JEE Advanced PYQ – Dimensions of Physical Quantities (Units & Measurements)

Q 1 :

A physical quantity $\vec{S}$ is defined as $\vec{S} = \frac{(\vec{E} \times \vec{B})}{μ_{0}},$ where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $μ_{0}$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantities? [2021]

$\frac{Energy}{Charge \times Current}$
$\frac{Force}{Length \times Time}$
$\frac{Energy}{Volume}$
$\frac{Power}{Area}$

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4

Select one or more options

(2, 4)

Given physical quantity,

$\vec{S} = [\vec{E} \times \vec{B}] \frac{1}{μ_{0}}$

Here $\vec{S}$ is known as poynting vector and it represents flow of energy per unit area per unit time

$\vec{S} = \frac{Energy}{Area \times Time} = \frac{Power}{Area} = \frac{Force}{Length \times Time}$

Q 2 :

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows:

$[position] = [X^{α}], [speed] = [X^{β}], [acceleration] = [X^{p}], [linear momentum] = [X^{q}], [force] = [X^{r}] .$ Then [2020]

$α + p = 2 β$
$p + q - r = β$
$p - q + r = α$
$p + q + r = β$

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Select one or more options

(1, 2)

$Given position, L = [X^{α}]$

$Speed, L T^{- 1} = [X^{β}]$

$Acceleration, L T^{- 2} = [X^{p}]$

$Linear momentum, M L T^{- 1} = [X^{q}]$

$Force, M L T^{- 2} = [X^{r}]$

$Position \div Speed = time, T = \frac{[X^{α}]}{[X^{β}]} = X^{α - β}$

$Acceleration = \frac{Speed}{Time} = \frac{X^{β}}{X^{α - β}} = X^{p}$

$X^{α - β + p} = X^{β} ∴$ $α + p = 2 β$

$Hence option (1) is correct.$

$Force = \frac{linear momentum}{time}$

$[X^{r}] = \frac{[X^{q}]}{[X^{α - β}]} \Rightarrow r = q + β - α$ $\Rightarrow r = q + β - (2 β - p)$

$\Rightarrow r = q - β + p \Rightarrow p + q - r = β$

$Hence option (2) is correct.$

Q 3 :

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimensions of L, which of the following statement(s) is/are correct? [2019]

The dimension of force is $L^{- 2}$
The dimension of linear momentum is $L^{- 1}$
The dimension of energy is $L^{- 2}$
The dimension of power is $L^{- 5}$

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2

3

4

Select one or more options

(1, 2, 3)

$According to question, dimensions of angular momentum [m v r] = M^{0} L^{0} T^{0} and of mass [m] = M^{0} L^{0} T^{0}$

$M L^{2} T^{- 1} = M L^{0} T^{0} \Rightarrow T = L^{2}$

Momentum $p = m v = \frac{m v r}{r} = \frac{M^{0} L^{0} T^{0}}{L} = \frac{L}{L^{2}} = L^{- 1}$

Energy $E = \frac{1}{2} m v^{2} = \frac{1}{2} \frac{{(m v)}^{2}}{m} = L^{- 2}$

Power $P = \frac{E}{t} = \frac{L^{- 2}}{T} = \frac{L^{- 2}}{L^{2}} = L^{- 4}$

Force $F = \frac{E}{x} = \frac{L^{- 2}}{L} = L^{- 3}$

Q 4 :

A length--scale $(l)$ depends on the permittivity $(ε)$ of a dielectric material, Boltzmann constant $(k_{B})$ , the absolute temperature $(T)$ , the number per unit volume $(n)$ of certain charged particles, and the charge $(q)$ carried by each of the particles. Which of the following expression(s) for $l$ is(are) dimensionally correct? [2016]

$l = \sqrt{(\frac{n q^{2}}{ε k_{B} T})}$
$l = \sqrt{(\frac{ε k_{B} T}{n q^{2}})}$
$l = \sqrt{(\frac{q^{2}}{ε n^{2 / 3} k_{B} T})}$
$l = \sqrt{(\frac{q^{2}}{ε n^{1 / 3} k_{B} T})}$

1

2

3

4

Select one or more options

(2, 4)

