Quick reference · 69 formulas · 15 chapters

Electricity
formulas

The formulas of the basic electricity course, chapter by chapter, with each quantity, its symbol and its unit. Type a word ("power", "kWh", "coil") to find a formula.

Search a formula The 8 essentials

U = R · IP = U · IW = P · t

Ohmvoltage

Powerin watts

Energyjoules, kWh

Search a formula

By word, or by choosing a chapter.

Basics · units

2 formulasThe lesson

Usual prefixes

k = 10³
m = 10⁻³
µ = 10⁻⁶
n = 10⁻⁹

Quantity	Symbol	Unit
Voltage	U	V
Current	I	A
Resistance	R	Ω
Power	P	W
Energy	W	J

Ampere-hour and kilowatt-hour

1 Ah = 3 600 C
1 kWh = 3,6 × 10⁶ J

Quantity	Symbol	Unit
Quantity of electricity	Q	C
Energy	W	J

Electrostatics

3 formulasThe lesson

Coulomb's law

F = k · q · q'd²

Quantity	Symbol	Unit
Force	F	N
Charge	q	C
Distance	d	m
Constant (9 × 10⁹)	k	N·m²/C²

Field between two plates

E = Ue

Quantity	Symbol	Unit
Electric field	E	V/m
Voltage	U	V
Plate spacing	e	m

Force on a charge

F = q · E

Quantity	Symbol	Unit
Force	F	N
Charge	q	C
Electric field	E	V/m

Voltage · current

3 formulasThe lesson

Quantity of electricity

Q = I · t

Quantity	Symbol	Unit
Quantity of electricity	Q	C
Current	I	A
Time	t	s

Current density

J = IS

Quantity	Symbol	Unit
Current density	J	A/mm²
Current	I	A
Cross-section	S	mm²

Cross-section of a round wire

S = π · d²4

Quantity	Symbol	Unit
Cross-section	S	mm²
Diameter	d	mm

Ohm's law · resistance

3 formulasThe lesson

Ohm's law

U = R · I

Quantity	Symbol	Unit
Voltage	U	V
Resistance	R	Ω
Current	I	A

Pouillet's law

R = ρ · lS

Quantity	Symbol	Unit
Resistance	R	Ω
Resistivity	ρ	Ω·m
Length	l	m
Cross-section	S	m²

Copper ρ ≈ 1.7 × 10⁻⁸ Ω·m; aluminium 2.8 × 10⁻⁸; iron 1.0 × 10⁻⁷.

Temperature effect

R = R₀ · (1 + α · Δθ)

Quantity	Symbol	Unit
Resistance at 20 °C	R₀	Ω
Temperature coefficient	α	/°C
Temperature change	Δθ	°C

Power, energy, efficiency

4 formulasThe lesson

Electrical power

P = U · I

Quantity	Symbol	Unit
Power	P	W
Voltage	U	V
Current	I	A

Joule heating

P = R · I² = U²R

Quantity	Symbol	Unit
Power	P	W
Resistance	R	Ω
Current	I	A

Energy

W = P · t

Quantity	Symbol	Unit
Energy	W	J
Power	P	W
Time	t	s

In kWh: power in kW × time in hours.

Efficiency

η = PuPa

Quantity	Symbol	Unit
Efficiency	η	-
Useful power	Pu	W
Absorbed power	Pa	W

Resistor networks

22 formulasThe lesson

Whatever the network

Ohm's law on each resistor

U_n = R_n · I_n
I_n = U_nR_n
R_n = U_nI_n

Quantity	Symbol	Unit
Voltage across resistor n	U_n	V
Resistor n	R_n	Ω
Current through it	I_n	A

Every resistor obeys Ohm's law with its own voltage and current, in series as in parallel.

Equivalent resistance

Réq = UI

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω
Voltage across the network	U	V
Total current	I	A

Req is the single resistor that, at the same voltage U, would carry the same current I.

