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Ohm's Law: Relating Voltage, Current, and Resistance

Ohm's law connects voltage across a component to the current through it. It is one of the most useful models in circuit analysis—but it is a model with conditions, not a promise that every device has constant resistance.

Learning objectives

After this lesson, you will be able to:

  • use V = IR and rearrange it without a formula triangle;
  • calculate voltage, current, resistance, and resistor power with correct units;
  • distinguish an ohmic resistor from a nonlinear device;
  • include source limits and component ratings in a practical calculation;
  • verify an Ohm's-law prediction with the interactive calculator.

The three quantities

Quantity Meaning SI unit
Voltage, V Energy transferred per unit charge between two points volt (V), where 1 V = 1 J/C
Current, I Rate of charge transfer ampere (A), where 1 A = 1 C/s
Resistance, R Ratio of voltage to current for an ohmic element under stated conditions ohm (Ω), where 1 Ω = 1 V/A

For an ideal resistor whose value is constant:

$$
V = IR
$$

The same relationship can be rearranged:

$$
I = \frac{V}{R}
\qquad
R = \frac{V}{I}
$$

The voltage is measured across the resistor; the current is measured through it.

A complete resistor circuit

The battery is the left branch and the resistor is the right branch. The upper and lower wires close one continuous loop, so the same current passes through both components.

BT1: Device:Battery value="5 V source" rotate=0
R1: Device:R value="1 kΩ load" rotate=0

layout direction=LR gap=150
group LOOP label="One closed loop" direction=LR gap=150 {
  BT1 R1
}

BT1.1 --> R1.1 color=#b91c1c
R1.2 --> BT1.2 color=#334155

For the ideal values shown:

$$
I = \frac{5\ \text{V}}{1000\ \Omega} = 0.005\ \text{A} = 5\ \text{mA}
$$

Track prefixes before calculating

Convert prefixes or use compatible engineering units:

Prefix Symbol Multiplier
milli m 10⁻³
micro µ 10⁻⁶
kilo k 10³
mega M 10⁶

A useful shortcut is V / kΩ = mA. For example, 12 V / 4.7 kΩ ≈ 2.55 mA.

Worked example: choose a resistor and its power rating

A 12 V source must produce approximately 8 mA through a resistor.

1. Calculate resistance

$$
R = \frac{12\ \text{V}}{0.008\ \text{A}} = 1500\ \Omega = 1.5\ \text{k}\Omega
$$

2. Calculate resistor power

Electrical power can be written as:

$$
P = VI = I^2R = \frac{V^2}{R}
$$

$$
P = 12\ \text{V} \times 0.008\ \text{A} = 0.096\ \text{W}
$$

A 0.125 W resistor is only slightly above the ideal dissipation. A 0.25 W part gives more thermal margin, subject to ambient temperature and manufacturer derating data.

Ohm's law is not universal constant resistance

An ideal resistor is ohmic: doubling its voltage doubles its current. Many devices are nonlinear:

  • an LED has a strongly nonlinear current-voltage curve;
  • a filament lamp's resistance rises as it heats;
  • a diode conducts very differently with polarity;
  • a battery's terminal voltage changes with load and state of charge;
  • a thermistor is intentionally temperature dependent.

At a particular operating point, R = V/I is a ratio. It does not prove that the device will have the same ratio at another voltage or temperature.

Source resistance matters

The earlier version of this lesson used a 9 V battery and 3 Ω load to predict 3 A. The arithmetic describes an ideal source, but a small 9 V battery cannot normally hold 9 V while delivering 3 A. Its internal resistance causes terminal-voltage drop and heating.

A simple source model is an ideal voltage source in series with internal resistance r:

$$
I = \frac{V_{OC}}{R_{load}+r}
$$

Never use an intentional short circuit to “measure the maximum current” of a battery.

Interactive calculator

Enter any two values and leave the third blank.

Processing...

Common mistakes

  • Mixing mA with A or kΩ with Ω without conversion.
  • Applying V = IR to an LED as though it were a fixed resistor.
  • Calculating current but ignoring source and component limits.
  • Confusing voltage across a component with current through it.
  • Selecting a resistor value without checking its power dissipation.
  • Treating the ideal answer as a guaranteed measurement.

Summary

  • For an ideal resistor, V = IR.
  • Units are part of the calculation.
  • Power and ratings must be checked after solving for current or resistance.
  • Nonlinear devices and real sources require models beyond a single constant resistance.

Next: Fundamental Electrical Quantities and Symbols.

Further reading