11.1 Ohm's Law, Power Formulas & Series/Parallel Circuits

11.1 Ohm's Law, Power Formulas & Series/Parallel Circuits

Exam Focus: E = I x R, the three power formulas, series versus parallel behavior, voltage division, combined resistance shortcuts, and the arithmetic habits that keep a theory item under 90 seconds.

The PSI bulletin lists electrical theory and general electrical trade knowledge among the subjects on the journeyman examination. These are the questions you cannot look up. The NEC contains no page that tells you the current through a 12-ohm heater on a 240-volt circuit — you have to know it. Fortunately the entire theory domain rests on a handful of relationships.

---

Ohm's Law

E = I × R, where E is electromotive force in volts, I is current in amperes, and R is resistance in ohms.

Rearranged:

$I = \frac{E}{R} \qquad R = \frac{E}{I} \qquad E = I \times R$

                    THE OHM'S LAW / POWER WHEEL
+-------------------------------------------------------------------------+
|                                                                         |
|     E = I x R          I = E / R          R = E / I                     |
|     P = E x I          P = I^2 x R        P = E^2 / R                   |
|                                                                         |
|     E = P / I          I = P / E          R = P / I^2                   |
|                                                                         |
|   Know any TWO of E, I, R, P and you can produce the other two.         |
+-------------------------------------------------------------------------+

The Three Power Formulas

$P = E \times I \qquad P = I^{2} \times R \qquad P = \frac{E^{2}}{R}$

Which one to reach for:

• You know volts and amps → P = E × I
• You know amps and ohms → P = I² × R (this is the heating formula; it is why doubling current quadruples heat)
• You know volts and ohms → P = E² / R

[!IMPORTANT] P = I²R is the one that explains the code. Conductor derating, ampacity correction, loose-connection failures, and overload protection all exist because heat rises with the square of current. A connection carrying twice its intended current dissipates four times the heat.

---

Series Circuits

A series circuit provides exactly one path for current.

Worked Example — Series

Three resistors of 4 Ω, 6 Ω, and 10 Ω are connected in series across 120 V.

1. $R_T = 4 + 6 + 10 = 20\ \Omega$
2. $I_T = \dfrac{120}{20} = 6\ \text{A}$ (the same 6 A flows through all three)
3. Voltage drops: $E_1 = 6 \times 4 = 24\ \text{V}$, $E_2 = 6 \times 6 = 36\ \text{V}$, $E_3 = 6 \times 10 = 60\ \text{V}$
4. Check: $24 + 36 + 60 = 120\ \text{V}$ ✔
5. Total power: $P = 120 \times 6 = 720\ \text{W}$

Trade connection: A loose terminal is a resistance placed in series with the load. It steals voltage from the load and converts it to heat right at the connection — which is why 110.14(D) requires terminations to be torqued to the manufacturer's value.

---

Parallel Circuits

A parallel circuit provides more than one path for current. Building wiring is overwhelmingly parallel: every receptacle on a circuit sees the same 120 V.

The Three Resistance Shortcuts

1. Equal resistances: $R_T = \dfrac{R}{N}$ — three 30 Ω branches give 10 Ω.
2. Exactly two resistances (product over sum): $R_T = \dfrac{R_1 \times R_2}{R_1 + R_2}$
3. Any number (reciprocal method): $R_T = \dfrac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}}$

Worked Example — Parallel

A 120 V circuit supplies a 24 Ω heater, a 40 Ω lamp bank, and a 60 Ω fan motor.

1. Branch currents: $I_1 = 120/24 = 5\ \text{A}$, $I_2 = 120/40 = 3\ \text{A}$, $I_3 = 120/60 = 2\ \text{A}$
2. Total current: $5 + 3 + 2 = 10\ \text{A}$
3. Total resistance: $R_T = 120/10 = 12\ \Omega$ — smaller than the 24 Ω smallest branch ✔
4. Total power: $P = 120 \times 10 = 1{,}200\ \text{W}$

[!CAUTION] The reasonableness check that catches most errors. If a parallel-resistance answer is larger than the smallest branch resistance, it is wrong — no exceptions. If a series-resistance answer is smaller than the largest resistor, it is wrong.

---

Kirchhoff's Two Laws, Stated for the Trade

• Current law: the current entering a node equals the current leaving it. This is why the neutral of a balanced multiwire branch circuit carries the difference and not the sum.
• Voltage law: the sum of the voltage drops around a closed loop equals the applied voltage. This is why the three series drops above had to total 120 V.

---

Exam Technique for Theory Items

1. Write down what you are given with units. Most wrong answers come from mixing watts with volt-amperes or ohms with amperes.
2. Identify the topology first. Series or parallel decides which rules apply before any arithmetic happens.
3. Use the check. Series drops must sum to the source; parallel currents must sum to the total.
4. Watch kilo prefixes. A 4.8 kW heater is 4,800 W. Dropping the factor of 1,000 turns a right method into a wrong answer.