JEE Mains Physics · Teaching notes

Units and Measurements — JEE Mains Physics

Units and Measurements has 190 past-year questions from 2021 to 2026, and 19 of them ask for a number rather than an option. Just over half are about dimensions: building a quantity's dimensions from the equation that defines it, then using them to find unknown constants and powers or to test an equation. Most of the rest are about errors and instruments, and there the arithmetic is short. Marks are lost on a sign or a factor: a zero error added instead of subtracted, a power left out of an error sum, or a time error divided by one period instead of the whole run.

Every subtopic, worked example, formula and trap in one printable document — answers shown, ready to share.

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Subtopic notes

Formula & revision sheet

15 formulas · 5 reference tables · 35 gotchas across all subtopics — the exam-eve cheat-sheet

Units, Significant Figures and Order of Magnitude

Formulas (1)

Significant figures in sums, products and means · Rounding rules $\text{sum or difference: fewest decimal places} \qquad \text{product or quotient: fewest significant figures}$

Reference tables (1)

Astronomical units of length, seconds of arc and order of magnitude6 rows

Unit or quantity	Value	How it is defined
Astronomical unit (AU)	$1.496 \times 10^{11}$ m	Mean distance from the Earth to the Sun
Light year (ly)	$9.46 \times 10^{15}$ m	Distance light travels in one year
Parsec (pc)	$3.08 \times 10^{16}$ m	Distance at which 1 AU subtends one second of arc The parsec is the largest of the three, about 3.26 light years.
Light second	$3.0 \times 10^{8}$ m	Distance light travels in one second
One second of arc (1″)	$4.85 \times 10^{-6}$ rad	One 3600th of a degree
One degree	$1.745 \times 10^{-2}$ rad	$\pi/180$ radian

In increasing size: astronomical unit, light year, parsec.

Watch out for (4)

A sum is rounded by decimal places, not by significant figures
→ Significant figures in sums, products and means
Leading zeros never count
→ Significant figures in sums, products and means
The parsec is bigger than the light year
→ Astronomical units of length, seconds of arc and order of magnitude
Convert the angle to radians first
→ Astronomical units of length, seconds of arc and order of magnitude

Dimensions of Mechanical and Thermal Quantities

Reference tables (3)

Dimensions of mechanical quantities15 rows

Quantity	Defining relation	Dimensions	SI unit
Force	$F = ma$	$MLT^{-2}$	newton (N)
Work, energy, torque	$W = Fs,\ \tau = rF$	$ML^{2}T^{-2}$	J (N m for torque)
Power	$P = W/t$	$ML^{2}T^{-3}$	watt (W)
Momentum, impulse	$p = mv,\ J = Ft$	$MLT^{-1}$	kg m/s or N s
Angular momentum, angular impulse	$L = mvr,\ \tau t$	$ML^{2}T^{-1}$	kg m²/s
Moment of inertia	$I = mr^{2}$	$ML^{2}$	kg m²
Pressure, stress, Young's modulus, bulk modulus	$F/A$	$ML^{-1}T^{-2}$	pascal (Pa)
Pressure gradient	$dP/dx$	$ML^{-2}T^{-2}$	Pa/m
Compressibility	$1/\text{bulk modulus}$	$M^{-1}LT^{2}$	Pa⁻¹
Surface tension, spring constant	$F/l,\ F/x$	$MT^{-2}$	N/m
Coefficient of viscosity	$F = \eta A\,dv/dx$	$ML^{-1}T^{-1}$	pascal-second (Pa s)
Intensity of a wave	power ÷ area	$MT^{-3}$	W/m²
Gravitational constant G	$F = Gm_1m_2/r^{2}$	$M^{-1}L^{3}T^{-2}$	N m²/kg²
Gravitational potential	energy ÷ mass	$L^{2}T^{-2}$	J/kg
Angular speed, frequency, velocity gradient	$\omega = \theta/t,\ dv/dx$	$T^{-1}$	s⁻¹

Each row follows from its defining relation; force, MLT⁻², is the starting point for most of them.

