Units of physical quantities: links between chapters

In this lesson you will learn

Derive the unit of any quantity from its defining formula and express it in SI base units
Recognise equivalent unit expressions (Wb = V · s = H · A = T · m²) and the quantity an expression defines
Check a formula with units and find exponents by dimensional analysis
Solve short problems that join two chapters (electricity + mechanics, current + heat)

One of the physics tasks in the 2026 entrance exam asked which of the given expressions are units of magnetic flux. Such questions are solved not by memory but with formulas: any unit can be derived from one or two formulas in a minute. In DİM’s test collection this topic is called Bölmələr arasında genetik əlaqə (links between chapters): units tie mechanics, electricity and magnetism together. In the lesson «What is physics? Measurement and SI units» you met the seven base units; here you will build the other units from them, check formulas with units and solve problems that join two chapters.

Base and derived units

SI has seven base units: the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd). All other units are derived units: they come from the formula that defines the quantity. The notation [X] means “the unit of X”: for example, [F] = N.

1
Choose a formula
Take a formula linking the quantity to quantities whose units you already know — usually its defining formula (e.g. B = F / (I · l)).
2
Substitute units
Replace each quantity by its unit; drop numerical factors (½, 2, π) — they have no unit.
3
Simplify
If needed, open up units such as N, J and V and simplify down to base units.
4
Check
Another formula must give the same unit: for example, Wb from both Φ = B · S and ε = ΔΦ / Δt.

Units of mechanics

1 N = 1 kg · m/s², 1 J = 1 N · m = 1 kg · m²/s², 1 W = 1 J/s = 1 kg · m²/s³, 1 Pa = 1 N/m² = 1 kg/(m · s²)1 N = 1 kg · m/s², 1 J = 1 N · m = 1 kg · m²/s², 1 W = 1 J/s = 1 kg · m²/s³, 1 Pa = 1 N/m² = 1 kg/(m · s²)

where:

Nnewton — from F = m · a
Jjoule (work and energy) — from W = F · s
Wwatt (power) — from P = W / t
Papascal (pressure) — from p = F / S

These four units underlie all the rest: the electrical units start from the joule and the watt.

Example 1: opening up mechanical units

1) Express the unit of a spring’s stiffness in SI base units.
2) Show that the unit of momentum equals N · s.
3) How many joules are there in 1 kW · h?

Show solution

1) k = F / Δl → N/m = (kg · m/s²) / m = kg/s².
2) p = m · v → kg · m/s; N · s = (kg · m/s²) · s = kg · m/s — the same (it matches F · Δt = Δp).
3) 1 kW · h = 1000 W · 3600 s = 3.6 · 10⁶ J — the unit an electricity meter counts in.

Example 2 (multiple choice): expressions equal to the joule

Which expressions are equal to the joule?
I. N · m II. W/s III. Pa · m³
A) only I B) I and II C) I and III D) II and III E) I, II, III

Show solution

I — work = F · s: N · m = J ✓.
II — W/s = (J/s)/s = J/s² ✗ (the joule is W · s, not W/s).
III — Pa · m³ = (N/m²) · m³ = N · m = J ✓ (work of a gas = p · ΔV).
Answer: option C (I and III).

Units of electricity and magnetism

Electrical units are built in a chain that starts from the ampere (a base unit), the joule and the watt. Each step uses one defining formula: q = I · t, U = W / q, R = U / I, C = q / U.

1 C = 1 A · s, 1 V = 1 J/C = 1 W/A, 1 Ω = 1 V/A, 1 F = 1 C/V1 C = 1 A · s, 1 V = 1 J/C = 1 W/A, 1 Ω = 1 V/A, 1 F = 1 C/V

where:

Ccoulomb (electric charge) — q = I · t
Vvolt (voltage, potential, emf) — U = W / q or U = P / I
Ωohm (resistance) — R = U / I
Ffarad (capacitance) — C = q / U

The volt comes out two ways — joule per coulomb or watt per ampere — with the same result.