$Dimensions of [n] = [L^{- 3}], [q] = [A T]$

$[\frac{q^{2}}{ε}] = [F r^{2}] = [U \times r] = [M L^{3} T^{- 2}]$

$[k_{B} T] = [U] = [M L^{2} T^{- 2}]$

$Dimension of l = [L]$

$(a) R.H.S. = \sqrt{(\frac{q^{2}}{ε}) \times \frac{1}{(k_{B} T)} \times n}$
$= \sqrt{[U \times r] \times \frac{1}{[U]} \times n} = \sqrt{[n] [r]} = \sqrt{[L^{- 3}] [L]} = [L^{- 1}]$

$(b) R.H.S. = \sqrt{\frac{ε (k_{B} T)}{n q^{2}}} = \sqrt{\frac{(k_{B} T)}{n (q^{2} / ε)}}$

$= \sqrt{\frac{[U]}{[n] [U \times r]}} = \sqrt{\frac{1}{[n] [r]}} = \sqrt{\frac{1}{[L^{- 3}] [L]}} = [L]$

$(c) R.H.S. = \sqrt{(\frac{q^{2}}{ε}) \times \frac{1}{(k_{B} T)} \times \frac{1}{n^{2 / 3}}}$

$= \sqrt{[U \times r] \times \frac{1}{[U]} \times \frac{1}{[L^{- 2}]}} = [L^{3 / 2}]$

$(d) R.H.S. = \sqrt{(\frac{q^{2}}{ε}) \times \frac{1}{(k_{B} T)} \times \frac{1}{n^{1 / 3}}}$

$= \sqrt{[U \times r] \times \frac{1}{[U]} \times \frac{1}{[n^{1 / 3}]}} = \sqrt{[r] \times \frac{1}{[n^{1 / 3}]}}$

$= \sqrt{[L] \frac{1}{[L^{- 1}]}} = [L]$

Q 5 :

A temperature difference can generate e.m.f. in some materials. Let $S$ be the e.m.f. produced per unit temperature difference between the ends of a wire, $σ$ the electrical conductivity and $k$ the thermal conductivity of the material of the wire. Taking $M, L, T, I$ and $K$ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $Z = \frac{S^{2} σ}{k}$ is: [2025]

$[M^{0} L^{0} T^{0} I^{0} K^{0}]$
$[M^{0} L^{0} T^{0} I^{0} K^{- 1}]$
$[M^{1} L^{2} T^{- 2} I^{- 1} K^{- 1}]$
$[M^{1} L^{2} T^{- 4} I^{- 1} K^{- 1}]$

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(2)

$Emf per unit temperature difference [S] = [M L^{2} T^{- 3} I^{- 1} K^{- 1}]$

$Electrical conductivity [σ] = [M^{- 1} L^{3} T^{3} I^{2}]$

$Thermal conductivity [K] = [M L T^{- 3} K^{- 1}]$

$[Z]$ $= \frac{S^{2} σ}{K} = \frac{{[M L^{2} T^{- 3} I^{- 1} K^{- 1}]}^{2} [M^{- 1} L^{- 3} T^{3} I^{2}]}{[M L T^{- 3} K^{- 1}]}$

$= \frac{[M^{2} L^{4} T^{- 6} I^{- 2} K^{- 2}] [M^{- 1} L^{- 3} T^{3} I^{2}]}{[M L T^{- 3} K^{- 1}]}$

$= \frac{[M L T^{- 3} K^{- 2}]}{[M L T^{- 3} K^{- 1}]}$

$∴$ $[Z] = [M^{0} L^{0} T^{0} I^{0} K^{- 1}]$

Q 6 :

A dimensionless quantity is constructed in terms of electronic charge $e$ , permittivity of free space $ε_{0}$ , Planck’s constant $h$ , and speed of light $c$ . If the dimensionless quantity is written as $e^{α} ε_{0}^{β} h^{γ} c^{δ}$ and $n$ is a non-zero integer, then $(α, β, γ, δ)$ is given by [2024]

$(2 n, - n, - n, - n)$
$(n, - n, - 2 n, - n)$
$(n, - n, - n, - 2 n)$
$(2 n, - n, - 2 n, - 2 n)$

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(1)

For dimensionless quantity

$e^{α} ε_{0}^{β} h^{γ} c^{δ} = M^{0} L^{0} T^{0} A^{0}$

Substituting dimensions of the physical quantities in the above relation and equating the dimensions of LHS and RHS

$\Rightarrow {(A T)}^{α} {(M^{- 1} L^{- 3} T^{4} A^{2})}^{β} {(M L^{2} T^{- 1})}^{γ} {(L T^{- 1})}^{δ} = A^{0} M^{0} L^{0} T^{0}$