Dimensions of thermal and modern-physics constants11 rows

Constant or quantity	Equation it comes from	Dimensions
Boltzmann constant $k_B$	$E = \tfrac{3}{2}k_BT$	$ML^{2}T^{-2}K^{-1}$
Gas constant R	$PV = nRT$	$ML^{2}T^{-2}K^{-1}\text{mol}^{-1}$
Specific heat capacity	$Q = mc\Delta T$	$L^{2}T^{-2}K^{-1}$
Latent heat	$Q = mL$	$L^{2}T^{-2}$ No temperature in latent heat, so it differs from specific heat.
Thermal conductivity	$Q/t = kA\,\Delta T/l$	$MLT^{-3}K^{-1}$
Stefan's constant $\sigma$	$P/A = \sigma T^{4}$	$MT^{-3}K^{-4}$
Planck's constant h	$E = h\nu$	$ML^{2}T^{-1}$
Work function	$h\nu = \phi + K_{\max}$	$ML^{2}T^{-2}$
Stopping potential	$eV_0 = K_{\max}$	$ML^{2}T^{-3}A^{-1}$
Rydberg constant	$1/\lambda = R(1/n_1^{2} - 1/n_2^{2})$	$L^{-1}$
Decay constant	$N = N_0e^{-\lambda t}$	$T^{-1}$

Each constant has the dimensions that balance its equation.

Pairs of quantities with the same dimensions8 rows

Pair	Dimensions of the first	Dimensions of the second	Same?
Torque and energy	$ML^{2}T^{-2}$	$ML^{2}T^{-2}$	Yes
Planck's constant and angular momentum	$ML^{2}T^{-1}$	$ML^{2}T^{-1}$	Yes
Stress and energy density	$ML^{-1}T^{-2}$	$ML^{-1}T^{-2}$	Yes
Velocity gradient and decay constant	$T^{-1}$	$T^{-1}$	Yes
Pressure × time and viscosity	$ML^{-1}T^{-1}$	$ML^{-1}T^{-1}$	Yes
Specific heat and latent heat	$L^{2}T^{-2}K^{-1}$	$L^{2}T^{-2}$	No The only difference is K⁻¹.
Linear momentum and torque	$MLT^{-1}$	$ML^{2}T^{-2}$	No
Surface tension and impulse	$MT^{-2}$	$MLT^{-1}$	No

Compare every power of M, L, T and K; one mismatch makes the pair different.

Watch out for (5)

Surface energy and surface tension are not the same entry
→ Dimensions of mechanical quantities
Viscosity has one power of time, pressure has two
→ Dimensions of mechanical quantities
Stopping potential is a voltage, not an energy
→ Dimensions of thermal and modern-physics constants
Specific heat carries K⁻¹, latent heat does not
→ Dimensions of thermal and modern-physics constants
Same dimensions do not mean the same quantity
→ Pairs of quantities with the same dimensions

Dimensions of Electric and Magnetic Quantities

Formulas (1)

Electromagnetic combinations with simple dimensions · Combinations to recognise $\frac{1}{\mu_0\varepsilon_0} = c^{2} \qquad \sqrt{\frac{\mu_0}{\varepsilon_0}} = [R] \qquad \tfrac{1}{2}\varepsilon_0E^{2} = \frac{B^{2}}{2\mu_0} = [\text{energy/volume}] \qquad [RC] = [L/R] = [\sqrt{LC}] = T$

Reference tables (1)

Dimensions of electric and magnetic quantities14 rows

Quantity	Defining relation	Dimensions
Charge	$q = It$	$AT$
Potential difference, emf	$V = W/q$	$ML^{2}T^{-3}A^{-1}$
Resistance	$R = V/I$	$ML^{2}T^{-3}A^{-2}$
Resistivity	$\rho = RA/l$	$ML^{3}T^{-3}A^{-2}$
Capacitance	$C = q/V$	$M^{-1}L^{-2}T^{4}A^{2}$
Self or mutual inductance	$U = \tfrac{1}{2}LI^{2}$	$ML^{2}T^{-2}A^{-2}$ A⁻², not A⁻¹: energy divided by current squared.
Electric field	$E = F/q$	$MLT^{-3}A^{-1}$
Magnetic field (induction) B	$F = qvB$	$MT^{-2}A^{-1}$
Magnetic flux	$\Phi = BA$	$ML^{2}T^{-2}A^{-1}$
Permittivity $\varepsilon_0$	$F = q_1q_2/(4\pi\varepsilon_0r^{2})$	$M^{-1}L^{-3}T^{4}A^{2}$
Permeability $\mu_0$	$B = \mu_0I/(2\pi r)$	$MLT^{-2}A^{-2}$
Magnetic moment	$m = IA$	$L^{2}A$
Magnetising field H, magnetisation	$H = B/\mu_0$	$L^{-1}A$
Electric dipole moment	$p = qd$	$LTA$

Charge is AT; every other entry follows from its defining relation.

