Ohm’s Law: Fundamentals, Formulas & Worked Examples

1827: German physicist Georg Simon Ohm publishes a landmark paper investigating how the potential difference (voltage) applied to a conductor governs the electric current that flows through it.
- His systematic experiments formalised the quantitative link between voltage, current, and resistance.
- The relationship subsequently became known as Ohm's Law and underpins virtually every calculation in basic circuit theory.

Textbook wording: “The current ( $I$ ) flowing through a conductor, provided its physical conditions (temperature, material, dimensions) remain constant, is:
- directly proportional to the potential difference ( $V$ ) across it, and
- inversely proportional to the resistance ( $R$ ) of the conductor.”
Equation set (all equivalent rearrangements):
- $V = I\,R$
- $I = \dfrac{V}{R}$
- $R = \dfrac{V}{I}$

$V$ (Potential Difference) → volts (V)
$I$ (Electric Current) → amperes (A). Some texts use $A$ as the symbol for current; examiners normally expect $I$ .
$R$ (Resistance) → ohms (Ω)
Remember the Greeks: Ω (omega) represents ohms; it is never used for anything else in DC-circuit calculations.

“Voltage is the push, resistance is the opposition, current is the result.”
- Increase $V$ with $R$ constant → $I$ rises proportionally.
- Increase $R$ with $V$ constant → $I$ falls proportionally.
Conductor must be ohmic (temperature and physical state unchanged) for the law to hold exactly. Deviations occur in devices like diodes, thermistors or filament lamps when they heat up.

Example 1 — Finding Resistance of a Heater Element
- Data: $V = 230\,\text{V}$ (mains supply), $I = 4\,\text{A}$ .
- Required: $R$ .
- Solution: $R = \dfrac{V}{I} = \dfrac{230\,\text{V}}{4\,\text{A}} = 57.5\,\Omega$ .
- Significance: Knowing element resistance lets engineers predict power draw $P = V I$ and required fuse ratings.
Example 2 — Current from a 12 V Battery
- Data: $V = 12\,\text{V},\; R = 3\,\Omega$ .
- Required: $I$ .
- Solution: $I = \dfrac{V}{R} = \dfrac{12\,\text{V}}{3\,\Omega} = 4\,\text{A}$ .
- Check: A car battery delivering 4 A to a load of 3 Ω will dissipate $P = V I = 48\,\text{W}$ .
Example 3 — Unknown Voltage Across a 4 Ω Resistor
- Data: $R = 4\,\Omega,\; I = 3\,\text{A}$ .
- Required: $V$ .
- Solution: $V = I R = 3\,\text{A} \times 4\,\Omega = 12\,\text{V}$ .
- Interpretation: The source is likely a small sealed-lead-acid battery or a regulated 12 V supply.

Builds directly on charge flow concepts: $I = \dfrac{\Delta Q}{\Delta t}$ .
Complements energy ideas: Work done moving charge $Q$ through voltage $V$ is $W = Q V$ ; combined with Ohm’s Law yields power formulations ( $P = I^2 R$ , $P = \dfrac{V^2}{R}$ ).

Domestic wiring: Fuse sizes and cable cross-sections calculated with $I = V/R$ to prevent overheating.
Electronics design: Resistor selection sets bias currents in transistor circuits.
Safety: Over-current protection (fuses, circuit-breakers) sized by estimating fault currents via Ohm’s Law.

Undersized wiring (high $R$ ) at household voltages can reach dangerous temperatures; engineers have a duty to apply Ohm’s Law conservatively.
Educational equity: Ohm’s accessible algebra makes electricity approachable for non-specialists, underscoring the importance of clear scientific communication.

Confusing symbols $A$ and $I$ . Examiners accept both, but be consistent.
Forgetting that resistance changes with temperature—Ohm’s Law works for ohmic conductors only.
Arithmetic slips: Always include units in intermediate steps; dimensional checks expose errors.

Memorise the triangle: place $V$ at the top, $I$ bottom left, $R$ bottom right. Cover the unknown to reveal the formula.
All rearrangements derive from $V = I R$ ; know at least one worked example for each variation.