Dimensional Analysis

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Let $[ε_{0}]$ denote the dimensional formula of the permittivity of the vacuum and $[μ_{0}]$ that of the permeability of the vacuum. If $M = mass$ , $L = length$ , $T = time$ and $I = electric current$ .

[ε_{0}] = [M^{- 1} L^{- 3} T^{2} I]

(B)

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

(C)

[μ_{0}] = [MLT^{- 2} I^{- 2}]

(D)

[μ_{0}] = [ML^{2} T^{- 1} I]

JEE Advanced 2015

LEVELJEE Advanced

Planck's constant $h$ , speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$ . Then, the correct options is/are

* Multiple Correct Options

(A)

M \propto c

(B)

M \propto G

(C)

L \propto h

(D)

L \propto G

JEE Advanced 2018

LEVELJEE Main

Comprehension Passage

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.

Question 1:

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

(A)

[E] = [B] [L] [T]

(B)

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

(C)

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

(D)

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

JEE Advanced 2018

LEVELJEE Main

Comprehension Passage

[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.

Question 1:

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

(A)

[μ_{0}] = [ε_{0}] [L]^{2} [T]^{- 2}

(B)

[μ_{0}] = [ε_{0}] [L]^{- 2} [T]^{2}

(C)

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

(D)

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

JEE Advanced 2015

LEVELJEE Advanced

In terms of potential difference $V$ , electric current $I$ , permittivity $ε_{0}$ , permeability $μ_{0}$ and speed of light $c$ , the dimensionally correct equations is/are

* Multiple Correct Options

(A)

μ_{0} I^{2} = ε_{0} V^{2}

(B)

ε_{0} I = μ_{0} V

(C)

I = ε_{0} c V

(D)

μ_{0} c I = ε_{0} V

JEE Advanced 1984

LEVELJEE Main

$L$ , $C$ and $R$ represent the physical quantities inductance, capacitance and resistance, respectively. The combinations which have the dimensions of frequency are

* Multiple Correct Options

(A)

\frac{1}{R C}

(B)

\frac{R}{L}

(C)

\frac{1}{L C}

(D)

\frac{C}{L}

JEE Main 2004

LEVELJEE Main

In the relation, $p = \frac{α}{β} e^{- \frac{α Z}{k θ}}$ $p$ is pressure, $Z$ is distance, $k$ is Boltzmann's constant and $θ$ is the temperature. The dimensional formula of $β$ will be

(A)

[M^{0} L^{2} T^{0}]

(B)

[M L^{2} T]

(C)

[M L^{0} T^{- 1}]

(D)

[M^{0} L^{2} T^{- 1}]

LEVELJEE Main

Write the dimensions of the following in terms of mass, time, length and charge. (a) Magnetic flux (b) Rigidity modulus

JEE Advanced 1980

LEVELBoard

Give the MKS units for each of the following quantities. (a) Young's modulus (b) Magnetic induction (c) Power of a lens

JEE Advanced 2007

LEVELJEE Advanced

Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II.

List-I

(P)

G M_{e} M_{s}

G

— universal gravitational constant,

M_{e}

— mass of the earth,

M_{s}

— mass of the sun.

(Q)

\frac{3 R T}{M}

R

— universal gas constant,

T

— absolute temperature,

M

— molar mass.

(R)

\frac{F ^{2}}{q ^{2} B ^{2}}

F

— force,

q

— charge,

B

— magnetic field.

(S)

\frac{G M _{e}}{R _{e}}

G

— universal gravitational constant,

M_{e}

— mass of the earth,

R_{e}

— radius of the earth.

List-II

(1)

(volt) (coulomb) (metre)

(2)

(kilogram) (metre)

^{3}

(second)

^{- 2}

(3)

(metre)

^{2}

(second)

^{- 2}

(4)

(farad) (volt)

^{2}

(kg)

^{- 1}

JEE Advanced 1986

LEVELJEE Main

The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair (s).