$∴$ $α + 2 β = 0, α + 4 β - γ - δ = 0, - β + γ = 0, - 3 β + 2 γ + δ = 0$

$∴$ $α = - 2 β, β = γ, γ = δ$

$∴$ $α = 2 n, β = - n, γ = - n, δ = - n$

Q 7 :

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$ , Planck's constant $h$ and the speed of light $c$ , as $Y = c^{α} h^{β} G^{γ} .$ Which of the following is the correct option? [2023]

$α = 7, β = - 1, γ = - 2$
$α = - 7, β = - 1, γ = - 2$
$α = 7, β = - 1, γ = 2$
$α = - 7, β = 1, γ = - 2$

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(1)

$Given, Y = c^{α} h^{β} G^{γ}$

$M L^{- 1} T^{- 2} = {(L T^{- 1})}^{α} {(M L^{2} T^{- 1})}^{β} {(M^{- 1} L^{3} T^{- 2})}^{γ}$

Equating powers

$1 = β - γ ...(i)$

$- 1 = α + 2 β + 3 γ ...(ii)$

$- 2 = - α - β - 2 γ ...(iii)$

$- 3 = β + γ ...(iv)$

$(By adding eq. (ii) & (iii))$

$Now by adding eq. (i) & (iv) we get,$

$- 2 = 2 β$

$∴$ $β = - 1, γ = - 2$

$- 1 = α - 2 - 6$ $∴$ $α = 7$

Q 8 :

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is 0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divisions. The measured readings are listed below. [2022]

Measurement condition Main scale reading Circular scale reading

Two arms of gauge touching each other without wire 0 division 4 divisions

Attempt-1: With wire 4 divisions 20 divisions

Attempt-2: With wire 4 divisions 16 divisions

What are the diameter and cross-sectional area of the wire measured using the screw gauge?

$2.22 \pm 0.02 mm, π (1.23 \pm 0.02) {mm}^{2}$
$2.22 \pm 0.01 mm, π (1.23 \pm 0.01) {mm}^{2}$
$2.14 \pm 0.02 mm, π (1.14 \pm 0.02) {mm}^{2}$
$2.14 \pm 0.01 mm, π (1.14 \pm 0.01) {mm}^{2}$

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(3)

$Least count = \frac{Pitch}{no. of division} = \frac{2 \times 0.5}{100} = 0.01 mm$

$Zero error = 4 \times L.C. = 0.04 mm$

$Attempt 1: d_{1} = MSR + CSR - zero error$

$d_{1} = 4 \times 0.5 + 20 \times 0.01 - 0.04 = 2.16 mm$

$Attempt 2: d_{2} = 4 \times 0.5 + 16 \times 0.01 - 0.04 = 2.12 mm$

$So, d_{mean} = \frac{2.16 + 2.12}{2} = 2.14 mm$

$Now, Δ d_{1} = 0.02 mm and Δ d_{2} = - 0.02 mm$

$Δ \bar{d} = \frac{| 0.02 | + | - 0.02 |}{2} = 0.01 mm$

$So, d = 2.14 \pm 0.02 mm$

$Area (A) = π R^{2} = π {(\frac{d}{2})}^{2} = π {(1.07)}^{2} = 1.14 π {mm}^{2}$

$\frac{Δ A}{A} = \frac{2 Δ R}{R} = \frac{2 Δ d}{d} = 2 \times \frac{0.02}{2.14}$

$\Rightarrow Δ A = \frac{2 \times 0.02}{2.14} \times 1.14 π = 0.02 π$

$So, area = π (1.14 \pm 0.02) {mm}^{2}$

Q 9 :

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$ , electron mass $m_{e}$ , Planck's constant $h$ , and Coulomb's constant $k = \frac{1}{4 π ε_{0}},$ where $ε_{0}$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B] = {[e]}^{α} {[m_{e}]}^{β} {[h]}^{γ} {[k]}^{δ} .$ The value of $α + β + γ + δ$ is _____ . [2022]

(4)

$[B] = {[e]}^{α} {[m_{e}]}^{β} {[h]}^{γ} {[k]}^{δ}$

$\Rightarrow M T^{- 2} A^{- 1} = A^{α} T^{α} M^{β} M^{γ} L^{2 γ} T^{- γ} M^{δ} L^{3 δ} A^{- 2 δ} T^{- 4 δ}$