Power in each resistor

P_n = U_n · I_n
P_n = R_n · I_n²
P_n = U_n²R_n

Quantity	Symbol	Unit
Power dissipated by resistor n	P_n	W

Power balance

P = P₁ + P₂ + ...
P = U · I

Quantity	Symbol	Unit
Total power	P	W

Powers always add up, in series, in parallel or in a mixed network.

In series

Equivalent resistance

Réq = R₁ + R₂ + ...

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω

Common and split quantities

I = I₁ = I₂ = ...
U = U₁ + U₂ + ...

Quantity	Symbol	Unit
Current (common)	I	A
Total voltage (split)	U	V

In series the current I is common to every resistor; the voltage U is split between them.

Voltage across each resistor

U = Réq · I
U₁ = R₁ · I
U_n = R_n · I

Quantity	Symbol	Unit
Voltage across resistor n	U_n	V
Resistor n	R_n	Ω
Current	I	A

Current

I = URéq
I₁ = U₁R₁
I_n = U_nR_n

Quantity	Symbol	Unit
Current	I	A
Total voltage	U	V
Equivalent resistance	Réq	Ω

Each resistance

Réq = UI
R₁ = U₁I
R_n = U_nI

Quantity	Symbol	Unit
Resistor n	R_n	Ω
Voltage across resistor n	U_n	V
Current	I	A

Voltage divider

U₁ = U · R₁R₁ + R₂

Quantity	Symbol	Unit
Voltage on R₁	U₁	V
Total voltage	U	V

In parallel

Equivalent resistance

1Réq = 1R₁ + 1R₂ + ...

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω

Two resistors: Req = R₁·R₂ / (R₁ + R₂).

Common and split quantities

U = U₁ = U₂ = ...
I = I₁ + I₂ + ...

Quantity	Symbol	Unit
Voltage (common)	U	V
Total current (split)	I	A

In parallel the voltage U is common to every branch; the current I is split between them.

Current in each branch

I = URéq
I₁ = UR₁
I_n = UR_n

Quantity	Symbol	Unit
Current in branch n	I_n	A
Voltage (common)	U	V
Resistance of branch n	R_n	Ω

The branch with the smallest resistance carries the largest current.

Voltage

U = Réq · I
U = R₁ · I₁ = R₂ · I₂ = ... = R_n · I_n

Quantity	Symbol	Unit
Voltage (common)	U	V
Total current	I	A
Equivalent resistance	Réq	Ω

Resistance of each branch

Réq = UI
R₁ = UI₁
R_n = UI_n

Quantity	Symbol	Unit
Resistance of branch n	R_n	Ω
Voltage (common)	U	V
Current in branch n	I_n	A

Current divider

I₁ = I · R₂R₁ + R₂

Quantity	Symbol	Unit
Current in R₁	I₁	A
Total current	I	A

Two branches: the current splits inversely to the resistances (I₁ gets R₂'s share).

n identical resistors

Réq = Rn

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω
Resistance of each branch	R	Ω
Number of resistors	n	-

Mixed network

Step-by-step reduction

R₂₃ = R₂ · R₃R₂ + R₃
Réq = R₁ + R₂₃

Quantity	Symbol	Unit
Equivalent resistance of R₂ ∥ R₃	R₂₃	Ω
Total equivalent resistance	Réq	Ω

Example: R₁ in series with R₂ ∥ R₃. Replace each series or parallel group by its equivalent resistance until a single resistor remains.

Other case: parallel then series

R₂₃ = R₂ + R₃
Réq = R₁ · R₂₃R₁ + R₂₃

Quantity	Symbol	Unit
Equivalent resistance of R₂ + R₃	R₂₃	Ω
Total equivalent resistance	Réq	Ω

R₁ in parallel with the R₂ + R₃ branch: reduce the series branch first, then the parallel.