Watch out for (4)

Inductance has A⁻², flux has A⁻¹
→ Dimensions of electric and magnetic quantities
Charge is AT, not a base quantity
→ Dimensions of electric and magnetic quantities
ε₀E² is energy per volume, not energy
→ Electromagnetic combinations with simple dimensions
μ₀ is not dimensionless
→ Electromagnetic combinations with simple dimensions

Dimensional Homogeneity and Unknown Constants

Formulas (3)

Constants in terms that are added or subtracted · Principle of homogeneity $[X] = [Y] = [Q] \text{ in } Q = X + Y \qquad \left[\frac{a}{V^{2}}\right] = [P],\ [b] = [V]$
Arguments of sine, exponential and logarithm are dimensionless · Argument rule $Q = A\sin(Bx) \Rightarrow [B] = [x]^{-1},\ [A] = [Q]$
Checking whether an equation is dimensionally correct · Homogeneity test $[\text{left side}] = [\text{right side}] = [\text{every added term}]$

Watch out for (4)

The constant inside a bracket takes the variable's dimensions
→ Constants in terms that are added or subtracted
a/V² is a pressure, so a is not a pressure
→ Constants in terms that are added or subtracted
The prefactor carries the whole dimension
→ Arguments of sine, exponential and logarithm are dimensionless
Dimensionally correct does not mean correct
→ Checking whether an equation is dimensionally correct

Deriving Relations and Changing Base Quantities

Formulas (2)

Finding unknown powers by equating dimensions · Exponent method $Q = k\,A^{a}B^{b}C^{c} \Rightarrow \text{equate the powers of } M,\ L,\ T$
Writing dimensions in new base quantities · Change of units $n_2 = n_1\left(\frac{M_1}{M_2}\right)^{a}\left(\frac{L_1}{L_2}\right)^{b}\left(\frac{T_1}{T_2}\right)^{c}$

Watch out for (2)

A base quantity on one side only forces its power to zero
→ Finding unknown powers by equating dimensions
Solve for the old base first
→ Writing dimensions in new base quantities

Propagation of Errors

Formulas (3)

Relative error of a product of powers · Maximum relative error $Q = \frac{a^{p}b^{q}}{c^{r}} \Rightarrow \frac{\Delta Q}{Q} = p\frac{\Delta a}{a} + q\frac{\Delta b}{b} + r\frac{\Delta c}{c}$
Percentage error from values given with their absolute errors · Absolute error from relative error $\Delta Q = Q\left(p\frac{\Delta a}{a} + q\frac{\Delta b}{b} + r\frac{\Delta c}{c}\right)$
Errors in sums, parallel combinations and means · Sums and reciprocal sums $\Delta(A \pm B) = \Delta A + \Delta B \qquad \frac{\Delta R}{R^{2}} = \frac{\Delta R_1}{R_1^{2}} + \frac{\Delta R_2}{R_2^{2}}$

Watch out for (6)

An error in the denominator is added, not subtracted
→ Relative error of a product of powers
A stated negative error still adds
→ Relative error of a product of powers
An exponential adds βΔt, not a percentage
→ Relative error of a product of powers
A diameter's error counts twice in an area
→ Percentage error from values given with their absolute errors
Errors add in a difference too
→ Errors in sums, parallel combinations and means
Do not add percentage errors in a sum
→ Errors in sums, parallel combinations and means

Errors in Laboratory Experiments

Formulas (2)

Timing many oscillations: pendulum and spring · Pendulum error $\frac{\Delta g}{g} = \frac{\Delta l}{l} + 2\frac{\Delta t}{t} \qquad (t = \text{total time of } n \text{ oscillations})$
Least count as the error of each reading · Errors from least counts $\Delta x = \text{LC} \qquad \Delta(x_2 - x_1) = 2\,\text{LC} \qquad \frac{\Delta f}{f^{2}} = \frac{\Delta u}{u^{2}} + \frac{\Delta v}{v^{2}}$