$\Rightarrow β + γ + δ = 1, - 2 = α - γ - 4 δ, - 1 = α - 2 δ$

$2 γ + 3 δ = 0$

$\Rightarrow α = 3, β = 2, γ = - 3, δ = 2$

$\Rightarrow α + β + γ + δ = 4$

Q 10 :

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways-engineer uses dimensions and assumes that the distance depends on the mass density $ρ$ of the fog, intensity (power/area) $S$ of the light from the signal and its frequency $f$ . The engineer finds that $d$ is proportional to $S^{1 / n}$ . The value of $n$ is ______. [2014]

(3)

$Let d \propto ρ^{x} s^{y} f^{z} = K ρ^{x} s^{y} f^{z} where K is a dimensionless constant.$

$M^{0} L T^{0} = M^{x} L^{- 3 x} M^{y} T^{- 3 y} T^{- z}$

Equating dimensions both sides

$M^{0} L T^{0} = M^{x + y} L^{- 3 x} T^{- 3 y - z}$

$∴$ $x + y = 0, - 3 x = 1$

$∴$ $x = - \frac{1}{3}$ and $y = \frac{1}{3}$

$∴$ $n = 3$

Q 11 :

Let $[ε_{0}]$ denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then: [2012]

$ε_{0} = [M^{- 1} L^{- 3} T^{2} A]$
$ε_{0} = [M^{- 1} L^{- 3} T^{4} A^{2}]$
$ε_{0} = [M^{1} L^{2} T^{1} A^{2}]$
$ε_{0} = [M^{1} L^{2} T^{1} A]$

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(2)

$As we know, F = \frac{1}{4 π ε_{0}} \frac{q_{1} q_{2}}{R^{2}} \Rightarrow ε_{0} = \frac{q_{1} q_{2}}{4 π F R^{2}}$

$Hence, ε_{0} = \frac{C^{2}}{N \cdot m^{2}} = \frac{{[A T]}^{2}}{[M L T^{- 2}] [L^{2}]} = [M^{- 1} L^{- 3} T^{4} A^{2}]$

Q 12 :

Which of the following set have different dimensions? [2005]

Pressure, Young’s modulus, Stress
EMF, Potential difference, Electric potential
Heat, Work done, Energy
Dipole moment, Electric flux, Electric field

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(4)

$Electric flux ϕ_{E} = \vec{E} \cdot \vec{S}$

$∴$ $Dimensionally ϕ_{E} \neq E (electric field)$

$Dipole moment, p = charge (q) \times distance (d)$

Q 13 :

Pressure depends on distance as, $P = \frac{α}{β} \exp (- \frac{α z}{k θ}),$ where $α, β$ are constants, $z$ is distance, $k$ is Boltzmann's constant and $θ$ is temperature. The dimension of $β$ are [2004]

$M^{0} L^{0} T^{0}$
$M^{- 1} L^{- 1} T^{- 1}$
$M^{0} L^{2} T^{0}$
$M^{- 1} L^{1} T^{2}$

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(3)

The power of an exponent is a number or constant.

Therefore, dimensionally $\frac{α z}{k θ} = [M^{0} L^{0} T^{0}]$

$∴$ $α = \frac{k θ}{z}$ $∴$ $α = \frac{[M L^{2} T^{- 2} θ^{- 1}] [θ]}{[L]} = [M L T^{- 2}]$

$As k = \frac{joule}{kelvin} = [M L^{2} T^{- 2} θ^{- 1}] and z = [L]$

$And, dimensionally P = \frac{α}{β} \Rightarrow β = \frac{α}{P}$

$∴$ $[β] = \frac{M L T^{- 2}}{M L^{- 1} T^{- 2}} = [M^{0} L^{2} T^{0}]$ $[∵ P = \frac{F}{A} = [M L^{- 1} T^{- 2}]]$

Q 14 :

A quantity $X$ is given by $ε_{0} L \frac{Δ V}{Δ t}$ where $ε_{0}$ is the permittivity of free space, $L$ is a length, $Δ V$ is a potential difference and $Δ t$ is a time interval. The dimensional formula for $X$ is the same as that of [2001]

resistance
charge
voltage
current

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(4)

$Capacitance, C = \frac{Δ q}{Δ V} = \frac{ε_{0} A}{L} \Rightarrow ε_{0} = \frac{(Δ q) L}{A (Δ V)}$

$∴$ $X = ε_{0} L \frac{Δ V}{Δ t} = \frac{(Δ q) L}{A (Δ V)} L \frac{Δ V}{Δ t}$

$= \frac{(Δ q)}{Δ V} \frac{L^{2}}{L^{2}} \frac{Δ V}{Δ t} = \frac{Δ q}{Δ t} = I (current)$

Q 15 :

The dimension of $(\frac{1}{2}) ε_{0} E^{2}$ ( $ε_{0}$ : permittivity of free space, E : electric field) [2000]

$M L T^{- 1}$
$M L^{2} T^{- 2}$
$M L^{- 1} T^{- 2}$
$M L^{2} T^{- 1}$

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(3)

Here $(\frac{1}{2}) ε_{0} E^{2}$ represents energy density, i.e., energy per unit volume.

