the readings.

Precautions: keep connections clean and tight; connect meters with the right polarity; use a small current and open the key between readings; read the meters straight on to avoid parallax; check both meters read zero before closing the key.

Ohm’s law numericals, solved

1. An electric iron draws 4.6 A from a 230 V supply. Find its resistance. R = V ÷ I = 230 ÷ 4.6 = 50 Ω.
2. A 9 V battery is connected across a 450 Ω resistor. What current flows? I = V ÷ R = 9 ÷ 450 = 0.02 A, or 20 mA.
3. A 10 Ω resistor carries 0.5 A. Find the potential difference across it and the power it dissipates. V = I × R = 0.5 × 10 = 5 V; P = I²R = 2.5 W.
4. A heater is rated 1,000 W at 230 V. Find its resistance and the current it draws. R = V² ÷ P = 230² ÷ 1,000 = 52.9 Ω; I = P ÷ V = 4.35 A.
5. The potential difference across a 20 Ω wire is doubled from 6 V to 12 V. What happens to the current? It doubles, from 0.3 A to 0.6 A, because the resistance is unchanged.

Resistance and resistivity: R = ρL/A

Resistance is not a property of copper. It is a property of a particular piece of copper, and it depends on what the conductor is made of, how long it is and how thick:

ρ (rho) is the resistivity of the material, L the length and A the cross-sectional area. Double the length and the resistance doubles; double the area and it halves. Annealed copper at 20 °C has a resistivity of 0.017241 Ω·mm²/m: the International Annealed Copper Standard’s conductivity of 58 MS/m turned upside down. Metals also become more resistive as they warm. Copper’s resistance rises by about 0.39 % for every degree Celsius near room temperature:

Worked example: 1,000 ft of 12 AWG copper

A 12 AWG conductor has an area of 6,530 circular mils, which is 3.309 mm². Take 1,000 ft, or 304.8 m, at 20 °C:

At 75 °C, the temperature the NEC uses for its conductor tables, that becomes 1.588 × (1 + 0.00393 × 55) = 1.931 Ω. NEC Chapter 9, Table 8 lists 1.93 Ω per 1,000 ft for solid, uncoated 12 AWG copper at 75 °C. A two-line formula and the code table agree to three figures.

Table 8 lists stranded 12 AWG at 1.98 Ω, 2.6 % higher, because the strands are laid in a helix and so are slightly longer than the cable. That is also where the voltage-drop constant K comes from: the resistance per foot times the area in circular mils is 12.6 Ω·cmil/ft for solid 12 AWG and 12.9 for stranded, the “K = 12.9” used for copper in many exam books.

Metric cable works the same way. The formula gives 6.9 Ω/km for 2.5 mm² copper at 20 °C. IEC 60228 caps a 2.5 mm² conductor at 7.41 Ω/km, about 7 % above the ideal figure, which leaves room for stranding and manufacturing tolerance.

Ohmic and non-ohmic devices

A device is ohmic when a graph of its current against its voltage is a straight line through zero: its resistance is the same at any voltage. Resistors, wire and heating elements at a steady temperature behave this way. Plenty of everyday parts don’t.

Lamp filaments

A tungsten filament’s resistance rises steeply as it heats from room temperature to white heat. From its rating, a 60 W, 120 V lamp has a hot resistance of 240 Ω and draws 0.5 A. Measure the same lamp cold and the reading is a small fraction of that, so for the first moments after switch-on the current is several times the running current. Ohm’s law still holds at any instant, with the resistance of that instant, but R is not a constant, so the line on the graph bends.

LEDs and diodes

An LED barely conducts until the voltage approaches its forward voltage, then its current climbs exponentially. A red 0603 chip LED is specified at 2.0 V typical at 20 mA, which Ohm’s law would call 100 Ω. That figure is meaningless a tenth of a volt either side: below it the LED is almost off, above it the current runs away. So an LED is never connected straight across a supply. A series resistor takes up the difference between the supply and the forward voltage, and the resistor, being ohmic, sets the current. Size the series resistor for an LED from the supply voltage and the LED’s forward voltage.

Other non-ohmic parts on the job: thermistors, whose resistance swings with temperature; the varistors in surge protectors, which conduct hard above a threshold voltage; and arcs, whose voltage falls as their current rises.

Limitations of Ohm’s law

• It holds only while temperature and other physical conditions stay constant; a heating conductor changes its resistance.
• It does not apply to non-ohmic devices: diodes, LEDs, transistors, thermistors, varistors and gas discharge lamps.
• Some devices conduct differently in each direction (a diode), so V ÷ I depends on the polarity.
• It describes one conductor at a time; circuits with several sources need Kirchhoff’s laws as well.
• On AC it holds for RMS values only when impedance Z replaces resistance.

