Dimension of a Physical Quantity

Q 21 :
Match List I with List II

List I List II

(A) Permeability of free space I. [ $M L^{2} T^{- 2}$ ]

(B) Magnetic field II. [ $M T^{- 2} A^{- 1}$ ]

(C) Magnetic moment III. [ $M L T^{- 2} A^{- 2}$ ]

(D) Torsional constant IV. [ $L^{2} A$ ]

Choose the correct answer from the options given below : [2025]

(A)-(I), (B)-(IV), (C)-(II), (D)-(III)
(A)-(II), (B)-(I), (C)-(III), (D)-(IV)
(A)-(IV), (B)-(III), (C)-(I), (D)-(II)
(A)-(III), (B)-(II), (C)-(IV), (D)-(I)

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(4)

$B = \frac{μ_{0} I}{2 π r}$

$\Rightarrow [μ_{0}] = [\frac{B \cdot r}{I}] = [\frac{M T^{- 2} A^{- 1} \times L}{A}] = [M L T^{- 2} A^{- 2}]$

magnetic field F = qvB

$B = [\frac{M L T^{- 2}}{A T L / T}] = [M T^{- 2} A^{- 1}]$

$[M] = [L^{2} A]$

$τ = C θ \Rightarrow C = [\frac{τ}{θ}] = [M L^{2} T^{- 2}]$

Q 22 :
The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

$E^{2} = p c^{2} + m^{2} c^{4}$
$E^{2} = p c^{2} + m^{2} c^{2}$
$E^{2} = p^{2} c^{2} + m^{2} c^{2}$
$E^{2} = p^{2} c^{2} + m^{2} c^{4}$

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(4)

We need to check the dimensions only.

$[E] = M^{1} L^{2} T^{- 2}$

$[P c] = M^{1} L^{1} T^{- 1} \cdot L^{1} T^{- 1} = M^{1} L^{2} T^{- 2}$

$[m c^{2}] = M^{1} L^{2} T^{- 2}$

So, $E^{2} = p^{2} c^{2} + m^{2} c^{4}$ (dimensionally)

Q 23 :
Match List I with List II

List I List II

A Angular Impulse (I) [ $M^{0} L^{2} T^{- 2}$ ]

B Latent Heat (II) [ $M L^{2} T^{- 3} A^{- 1}$ ]

C Electrical Resistivity (III) [ $M L^{2} T^{- 1}$ ]

D Electromotive Force (IV) [ $M L^{3} T^{- 3} A^{- 2}$ ]

Choose the correct answer from the options given below : [2025]

(A)-(III), (B)-(I), (C)-(IV), (D)-(II)
(A)-(I), (B)-(III), (C)-(IV), (D)-(II)
(A)-(III), (B)-(I), (C)-(II), (D)-(IV)
(A)-(II), (B)-(I), (C)-(IV), (D)-(III)

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(1)

Angular impulse = [ $M L^{2} T^{- 1}$ ]

Latent Heat = [ $M^{0} L^{2} T^{- 2}$ ]

Electrical Resistivity = [ $M L^{3} T^{- 3} A^{- 2}$ ]

Electromotive Force = [ $M L^{2} T^{- 3} A^{- 1}$ ]

Q 24 :
The pair of physical quantities not having same dimensions is [2025]

Torque and energy
Surface tension and impulse
Angular momentum and Planck's constant
Pressure and Young's modulus

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(2)

[Angular momentum] = $M L^{2} T^{- 1}$

[Planck's Constant] = $M L^{2} T^{- 1}$

[Torque] = $M L^{2} T^{- 2}$

[Energy] = $M L^{2} T^{- 2}$

[Surface tension] = $M T^{- 2}$

[Impulse] = $M L T^{- 1}$

[Pressure] = $M L^{- 1} T^{- 2}$

[Young's modulus] = $M L^{- 1} T^{- 2}$

Q 25 :
The expression given below shows the variation of velocity (v) with time (t), $v = A t^{2} + \frac{B t}{C + t}$ . The dimension of ABC is : [2025]

$[M^{0} L^{2} T^{- 3}]$
$[M^{0} L^{1} T^{- 3}]$
$[M^{0} L^{1} T^{- 2}]$
$[M^{0} L^{2} T^{- 2}]$

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(1)

$v = A t^{2} + \frac{B t}{C + t}$

$[v] = [A] t^{2} = [\frac{B t}{C + t}]$

[A] = $L T^{- 3}$

[C] = T

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

[ABC] = $L^{2} T^{- 3}$

Q 26 :
Match List I with List II.