Total current

I = URéq

Quantity	Symbol	Unit
Current delivered by the generator	I	A
Generator voltage	U	V

Back down to each resistor

U₁ = R₁ · I
U₂₃ = R₂₃ · I
I₂ = U₂₃R₂
I₃ = U₂₃R₃

Quantity	Symbol	Unit
Voltage on R₁	U₁	V
Voltage shared by R₂ and R₃	U₂₃	V
Current in R₂	I₂	A

Walk back the other way: a series group gets the common current (U = R · I), a parallel group gets the common voltage (I = U / R). Check: U₁ + U₂₃ = U and I₂ + I₃ = I.

Power line

Line voltage drop

ΔU = 2 · R_l · I

Quantity	Symbol	Unit
Voltage drop	ΔU	V
Resistance of one wire	R_l	Ω
Current	I	A

Real generators · loads

4 formulasThe lesson

Real generator

U = E − r · I

Quantity	Symbol	Unit
EMF	E	V
Internal resistance	r	Ω
Terminal voltage	U	V

Active load

U = E' + r' · I

Quantity	Symbol	Unit
Back-EMF	E'	V
Internal resistance	r'	Ω

Generalised Ohm's law

I = ΣE − ΣE'ΣR

Quantity	Symbol	Unit
Current	I	A
Sum of loop resistances	ΣR	Ω

Characteristic

U = f(I) : droite de pente −r, ordonnée à l'origine E

Kirchhoff's laws

2 formulasThe lesson

Junction rule

Σ I entrants = Σ I sortants

Quantity	Symbol	Unit
Current	I	A

Loop rule

Σ E − Σ R · I = 0

Quantity	Symbol	Unit
EMF	E	V
Voltage drop	R·I	V

Capacitors

4 formulasThe lesson

Capacitance

Q = C · U

Quantity	Symbol	Unit
Charge	Q	C
Capacitance	C	F
Voltage	U	V

Stored energy

W = ½ · C · U²

Quantity	Symbol	Unit
Energy	W	J

Combinations

parallèle : Céq = C₁ + C₂
série : 1Céq = 1C₁ + 1C₂

Quantity	Symbol	Unit
Equivalent capacitance	Céq	F

RC time constant

τ = R · C

Quantity	Symbol	Unit
Time constant	τ	s
Resistance	R	Ω
Capacitance	C	F

3-digit marking: two digits then number of zeros, in pF (104 = 100 000 pF = 100 nF).

Magnetism · electromagnetism

3 formulasThe lesson

Magnetic flux

Φ = B · S

Quantity	Symbol	Unit
Flux	Φ	Wb
Magnetic flux density	B	T
Area	S	m²

Field of a long coil

B = µ₀ · N · Il

Quantity	Symbol	Unit
Vacuum permeability (4π × 10⁻⁷)	µ₀	T·m/A
Number of turns	N	-
Length	l	m

Laplace force

F = B · I · l

Quantity	Symbol	Unit
Force	F	N
Current	I	A
Conductor length	l	m

Induction · self-induction

5 formulasThe lesson

Faraday's law

E = −N · ΔΦΔt

Quantity	Symbol	Unit
Induced EMF	E	V
Number of turns	N	-
Flux	Φ	Wb
Time	t	s

Cut flux

E = B · l · v

Quantity	Symbol	Unit
Magnetic flux density	B	T
Length	l	m
Speed	v	m/s

Self-induction

E = −L · ΔIΔt

Quantity	Symbol	Unit
Inductance	L	H
Current	I	A

Energy of a coil

W = ½ · L · I²

Quantity	Symbol	Unit
Energy	W	J

RL time constant

τ = LR

Quantity	Symbol	Unit
Time constant	τ	s

Alternating current

3 formulasThe lesson

Sinusoidal quantity

u(t) = Umax · sin(ω · t + φ)

Quantity	Symbol	Unit
Peak value	Umax	V
Angular frequency	ω	rad/s
Initial phase	φ	rad