Example 3: electrical units in base units

1) Express the volt in SI base units.
2) Express the ohm in base units.
3) What is the unit of the product Ω · F? Which quantity is R · C?

Show solution

1) V = W/A = (kg · m²/s³) / A = kg · m²/(A · s³).
2) Ω = V/A = kg · m²/(A² · s³).
3) Ω · F = (V/A) · (C/V) = C/A = (A · s)/A = s: R · C is a time — it describes how fast a capacitor discharges.

1 T = 1 N/(A · m), 1 Wb = 1 T · m² = 1 V · s, 1 H = 1 Wb/A = 1 Ω · s1 T = 1 N/(A · m), 1 Wb = 1 T · m² = 1 V · s, 1 H = 1 Wb/A = 1 Ω · s

where:

Ttesla (magnetic induction) — from the Ampère force, B = F / (I · l)
Wbweber (magnetic flux) — Φ = B · S; from ε = ΔΦ / Δt, Wb = V · s
Hhenry (inductance) — L = Φ / I; from ε = L · ΔI / Δt, H = V · s/A = Ω · s

All magnetic units follow from the Ampère force and the law of electromagnetic induction: once you know the weber, the tesla and the henry follow easily.

Example 4: magnetic units

1) Express the tesla in SI base units.
2) Show with two formulas that Wb = V · s.
3) What is the unit of the expression L / R?

Show solution

1) T = N/(A · m) = (kg · m/s²) / (A · m) = kg/(A · s²).
2) Φ = B · S: T · m² = N · m/A = J/A = (V · C)/A = V · A · s / A = V · s. Another way: ε = ΔΦ / Δt → Φ = ε · Δt → V · s ✓.
3) H/Ω = (Ω · s)/Ω = s — L / R is also a time (how fast the current builds up in a coil).

Quantity	Formula	Unit	Equivalent expressions	In SI base units
Electric charge	q = I · t	C	A · s = F · V	A · s
Voltage	U = W / q	V	J/C = W/A = Ω · A	kg · m²/(A · s³)
Resistance	R = U / I	Ω	V/A = W/A² = V²/W	kg · m²/(A² · s³)
Capacitance	C = q / U	F	C/V = s/Ω	A² · s⁴/(kg · m²)
Magnetic induction	B = F / (I · l)	T	N/(A · m) = Wb/m² = V · s/m²	kg/(A · s²)
Magnetic flux	Φ = B · S	Wb	T · m² = V · s = H · A	kg · m²/(A · s²)
Inductance	L = Φ / I	H	Wb/A = Ω · s = J/A²	kg · m²/(A² · s²)

The same unit can be written in several ways — this is exactly what DİM tests. Watch the powers: Wb/m², A², s³.

Example 5 (multiple choice): which quantity does the expression define?

Which physical quantity is defined by the expression q² / (2W), where q is the charge of a capacitor and W the energy of its electric field?
A) voltage B) capacitance C) field strength D) inductance E) resistance

Show solution

The energy of a capacitor is W = q² / (2C) → C = q² / (2W): it is the capacitance (B).
Unit check: C²/J = C²/(V · C) = C/V = F ✓.
The same way: U²/R → V²/Ω = V · A = W — power; √(L · C) → √(Ω · s · s/Ω) = s — a time (Thomson’s formula).

Example 6 (coded answer): matching

Match each unit with an equivalent expression.
1. Wb 2. T 3. H
a) V · s/A b) V · s c) N/(A · m) d) V/m e) C/V

Show solution

1. Wb = V · s → b.
2. T = N/(A · m) → c.
3. H = Wb/A = V · s/A → a.
d) V/m is the unit of field strength and e) C/V that of capacitance (spare options).
The code on the answer sheet is 1-b; 2-c; 3-a.

Interactive

Loading simulation…

Check each expression with a formula: for example, V · A = W because P = U · I.

Checking a formula with units

Units also help you catch mistakes. A correct formula obeys three rules:

Both sides of an equation must have the same unit.
Only quantities with the same unit can be added or subtracted: v₀ + at is m/s + m/s.
The argument of sin, cos, an exponent or a logarithm must be unitless: ωt is a number (radians).