Watch out for (4)

Divide the resolution by the total time, not by the period
→ Timing many oscillations: pendulum and spring
The period enters g squared
→ Timing many oscillations: pendulum and spring
A distance read from two marks carries twice the least count
→ Least count as the error of each reading
The diameter's error counts twice in Young's modulus
→ Least count as the error of each reading

Vernier Callipers, Screw Gauge and Zero Error

Formulas (3)

Least count of a vernier and a screw gauge · Least count $\text{vernier: } \text{LC} = 1\,\text{MSD} - 1\,\text{VSD} = \left(1 - \frac{m}{N}\right)\text{MSD} \qquad \text{screw gauge: } \text{LC} = \frac{\text{pitch}}{\text{circular divisions}}$
Zero error and the corrected reading · Corrected reading $\text{true reading} = \text{MSR} + n \times \text{LC} - (\text{zero error})$
Taking a reading and using it in a calculation · Reading $\text{reading} = \text{MSR} + n \times \text{LC}$

Watch out for (6)

The vernier constant is MSD minus VSD, not MSD times divisions
→ Least count of a vernier and a screw gauge
Find the pitch from distance per rotation
→ Least count of a vernier and a screw gauge
Subtract a positive zero error, never add it
→ Zero error and the corrected reading
Below the line is positive for a screw gauge
→ Zero error and the corrected reading
A vernier zero to the left: the textbook count and the keys' count differ
→ Zero error and the corrected reading
Match the units before adding
→ Taking a reading and using it in a calculation

PYQ weightage by concept

20 concepts · 190 PYQs — where the marks actually sit, so you know what to drill first

Units, Significant Figures and Order of Magnitude10 PYQs · 5%

Concept	PYQs	Share
Significant figures in sums, products and means	5	3%
Astronomical units of length, seconds of arc and order of magnitude	5	3%

Dimensions of Mechanical and Thermal Quantities35 PYQs · 18%

Concept	PYQs	Share
Dimensions of mechanical quantities	19	10%
Dimensions of thermal and modern-physics constants	10	5%
Pairs of quantities with the same dimensions	6	3%

Dimensions of Electric and Magnetic Quantities25 PYQs · 13%

Concept	PYQs	Share
Electromagnetic combinations with simple dimensions	14	7%
Dimensions of electric and magnetic quantities	11	6%

Dimensional Homogeneity and Unknown Constants26 PYQs · 14%

Concept	PYQs	Share
Constants in terms that are added or subtracted	14	7%
Arguments of sine, exponential and logarithm are dimensionless	8	4%
Checking whether an equation is dimensionally correct	4	2%

Deriving Relations and Changing Base Quantities14 PYQs · 7%

Concept	PYQs	Share
Writing dimensions in new base quantities	8	4%
Finding unknown powers by equating dimensions	6	3%

Propagation of Errors27 PYQs · 14%

Concept	PYQs	Share
Relative error of a product of powers	15	8%
Percentage error from values given with their absolute errors	7	4%
Errors in sums, parallel combinations and means	5	3%

Errors in Laboratory Experiments16 PYQs · 8%

Concept	PYQs	Share
Least count as the error of each reading	11	6%
Timing many oscillations: pendulum and spring	5	3%

Vernier Callipers, Screw Gauge and Zero Error37 PYQs · 19%

Concept	PYQs	Share
Zero error and the corrected reading	17	9%
Least count of a vernier and a screw gauge	16	8%
Taking a reading and using it in a calculation	4	2%

Test yourself on Units and Measurements

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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Units, Significant Figures and Order of Magnitude

Dimensions of Mechanical and Thermal Quantities

Dimensions of Electric and Magnetic Quantities

Dimensional Homogeneity and Unknown Constants

Deriving Relations and Changing Base Quantities

Propagation of Errors

Errors in Laboratory Experiments

Vernier Callipers, Screw Gauge and Zero Error

Formula & revision sheet

PYQ weightage by concept

Test yourself on Units and Measurements

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