Angular frequency, frequency, period

ω = 2 · π · f
f = 1T

Quantity	Symbol	Unit
Frequency	f	Hz
Period	T	s

RMS value

Ueff = Umax√2

Quantity	Symbol	Unit
RMS value	Ueff	V
Peak value	Umax	V

RLC circuits

4 formulasThe lesson

Reactances

X_L = L · ω
X_C = 1C · ω

Quantity	Symbol	Unit
Reactance	X	Ω
Inductance	L	H
Capacitance	C	F
Angular frequency	ω	rad/s

Series impedance

Z = √(R² + (X_L − X_C)²)

Quantity	Symbol	Unit
Impedance	Z	Ω
Resistance	R	Ω

Ohm's law in AC

U = Z · I

Quantity	Symbol	Unit
RMS voltage	U	V
RMS current	I	A

Phase shift

tan φ = X_L − X_CR
cos φ = RZ

Quantity	Symbol	Unit
Phase shift	φ	rad

AC power

3 formulasThe lesson

The three powers

S = U · I
P = U · I · cos φ
Q = U · I · sin φ

Quantity	Symbol	Unit
Apparent power	S	VA
Active power	P	W
Reactive power	Q	var

Power triangle

S² = P² + Q²
cos φ = PS

Quantity	Symbol	Unit
Power factor	cos φ	-

Power factor correction

C = P · (tan φ₁ − tan φ₂)U² · ω

Quantity	Symbol	Unit
Capacitance to add	C	F
Phase shifts before / after	φ₁, φ₂	rad

Measurement · instruments

4 formulasThe lesson

Reading a dial

valeur = divisions luesdivisions totales × calibre

Ammeter shunt

R_s = r_A · iI − i

Quantity	Symbol	Unit
Shunt resistance	R_s	Ω
Ammeter resistance	r_A	Ω
Full-scale current	i	A
Total current to measure	I	A

Series multiplier (voltmeter)

R_add = Ui − r_V

Quantity	Symbol	Unit
Series multiplier	R_add	Ω
Voltmeter resistance	r_V	Ω

Volt-ammeter method

R = UI

Quantity	Symbol	Unit
Measured resistance	R	Ω

Downstream: the voltmeter is across R alone (error from the current shunted by V); upstream: the voltmeter spans A (error from the drop in A).

The 8 essential formulas

The ones that come up in almost every problem.

U = R · IOhm's lawOhm's law P = U · IPowerElectric power W = P · tEnergyElectric power Q = I · tElectric chargeVoltage and current Réq = R₁ + R₂ + ...Resistors in seriesSeries and parallel circuits 1Réq = 1R₁ + 1R₂ + ...Resistors in parallelSeries and parallel circuits U = E − r · IReal sourceEMF and internal resistance Ueff = Umax√2RMS valueAlternating current

Remembering formulas

Less memorizing, more logic.

1One formula, three forms

U = R × I also gives R = U / I and I = U / R: just remember one.

2The triangle

U on top, R and I at the bottom: cover the quantity you want, and the formula is what's left.

3Check the units

Volts divided by ohms give amps: if the unit doesn't fit, the formula is wrong.

4Do problems

A formula used ten times sticks without effort.

Frequently asked questions about formulas

What is the formula for electric power?

P = U × I: power in watts is the voltage in volts multiplied by the current in amps. With Ohm's law, you also get P = R × I² and P = U² / R.

How do you calculate energy used in kWh?

W = P × t with P in kilowatts and t in hours. A 2 kW heater switched on for 3 hours uses 6 kWh. 1 kWh is 3.6 million joules.

Which formulas should you know by heart?

Ohm's law, power, energy, resistors in series and in parallel: the eight essential formulas above. The others can be derived from them or looked up in this formula sheet.

How do you check that a formula is right?

With the units: each side of the = sign must give the same unit. Watts divided by volts give amps, so I = P / U is consistent.

What does the dot mean in formulas?

The dot "·" means "multiplied by". U = R · I reads "U equals R times I".

Simulated time00:00:00

Q = I·t-

W = P·t-