* Multiple Correct Options

(A)

Torque and work

(B)

Angular momentum and work

(C)

Energy and Young's modulus

(D)

Light year and wavelength

JEE Advanced 1995

LEVELJEE Main

The pairs of physical quantities that have the same dimensions is (are )

* Multiple Correct Options

(A)

Reynolds number and coefficient of friction

(B)

Curie and frequency of a light wave

(C)

Latent heat and gravitational potential

(D)

Planck's constant and torque

JEE Advanced 2011

LEVELJEE Advanced

Comprehension Passage

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let

N

be the number density of free electrons, each of mass

m

. When the electrons are subjected to an electric field, they are displaced relatively away from the heavy positive ions. If the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency

ω_{p}

, which is called the plasma frequency. To sustain the oscillations, a time varying electric field needs to be applied that has an angular frequency

ω

, where a part of the energy is absorbed and a part of it is reflected. As

ω

approaches

ω_{p}

, all the free electrons are set to resonance together and all the energy is reflected. This is the explanation of high reflectivity of metals.

Question 1:

Taking the electronic charge as $e$ and the permittivity as $ε_{0}$ , use dimensional analysis to determine the correct expression for $ω_{p}$ .

(A)

\frac{N e}{m ε _{0}}

(B)

\frac{m ε _{0}}{N e}

(C)

\frac{N e ^{2}}{m ε _{0}}

(D)

\frac{m ε _{0}}{N e ^{2}}

Question 2:

Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $N \approx 4 \times 1 0^{27} m^{- 3}$ . Take $ε_{0} \approx 1 0^{- 11}$ and $m \approx 1 0^{- 30}$ , where these quantities are in proper SI units.

(A)

800 nm

(B)

600 nm

(C)

300 nm

(D)

200 nm

JEE Main 2005

LEVELBoard

Which of the following sets have different dimensions?

(A)

Pressure, Young's modulus, Stress

(B)

Emf, Potential difference, Electric potential

(C)

Heat, Work done, Energy

(D)

Dipole moment, Electric flux, Electric field

JEE Advanced 1981

LEVELJEE Main

A gas bubble, from an explosion under water, oscillates with a period $T$ proportional to $p^{a} d^{b} E^{c}$ , where $p$ is the static pressure, $d$ is the density of water and $E$ is the total energy of the explosion. Find the values of $a, b$ and $c$ .

JEE Main 2001

LEVELJEE Main

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

Column I gives three physical quantities. Select the appropriate units for the choices given in Column II. Some of the physical quantities may have more than one choice.

List-I

List-II

^{2}

-joule

^{- 1}

(3)

Coulomb (volt)

^{- 1}

(4)

Newton (ampere metre)

^{- 1}

(5)

volt-second (ampere)

^{- 1}

JEE Advanced 1993

LEVELJEE Main

Match the physical quantities given in Column I with dimensions expressed in terms of mass ( $M$ ), length ( $L$ ), time ( $T$ ), and charge ( $Q$ ) given in Column II and write the correct answer against the matched quantity in a tabular form in your answer book.

List-I

List-II

(1)

[M L^{2} T^{- 2}]

(2)

[M L^{2} Q^{- 2}]

(3)

[M L^{2} T^{- 1}]

(4)

[M L^{3} T^{- 1} Q^{- 2}]

(5)

[M^{- 1} L^{- 2} T^{2} Q^{2}]

(6)

[L^{2} T^{- 2}]

LEVELJEE Main

The dimensions of $\frac{1}{2} ε_{0} E^{2}$ ( $ε_{0}$ : permittivity of free space; $E$ : electric field) is

(A)

[MLT^{- 1}]

(B)

[ML^{2} T^{- 2}]

(C)

[MLT^{- 2}]

(D)

[ML^{2} T^{- 1}]

JEE Advanced 2014

LEVELJEE Main

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railway engineer uses dimensional analysis 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

LEVELJEE Main

In the formula $X = 3 Y Z^{2}$ , $X$ and $Z$ have dimensions of capacitance and magnetic induction, respectively. What are the dimensions of $Y$ in MKSQ system?

