Units and Measurements — JEE Mains Physics Formulas and Traps

Units, Significant Figures and Order of Magnitude

Significant figures in sums, products and means

Rounding rules

\text{sum or difference: fewest decimal places} \qquad \text{product or quotient: fewest significant figures}

Astronomical units of length, seconds of arc and order of magnitude

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.

Common traps

A sum is rounded by decimal places, not by significant figures

When numbers are added, the answer keeps the fewest decimal places, even if that leaves it with more significant figures than some term. 101.1 + 0.25 = 101.35, reported as 101.4 with four significant figures, although 0.25 has only two.

Leading zeros never count

In 0.0010010 the first three zeros only place the decimal point. The significant figures are 1, 0, 0, 1, 0 — five of them. The trailing zero counts because the number has a decimal point.

The parsec is bigger than the light year

A parsec is about 3.26 light years. The order is AU < light year < parsec. A statement that puts the parsec between the AU and the light year is false.

Convert the angle to radians first

In d = θD the angle must be in radians. Seconds of arc are multiplied by 4.85 × 10⁻⁶; leaving the angle in seconds or degrees gives a size wrong by a factor of thousands.

Dimensions of Mechanical and Thermal Quantities

Learn this subtopic in the notes

Dimensions of mechanical quantities

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 constants

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 dimensions

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.

Common traps

Surface energy and surface tension are not the same entry

Surface tension is force per length, MT⁻². A match list that names 'surface energy' means an energy, ML²T⁻². Read the exact words before matching.

Viscosity has one power of time, pressure has two

Viscosity is ML⁻¹T⁻¹ and pressure is ML⁻¹T⁻². The two differ only in the power of T, and match lists put them side by side to catch a slip.

Stopping potential is a voltage, not an energy

The stopping potential V₀ satisfies eV₀ = K_max, so V₀ itself is energy per charge, ML²T⁻³A⁻¹. Only eV₀ is an energy. The work function, by contrast, is an energy, ML²T⁻².

Specific heat carries K⁻¹, latent heat does not

Specific heat is energy per mass per kelvin, L²T⁻²K⁻¹. Latent heat is energy per mass, L²T⁻². They are a standard 'different dimensions' pair.

Same dimensions do not mean the same quantity

Torque and work are both ML²T⁻², but torque is not an energy: it is a turning effect, and it is not measured in joules. A statement that they 'have the same dimensions' is true; one that they 'are the same quantity' is not.

Dimensions of Electric and Magnetic Quantities

Learn this subtopic in the notes

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

Dimensions of electric and magnetic quantities

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.

Common traps

Inductance has A⁻², flux has A⁻¹

Magnetic flux is ML²T⁻²A⁻¹ and inductance is flux per current, ML²T⁻²A⁻². The two entries differ only in the power of A, and a match list places them side by side.

Charge is AT, not a base quantity

The base quantity is current. When a question uses charge Q as a base instead, rewrite A as QT⁻¹: capacitance then becomes M⁻¹L⁻²T²Q².

ε₀E² is energy per volume, not energy

½ε₀E² is the energy stored per unit volume, so its dimensions are ML⁻¹T⁻², the same as pressure. Writing ML²T⁻² misses the division by volume.

μ₀ is not dimensionless

μ₀ has dimensions MLT⁻²A⁻². The dimensionless ones are ratios: relative permeability μ/μ₀, dielectric constant, power factor cos φ and the quality factor.

Dimensional Homogeneity and Unknown Constants

Learn this subtopic in the notes

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}]

Common traps

The constant inside a bracket takes the variable's dimensions

In V − b, the constant b is a volume. It is not found from RT; it is fixed by the quantity it is subtracted from. The same holds for c in t + c, which is a time.

a/V² is a pressure, so a is not a pressure

The term a/V² has pressure's dimensions, so a itself is pressure × volume², ML⁵T⁻². Giving a the dimensions of pressure drops the V².

The prefactor carries the whole dimension

In Q = A sin(Bx), the sine is a pure number, so A has exactly the dimensions of Q. Giving A some of B's dimensions, as if the sine passed them on, is the usual slip.

Dimensionally correct does not mean correct

T = π√(l/g) passes the dimension test but is wrong by a factor 2. A dimension check can only rule an equation out.

Deriving Relations and Changing Base Quantities

Learn this subtopic in the notes

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}

Common traps

A base quantity on one side only forces its power to zero

If only one quantity on the right carries mass, and the left side has no mass, that quantity's power must be zero. Dropping the M equation because 'nothing is heavy' loses this result.

Solve for the old base first

Writing the new quantities in M, L and T is the easy half. The answer needs the reverse: M, L and T in the new quantities. Substituting before inverting gives the reciprocal powers.