$∴$ $[\frac{1}{2} ε_{0} E^{2}] = \frac{[Energy]}{[Volume]} = \frac{M L^{2} T^{- 2}}{L^{3}} = [M L^{- 1} T^{- 2}]$

Q 16 :

Match List I with List II and select the correct answer using the codes given below the lists: [2013]

List I List II

P. Boltzmann constant 1. $[{ML}^{2} T^{- 1}]$

Q. Coefficient of viscosity 2. $[{ML}^{- 1} T^{- 1}]$

R. Planck constant 3. $[{MLT}^{- 3} K^{- 1}]$

S. Thermal conductivity 4. $[{ML}^{2} T^{- 2} K^{- 1}]$

$P \to 3, Q \to 1, R \to 2, S \to 4$
$P \to 3, Q \to 2, R \to 1, S \to 4$
$P \to 4, Q \to 2, R \to 1, S \to 3$
$P \to 4, Q \to 1, R \to 2, S \to 3$

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(3)

$Boltzmann constant [K_{B}] = \frac{u}{θ}$ $(∵ u = \frac{1}{2} K_{B} T)$

$∴$ $[K_{B}] = M L^{2} T^{- 2} K^{- 1}$

$Coefficient of viscosity [η] = \frac{F}{6 π r v} = \frac{M L T^{- 2}}{L \times L T^{- 1}} = M L^{- 1} T^{- 1}$

$Planck constant, h = \frac{E}{ν} = \frac{M L^{2} T^{- 2}}{T^{- 1}} = M L^{2} T^{- 1}$

$Thermal conductivity K_{conductivity} = \frac{H ℓ}{t A Δ T} = \frac{M L^{2} T^{- 2} \times L}{T \times L^{2} \times K} = M L T^{- 3} K^{- 1}$

Q 17 :

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $[ε_{0}]$ and $[μ_{0}]$ stand for dimensions of the permittivity and permeability of free space respectively. $[L]$ and $[T]$ are dimensions of length and time respectively. All the quantities are given in SI units. [2018]

Q. The relation between $[E]$ and $[B]$ is

$[E] = [B] [L] [T]$
$[E] = [B] {[L]}^{- 1} [T]$
$[E] = [B] [L] {[T]}^{- 1}$
$[E] = [B] {[L]}^{- 1} {[T]}^{- 1}$

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(3)

$Using, C = \frac{E}{B} where C = speed of light$

$∴$ $E = C B = L T^{- 1} B$

Q 18 :

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $[ε_{0}]$ and $[μ_{0}]$ stand for dimensions of the permittivity and permeability of free space respectively. $[L]$ and $[T]$ are dimensions of length and time respectively. All the quantities are given in SI units. [2018]

Q. The relation between $[ε_{0}]$ and $[μ_{0}]$ is

$[μ_{0}] = [ε_{0}] {[L]}^{2} {[T]}^{- 2}$
$[μ_{0}] = [ε_{0}] {[L]}^{- 2} {[T]}^{2}$
$[μ_{0}] = {[ε_{0}]}^{- 1} {[L]}^{2} {[T]}^{- 2}$
$[μ_{0}] = {[ε_{0}]}^{- 1} {[L]}^{- 2} {[T]}^{2}$

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(4)

We know that

$C = \frac{1}{\sqrt{μ_{0} ε_{0}}}$ $∴$ $C^{2} = \frac{1}{μ_{0} ε_{0}}$

$∴$ $μ_{0} = ε_{0}^{- 1} L^{- 2} T^{2} [∵ C = L T^{- 1}]$

Topic Question Set

Q 3 :

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimensions of L, which of the following statement(s) is/are correct? [2019]

Q 7 :

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$ , Planck's constant $h$ and the speed of light $c$ , as $Y = c^{α} h^{β} G^{γ} .$ Which of the following is the correct option? [2023]

Q 11 :