List I List II

(A) Young's Modulus (I) $M L^{- 1} T^{- 1}$

(B) Torque (II) $M L^{- 1} T^{- 2}$

(C) Coefficient of Viscosity (III) $M^{- 1} L^{3} T^{- 2}$

(D) Gravitational Constant (IV) $M L^{2} T^{- 2}$

Choose the correct answer from the options given below : [2025]

(A)-(I), (B)-(III), (C)-(II), (D)-(IV)
(A)-(II), (B)-(I), (C)-(IV), (D)-(III)
(A)-(IV), (B)-(II), (C)-(III), (D)-(I)
(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

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(4)

(A) $[Y] = \frac{F}{A (\frac{△ l}{l})} \Rightarrow \frac{M L T^{- 2}}{L^{2}} = M L^{- 1} T^{- 2}$

(B) Torque $(\vec{τ}) = \vec{r} \times \vec{F} = L \times M L T^{- 2} = M L^{2} T^{- 2}$

(C) Coefficient of viscosity $\Rightarrow F = η A \frac{d V}{d t}$

$[η] = \frac{M L T^{- 2}}{L^{2}} \times T = M L^{- 1} T^{- 1}$

(D) Gravitational constant (G)

$F = \frac{G M_{1} M_{2}}{r^{2}}$

$[G] = \frac{F \cdot r^{2}}{m_{1} m_{2}} = \frac{M L T^{- 2} \times L^{2}}{M^{2}} = M^{- 1} L^{3} T^{- 2}$

Q 27 :
The equation for real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T$ , where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $a b^{- 2}$ is equivalent to that of : [2025]

Planck's constant
Compressibility
Strain
Energy density

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(4)

Given, $[P + \frac{a}{V^{2}}] (V - b) = R T$

$[P] = [\frac{a}{V^{2}}] \Rightarrow [a] = [P] [V^{2}]$

$[a] = [P] [V^{2}] = [M L^{- 1} T^{- 2}] [L^{6}] = M L^{5} T^{- 2}$

$[b] = [V] = L^{3}$

$[a b^{- 2}] = [M L^{5} T^{- 2} L^{- 6}] = [M L^{- 1} T^{- 2}]$

Dimension of energy density.

Q 28 :
Match List I with List II

List I List II

(A) Coefficient of viscosity (I) [ $M L^{0} T^{- 3}$ ]

(B) Intensity of wave (II) [ $M L^{- 2} T^{- 2}$ ]

(C) Pressure gradient (III) [ $M^{- 1} L T^{2}$ ]

(D) Compressibility (IV) [ $M L^{- 1} T^{- 1}$ ]

Choose the correct answer from the options given below : [2025]

(A)-(I), (B)-(IV), (C)-(III), (D)-(II)
(A)-(IV), (B)-(I), (C)-(II), (D)-(III)
(A)-(IV), (B)-(II), (C)-(I), (D)-(III)
(A)-(II), (B)-(III), (C)-(IV), (D)-(I)

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(2)

$η \equiv \frac{F l}{A v} \equiv [M L^{- 1} T^{- 1}]$	A $\to$ IV
$I \equiv \frac{P}{A} \equiv [M T^{- 3}]$	B $\to$ I
$\frac{d P}{d x} \equiv [M L^{- 2} T^{- 2}]$	C $\to$ II
$K \equiv \frac{1}{B} \equiv [M^{- 1} L T^{2}]$	D $\to$ III

Q 29 :
Given a charge q, current I and permeability of vacuum $μ_{0}$ . Which of the following quantity has the dimension of momentum? [2025]

$q I / μ_{0}$
$q μ_{0} I$
$q^{2} μ_{0} I$
$q μ_{0} / I$

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(2)

Momentum = mv

$r = \frac{m v}{q B}$

$[m v] \equiv [q B r] \Rightarrow [m v] \equiv [q \frac{μ_{0} I}{2 π r} r]$

$\Rightarrow [m v] \equiv [q μ_{0} I]$

Q 30 :
If $μ_{0}$ and $ε_{0}$ are the permeability and permittivity of free space, respectively. then the dimension of $(\frac{1}{μ_{0} ε_{0}})$ is : [2025]

$L / T^{2}$
$L^{2} / T^{2}$
$T^{2} / L$
$T^{2} / L^{2}$

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(2)

$C = \frac{1}{\sqrt{μ_{0} ε_{0}}}$

$μ_{0} ε_{0} \equiv \frac{1}{C^{2}} \equiv [L^{- 2} T^{2}]$

$∴ [\frac{1}{μ_{0} ε_{0}}] = \frac{L^{2}}{T^{2}}$

Topic Question Set

Q 22 :
The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

Q 24 :
The pair of physical quantities not having same dimensions is [2025]

Q 25 :
The expression given below shows the variation of velocity (v) with time (t), $v = A t^{2} + \frac{B t}{C + t}$ . The dimension of ABC is : [2025]