Ohm’s law on AC: impedance

On AC, coils and capacitors oppose the current as well as resistance does. That opposition, reactance X, depends on frequency. Resistance and reactance combine at right angles into impedance Z, and Ohm’s law becomes V = I × Z with RMS values:

The catch is that an ohmmeter measures only R. Take a coil that reads 40 Ω and has 1.2 H of inductance. At 60 Hz its reactance is 452.4 Ω and its impedance 454.2 Ω, so on 120 V AC it draws 0.264 A. On 120 V DC only the resistance limits the current: 3 A, and 360 W of heat in a winding built for a small fraction of that. That is why AC coils burn out on DC, and why you can’t predict a motor’s or a transformer’s current from a resistance reading.

Because the current in a reactive load is out of step with the voltage, volts times amps is no longer watts. The share that is real power is the power factor; for this coil it is only 0.088.

How electricians use Ohm’s law

Checking a heating element

A nameplate gives watts and volts, so R = V²/P predicts what a healthy element should read. The 4,500 W, 240 V element above should measure close to 12.8 Ω, a little less when cold. A reading near that is a good element; an open circuit (OL) is a broken one; and any continuity from a terminal to the metal sheath means the insulation has failed and the element is a shock hazard.

Previewing voltage drop

The wire is a resistance in series with the load, and the current flows out and back. A 16 A load at the end of 100 ft of 12 AWG solid copper, as in NM cable, sees 200 ft of conductor: 0.386 Ω at 75 °C. The wire drops V = IR = 6.18 V, which is 5.1 % of 120 V, and turns 98.8 W into heat along the way. The NEC’s informational note on branch-circuit voltage drop (210.19 Informational Note in the 2023 edition) suggests keeping a branch circuit within 3 %, which at this current limits the run to about 58 ft. The voltage drop calculator repeats this for any size, material and length.

Finding heat at a bad connection

A termination that has worked loose may add only a fraction of an ohm, but P = I²R concentrates the heat in a small space. At 20 A, a connection of 0.05 Ω dissipates 20 W inside a terminal designed to run almost cool. The voltage across it is only 1 V, which is why measuring the voltage across connections under load, or a thermal camera, finds bad joints that a continuity test passes.

Measuring volts, amps and ohms

A multimeter applies Ohm’s law internally, and each function connects to the circuit differently:

• Voltage: across the part, in parallel. The meter’s high input resistance means it draws almost no current and barely disturbs the circuit.
• Current: in series, by breaking the circuit, or with a clamp meter around a single conductor. On its amps setting a meter is close to a short circuit; connected across a supply, it blows its fuse or worse.
• Resistance: only on a de-energized, isolated part. The meter drives a small known current and measures the voltage it produces, so an outside voltage or a parallel path through other components gives a false reading.

Compare cold readings with hot calculations carefully: heating elements read close to their working value, lamps far below it. And on a de-energized conductor run beside live ones, a high-impedance meter can show a “ghost” voltage coupled from its neighbours. A low-impedance setting loads that coupling down, which tells you whether the voltage is real.

Ohm’s law and electric shock

The body is a resistance, and the current through it, not the voltage as such, does the harm. OSHA’s table of effects, based on Kouwenhoven’s research, puts a faint tingle at about 1 mA, the “let-go” range, where muscles clamp onto the conductor, at 6–25 mA for women and 9–30 mA for men, and respiratory arrest from 50 mA. Ohm’s law turns those into resistances: 120 V drives 50 mA through any path of 2,400 Ω or less, and 230 V drives it through 4,600 Ω. Skin resistance drops sharply when it is wet, and OSHA notes that the current itself raises blisters that lower it further.

Hence the habits: isolate, lock out and prove dead before touching; treat a conductor as live until it is tested; and use GFCI protection where water is present. A Class A GFCI trips at a few milliamps, around the bottom of the let-go range; see GFCI requirements by NEC edition for where the NEC requires one.

Georg Ohm and the 1827 book

Georg Simon Ohm published the law in Berlin in 1827, in a short book titled Die galvanische Kette, mathematisch bearbeitet (“The galvanic circuit investigated mathematically”), whose preface is dated 1 May 1827. He treated circuits driven both by chemical cells and by thermocouples, and stated the rule in words: the current in a circuit is directly proportional to the sum of its voltages (“tensions”) and inversely proportional to the circuit’s total “reduced length”.

That reduced length is the part that has aged best. Ohm defined it, for each uniform section of the circuit, as the section’s length divided by the product of its conductivity and its cross-section, and added the sections up. In today’s notation that is R = ρL/A, the same formula that reproduced the NEC’s 12 AWG figure above.

The SI unit of resistance is named after him. Since the 2019 revision of the SI, the Planck constant h and the elementary charge e have exact values, so the von Klitzing constant h/e² = 25,812.807 Ω is exact too. National measurement laboratories realize the ohm with the quantum Hall effect, which produces resistances at exact fractions of that constant.