(A)

[M^{- 3} L^{- 1} T^{3} Q^{4}]

(B)

[M^{- 3} L^{- 2} T^{4} Q^{4}]

(C)

[M^{- 2} L^{- 2} T^{4} Q^{4}]

(D)

[M^{- 3} L^{- 2} T^{4} Q]

JEE Advanced 2016

LEVELJEE Advanced

A length-scale ( $l$ ) depends on the permittivity ( $ε$ ) of a dielectric material, Boltzmann's 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?

* Multiple Correct Options

(A)

l = \frac{n q ^{2}}{ε k _{B} T}

(B)

l = \frac{ε k _{B} T}{n q ^{2}}

(C)

l = \frac{q ^{2}}{ε n ^{2/3} k _{B} T}

(D)

l = \frac{q ^{2}}{ε n ^{1/3} k _{B} T}

JEE Advanced 2025

LEVELJEE Main

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 $κ$ 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 $Z = \frac{S ^{2} σ}{κ}$ is :-

(A)

[M^{0} L^{0} T^{0} I^{0} K^{0}]

(B)

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

(C)

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

(D)

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

JEE Advanced 2026

LEVELJEE Main

In a new system of units, the units of mass, length, time and current are $5 kg$ , $5 m$ , $5 s$ and $5 A$ , respectively. If $μ_{0}$ and $ϵ_{0}$ are the permeability and permittivity of free space, respectively, then in this new system of units, the magnitude of one SI unit of $μ_{0} / ϵ_{0}$ , is :

JEE Advanced 2019

LEVELJEE Main

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

* Multiple Correct Options

(A)

The dimension of force is

L^{- 3}

(B)

The dimension of energy is

L^{- 2}

(C)

The dimension of power is

L^{- 5}

(D)

The dimension of linear momentum is

L^{- 1}

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Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The relation between [E] and [B] is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The relation between [ε0​] and [μ0​] is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In terms of potential difference V, electric current I, permittivity ε0​, permeability μ0​ and speed of light c, the dimensionally correct equations is/are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L, C and R represent the physical quantities inductance, capacitance and resistance, respectively. The combinations which have the dimensions of frequency are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In the relation, p=βα​e−kθαZ​ p is pressure, Z is distance, k is Boltzmann's constant and θ is the temperature. The dimensional formula of β will be

Write the dimensions of the following in terms of mass, time, length and charge. (a) Magnetic flux (b) Rigidity modulus

Give the MKS units for each of the following quantities. (a) Young's modulus (b) Magnetic induction (c) Power of a lens

Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II.

List-I

List-II

The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair (s).

The pairs of physical quantities that have the same dimensions is (are )

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Taking the electronic charge as e and the permittivity as ε0​, use dimensional analysis to determine the correct expression for ωp​.

Which of the following sets have different dimensions?

Column I gives three physical quantities. Select the appropriate units for the choices given in Column II. Some of the physical quantities may have more than one choice.

List-I

List-II

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The dimensions of 21​ε0​E2 (ε0​ : permittivity of free space; E : electric field) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In the formula X=3YZ2, X and Z have dimensions of capacitance and magnetic induction, respectively. What are the dimensions of Y in MKSQ system?

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

Planck's constant $h$ , speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$ . Then, the correct options is/are

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

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

In terms of potential difference $V$ , electric current $I$ , permittivity $ε_{0}$ , permeability $μ_{0}$ and speed of light $c$ , the dimensionally correct equations is/are

$L$ , $C$ and $R$ represent the physical quantities inductance, capacitance and resistance, respectively. The combinations which have the dimensions of frequency are

In the relation, $p = \frac{α}{β} e^{- \frac{α Z}{k θ}}$ $p$ is pressure, $Z$ is distance, $k$ is Boltzmann's constant and $θ$ is the temperature. The dimensional formula of $β$ will be

Taking the electronic charge as $e$ and the permittivity as $ε_{0}$ , use dimensional analysis to determine the correct expression for $ω_{p}$ .

The dimensions of $\frac{1}{2} ε_{0} E^{2}$ ( $ε_{0}$ : permittivity of free space; $E$ : electric field) is

In the formula $X = 3 Y Z^{2}$ , $X$ and $Z$ have dimensions of capacitance and magnetic induction, respectively. What are the dimensions of $Y$ in MKSQ system?