Propagation of Errors

Learn this subtopic in the notes

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}}

Common traps

An error in the denominator is added, not subtracted

Dividing by c does not cancel c's error. The maximum relative error of a/c is Δa/a + Δc/c. Subtracting gives a smaller number, which is usually one of the options.

A stated negative error still adds

For the maximum error, a '1% negative error' counts as 1%. Signs matter only for the most likely error, which JEE questions do not ask for.

An exponential adds βΔt, not a percentage

In E = α³e^(−βt), the exponential contributes β·Δt to ΔE/E. With β = 0.2 s⁻¹ and Δt = 0.5 s, that is 0.1, or 10%, whatever the value of t.

A diameter's error counts twice in an area

Area is πd²/4, so its relative error is 2Δd/d. Using Δd/d once, as if the area grew in proportion to d, halves that contribution.

Errors add in a difference too

For Z = A − B, ΔZ = ΔA + ΔB. Subtracting the errors assumes they cancel, but the maximum error comes when they do not.

Do not add percentage errors in a sum

For k = k₁ + k₂, add the ABSOLUTE errors and then divide by k. Adding the two percentage errors overstates the answer: (10 ± 0.3) + (30 ± 0.3) is 40 ± 0.6, which is 1.5%, not 3% + 1% = 4%.

Errors in Laboratory Experiments

Learn this subtopic in the notes

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}}

Common traps

Divide the resolution by the total time, not by the period

If 40 oscillations take 80 s on a 1 s watch, ΔT/T is 1/80, not 1/2. The watch was read once over the whole run, so its error is shared by every oscillation.

The period enters g squared

In g = 4π²l/T² the period has power 2, so its relative error counts twice. Forgetting the 2 gives the 'length error + period error' distractor.

A distance read from two marks carries twice the least count

On an optical bench the object distance is (lens mark − object mark). Each mark is uncertain by one least count, so the distance is uncertain by two. Using one least count halves the error.

The diameter's error counts twice in Young's modulus

Y = 4FL/(πd²ΔL) has d squared, so 2Δd/d enters. A fine screw gauge can still give the largest term because the diameter itself is so small.

Vernier Callipers, Screw Gauge and Zero Error

Learn this subtopic in the notes

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}

Common traps

The vernier constant is MSD minus VSD, not MSD times divisions

The least count is the DIFFERENCE between one main-scale and one vernier division. It is not one MSD multiplied by the number of vernier divisions, a statement that appears as a false option.

Find the pitch from distance per rotation

If five rotations move the spindle 2.5 mm, the pitch is 0.5 mm, not 2.5 mm. Divide by the number of rotations before dividing by the circular divisions.

Subtract a positive zero error, never add it

A positive zero error means the instrument already reads more than zero with nothing between the jaws, so every reading is too large. Subtract it. Adding it is the most common wrong option.

Below the line is positive for a screw gauge

When the studs touch and the circular-scale zero sits below the reference line, the screw has gone past zero: the error is positive. Above the line, it has not reached zero: the error is negative.

A vernier zero to the left: the textbook count and the keys' count differ

When the vernier zero is left of the main zero, the textbook zero error is −(N − n) × LC for an N-division vernier whose n-th division coincides: count back from the vernier's last mark. Two JEE keys instead took it as −n × LC, the right-side count with a minus sign. Work the textbook count first; if no option matches it and −n × LC is offered, that is the count the paper used.

Match the units before adding

A main-scale reading in cm and a least count in mm cannot be added directly. 2.1 cm + 5 × 0.1 mm is 2.15 cm, not 2.6 cm.

Units and Measurements formulas

Units, Significant Figures and Order of Magnitude

Significant figures in sums, products and means

Astronomical units of length, seconds of arc and order of magnitude

Common traps

Dimensions of Mechanical and Thermal Quantities

Dimensions of mechanical quantities

Dimensions of thermal and modern-physics constants

Pairs of quantities with the same dimensions

Common traps

Dimensions of Electric and Magnetic Quantities

Electromagnetic combinations with simple dimensions

Dimensions of electric and magnetic quantities

Common traps

Dimensional Homogeneity and Unknown Constants

Constants in terms that are added or subtracted

Arguments of sine, exponential and logarithm are dimensionless

Checking whether an equation is dimensionally correct

Common traps

Deriving Relations and Changing Base Quantities

Finding unknown powers by equating dimensions

Writing dimensions in new base quantities

Common traps

Propagation of Errors

Relative error of a product of powers

Percentage error from values given with their absolute errors

Errors in sums, parallel combinations and means

Common traps

Errors in Laboratory Experiments

Timing many oscillations: pendulum and spring

Least count as the error of each reading

Common traps

Vernier Callipers, Screw Gauge and Zero Error

Least count of a vernier and a screw gauge

Zero error and the corrected reading

Taking a reading and using it in a calculation

Common traps

More JEE Mains Physics formula sheets