Q 27 :
The equation for real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T$ , where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $a b^{- 2}$ is equivalent to that of : [2025]

Q 29 :
Given a charge q, current I and permeability of vacuum $μ_{0}$ . Which of the following quantity has the dimension of momentum? [2025]

Q 30 :
If $μ_{0}$ and $ε_{0}$ are the permeability and permittivity of free space, respectively. then the dimension of $(\frac{1}{μ_{0} ε_{0}})$ is : [2025]

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	List I		List II
(A)	Permeability of free space	I.	[ $M L^{2} T^{- 2}$ ]
(B)	Magnetic field	II.	[ $M T^{- 2} A^{- 1}$ ]
(C)	Magnetic moment	III.	[ $M L T^{- 2} A^{- 2}$ ]
(D)	Torsional constant	IV.	[ $L^{2} A$ ]

	List I		List II
A	Angular Impulse	(I)	[ $M^{0} L^{2} T^{- 2}$ ]
B	Latent Heat	(II)	[ $M L^{2} T^{- 3} A^{- 1}$ ]
C	Electrical Resistivity	(III)	[ $M L^{2} T^{- 1}$ ]
D	Electromotive Force	(IV)	[ $M L^{3} T^{- 3} A^{- 2}$ ]

	List I		List II
(A)	Young's Modulus	(I)	$M L^{- 1} T^{- 1}$
(B)	Torque	(II)	$M L^{- 1} T^{- 2}$
(C)	Coefficient of Viscosity	(III)	$M^{- 1} L^{3} T^{- 2}$
(D)	Gravitational Constant	(IV)	$M L^{2} T^{- 2}$

	List I		List II
(A)	Coefficient of viscosity	(I)	[ $M L^{0} T^{- 3}$ ]
(B)	Intensity of wave	(II)	[ $M L^{- 2} T^{- 2}$ ]
(C)	Pressure gradient	(III)	[ $M^{- 1} L T^{2}$ ]
(D)	Compressibility	(IV)	[ $M L^{- 1} T^{- 1}$ ]

Topic Question Set

Q 21 : Match List I with List II List I List II (A) Permeability of free space I. [ML2T–2] (B) Magnetic field II. [MT–2A–1] (C) Magnetic moment III. [MLT–2A–2] (D) Torsional constant IV. [L2A] Choose the correct answer from the options given below : [2025]

Q 22 : The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

Q 23 : Match List I with List II List I List II A Angular Impulse (I) [M0L2T–2] B Latent Heat (II) [ML2T–3A–1] C Electrical Resistivity (III) [ML2T–1] D Electromotive Force (IV) [ML3T–3A–2] Choose the correct answer from the options given below : [2025]

Q 24 : The pair of physical quantities not having same dimensions is [2025]

Q 25 : The expression given below shows the variation of velocity (v) with time (t), v=At2+BtC+t. The dimension of ABC is : [2025]

Q 26 : Match List I with List II. List I List II (A) Young's Modulus (I) ML–1T–1 (B) Torque (II) ML–1T–2 (C) Coefficient of Viscosity (III) M–1L3T–2 (D) Gravitational Constant (IV) ML2T–2 Choose the correct answer from the options given below : [2025]

Q 27 : The equation for real gas is given by (P+aV2)(V–b)=RT, where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of ab–2 is equivalent to that of : [2025]

Q 28 : Match List I with List II List I List II (A) Coefficient of viscosity (I) [ML0T–3] (B) Intensity of wave (II) [ML–2T–2] (C) Pressure gradient (III) [M–1LT2] (D) Compressibility (IV) [ML–1T–1] Choose the correct answer from the options given below : [2025]

Q 29 : Given a charge q, current I and permeability of vacuum μ0. Which of the following quantity has the dimension of momentum? [2025]

Q 30 : If μ0 and ε0 are the permeability and permittivity of free space, respectively. then the dimension of (1μ0ε0) is : [2025]

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Q 22 :
The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation. [2025]

Q 24 :
The pair of physical quantities not having same dimensions is [2025]

Q 25 :
The expression given below shows the variation of velocity (v) with time (t), $v = A t^{2} + \frac{B t}{C + t}$ . The dimension of ABC is : [2025]

Q 27 :
The equation for real gas is given by $(P + \frac{a}{V^{2}}) (V - b) = R T$ , where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $a b^{- 2}$ is equivalent to that of : [2025]

Q 29 :
Given a charge q, current I and permeability of vacuum $μ_{0}$ . Which of the following quantity has the dimension of momentum? [2025]

Q 30 :
If $μ_{0}$ and $ε_{0}$ are the permeability and permittivity of free space, respectively. then the dimension of $(\frac{1}{μ_{0} ε_{0}})$ is : [2025]