A unit check cannot catch a wrong number factor (½, 2π): v = gh fails the check, while v = √(gh) passes it but still has the wrong factor (the correct one is √(2gh)).

Example 7: check a formula and find the exponents

1) Check that the formula v = √(2gh) is correct in units.
2) A student wrote s = v₀t + at/2 for uniformly accelerated motion. Find the mistake with units.
3) The period of a simple pendulum depends only on l and g: T ~ lᵃ · gᵇ. Find a and b.

Show solution

1) √(m/s² · m) = √(m²/s²) = m/s ✓.
2) v₀t → m/s · s = m, but at/2 → m/s² · s = m/s ≠ m: the second term is wrong (it should be at²/2).
3) s = mᵃ · (m/s²)ᵇ = mᵃ⁺ᵇ · s⁻²ᵇ. For seconds −2b = 1 → b = −½; for metres a + b = 0 → a = ½. So T ~ √(l/g) — the exact formula is T = 2π√(l/g); units cannot give the 2π.

Cross-chapter tasks

DİM sometimes joins two chapters in one task: a charged ball hovering in an electric field (mechanics + electrostatics), water heated by a current (work of current + quantity of heat). The method is always the same: write each chapter’s formula separately, equate the shared quantity — a force or an energy — and check the unit at the end.

q · E = m · g, η · P · t = c · m · Δt

where:

q · E = m · ga charged body in equilibrium in an electric field: the electric force balances gravity (E is the field strength, N/C = V/m)
η · P · t = c · m · Δtheat balance of a heater: the share η of the energy released at power P during time t warms the water by Δt

Unit check: C · N/C = N and W · s = J — both sides match.

Example 8: a charged ball in an electric field

A charged ball of mass 0.2 g hovers in equilibrium in a uniform electric field of strength 4 · 10⁴ V/m directed vertically downwards. Find the ball’s charge and the number of extra electrons on it (e = 1.6 · 10⁻¹⁹ C, g = 10 m/s²).

Show solution

Equilibrium: q · E = m · g → q = m · g / E = 2 · 10⁻⁴ · 10 / (4 · 10⁴) = 5 · 10⁻⁸ C.
Unit check: [q] = kg · (m/s²) / (V/m) = N/(N/C) = C ✓.
Number of extra electrons: N = q / e = 5 · 10⁻⁸ / (1.6 · 10⁻¹⁹) ≈ 3.1 · 10¹¹. The ball is negative, so the electric force points against the field — upwards.

Example 9 (coded answer): an electric kettle

An electric kettle with a power of 2 kW and an efficiency of 84% heats 1.5 kg of water from 20 °C to 100 °C. Calculate the time needed, in seconds (c = 4200 J/(kg · °C)).

Show solution

Heat needed by the water: Q = c · m · Δt = 4200 · 1.5 · 80 = 504 000 J.
η · P · t = Q → t = Q / (η · P) = 504 000 / (0.84 · 2000) = 300 s.
Unit check: J / W = J / (J/s) = s ✓.
The code on the answer sheet is 300.

Fill in the unit

1.1 Wb = 1 V ·
2.1 T = 1 N/(A · )
3.1 F = 1 C /
4.1 H = 1 Ω ·

Key points

A derived unit comes from the quantity’s defining formula: [B] = N/(A · m) = T.
Key equivalents: J = N · m = W · s = V · C; V = W/A; Wb = T · m² = V · s = H · A; H = Ω · s.
To see which quantity an expression defines, solve a known formula for that quantity: q²/(2W) = C, U²/R = P.
Both sides of a formula, and all terms of a sum, must have the same unit; a unit check does not reveal number factors.
In a cross-chapter problem equate the shared quantity (a force or an energy): qE = mg, ηPt = cmΔt.

Check yourself

12 questions. Every correct answer earns XP.

1 / 12

Which unit is NOT an SI base unit?