Let $[ε_{0}]$ denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then: [2012]

Q 12 :

Which of the following set have different dimensions? [2005]

Q 13 :

Pressure depends on distance as, $P = \frac{α}{β} \exp (- \frac{α z}{k θ}),$ where $α, β$ are constants, $z$ is distance, $k$ is Boltzmann's constant and $θ$ is temperature. The dimension of $β$ are [2004]

Q 14 :

A quantity $X$ is given by $ε_{0} L \frac{Δ V}{Δ t}$ where $ε_{0}$ is the permittivity of free space, $L$ is a length, $Δ V$ is a potential difference and $Δ t$ is a time interval. The dimensional formula for $X$ is the same as that of [2001]

Q 15 :

The dimension of $(\frac{1}{2}) ε_{0} E^{2}$ ( $ε_{0}$ : permittivity of free space, E : electric field) [2000]

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Measurement condition	Main scale reading	Circular scale reading
Two arms of gauge touching each other without wire	0 division	4 divisions
Attempt-1: With wire	4 divisions	20 divisions
Attempt-2: With wire	4 divisions	16 divisions

	List I		List II
P.	Boltzmann constant	1.	$[{ML}^{2} T^{- 1}]$
Q.	Coefficient of viscosity	2.	$[{ML}^{- 1} T^{- 1}]$
R.	Planck constant	3.	$[{MLT}^{- 3} K^{- 1}]$
S.	Thermal conductivity	4.	$[{ML}^{2} T^{- 2} K^{- 1}]$

Topic Question Set

Q 1 : A physical quantity S→ is defined as S→=(E→×B→)μ0, where E→ is electric field, B→ is magnetic field and μ0 is the permeability of free space. The dimensions of S→ are the same as the dimensions of which of the following quantities? [2021]

Q 3 : Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimensions of L, which of the following statement(s) is/are correct? [2019]

Q 6 : A dimensionless quantity is constructed in terms of electronic charge e, permittivity of free space ε0, Planck’s constant h, and speed of light c. If the dimensionless quantity is written as eαε0βhγcδ and n is a non-zero integer, then (α,β,γ,δ) is given by [2024]

Q 7 : Young's modulus of elasticity Y is expressed in terms of three derived quantities, namely, the gravitational constant G, Planck's constant h and the speed of light c, as Y=cαhβGγ. Which of the following is the correct option? [2023]

Q 11 : Let [ε0] denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then: [2012]

Q 12 : Which of the following set have different dimensions? [2005]

Q 13 : Pressure depends on distance as, P=αβexp(-αzkθ), where α,β are constants, z is distance, k is Boltzmann's constant and θ is temperature. The dimension of β are [2004]

Q 14 : A quantity X is given by ε0LΔVΔt where ε0 is the permittivity of free space, L is a length, ΔV is a potential difference and Δt is a time interval. The dimensional formula for X is the same as that of [2001]

Q 15 : The dimension of (12)ε0E2 (ε0 : permittivity of free space, E : electric field) [2000]

Q 16 : Match List I with List II and select the correct answer using the codes given below the lists: [2013] List I List II P. Boltzmann constant 1. [ML2T-1] Q. Coefficient of viscosity 2. [ML-1T-1] R. Planck constant 3. [MLT-3K-1] S. Thermal conductivity 4. [ML2T-2K-1]

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Q 3 :

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimensions of L, which of the following statement(s) is/are correct? [2019]

Q 7 :

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$ , Planck's constant $h$ and the speed of light $c$ , as $Y = c^{α} h^{β} G^{γ} .$ Which of the following is the correct option? [2023]

Q 11 :

Let $[ε_{0}]$ denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then: [2012]

Q 12 :

Which of the following set have different dimensions? [2005]

Q 13 :

Pressure depends on distance as, $P = \frac{α}{β} \exp (- \frac{α z}{k θ}),$ where $α, β$ are constants, $z$ is distance, $k$ is Boltzmann's constant and $θ$ is temperature. The dimension of $β$ are [2004]

Q 14 :

A quantity $X$ is given by $ε_{0} L \frac{Δ V}{Δ t}$ where $ε_{0}$ is the permittivity of free space, $L$ is a length, $Δ V$ is a potential difference and $Δ t$ is a time interval. The dimensional formula for $X$ is the same as that of [2001]

Q 15 :

The dimension of $(\frac{1}{2}) ε_{0} E^{2}$ ( $ε_{0}$ : permittivity of free space, E : electric field) [2000]