Which of the following equations is dimensionally correct?
$(I) v=\sqrt{\dfrac{P}{\rho}}(II)v=\sqrt{\dfrac{mgl}{I}}~~(III)v=\dfrac{Pr^2}{2\eta l}$
(where $v=$ speed, $P=$ pressure; $r,$ $l$ are lengths; $\rho=$ density, $m=$ mass, $g=$ acceleration due to gravity, $I=$ moment of inertia, and $\eta=$ coefficient of viscosity)

1. $I~ \text{and}~II$

2. $I~ \text{and}~III$

3. $II~ \text{and}~III$

4. $I,~II~\text{and}~III$

Subtopic: Dimensions |

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Q4:

The acceleration due to gravity on the surface of the earth is $g=10$ m/s². The value in km/(minute)² is:

1.	$36$	2.	$0.6$
3.	$\dfrac{10}{6}$	4.	$3.6$

Subtopic: Dimensions |

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Q5:

Find the unit of the ratio $\dfrac{E}{gB}$ where $E$ = electric field, $B$ = magnetic field, $g$ = acceleration due to gravity.
1. $\text{s}^{-1}$
2. $\text{s}$
3. $\text{m/s}$
4. $\text{s/m}^2$

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Q2:

The diffusion equation is given by:

\frac{\partial n}{\partial t} = D \frac{\partial^{2} n}{\partial x^{2}}

$\dfrac{\partial n}{\partial t}=D\dfrac{\partial^2 n}{\partial x^2}$
where

n (x, t)

$n(x,t)$ is the concentration of particles per unit volume at position

x

$x$ , at time

t

$t$ . The dimensions of the diffusion constant

D,

$D,$ are:
1.

L T^{- 1}

$LT^{-1}$
2.

L^{- 1} T

$L^{-1}T$
3.

L^{5} T^{- 1}

$L^{5}T^{-1}$
4.

L^{2} T^{- 1}

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

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Q6:

The quantity "Action" (

S

$S$ ) is defined by the equation:

S = \int_{0}^{t} (K - U)^{} d t

$S=\int_{0}^{t}(K-U)^{} d t$
where

K

$K$ is the kinetic energy,

U

$U$ is the potential energy and the integral is over the time,

t

$t$ during the motion.
The proper unit of action will be:
1. kg-m/s
2. kg-m²
3. kg-m²-s
4. kg-m²/s

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Q7:

The dimension of which group of quantities is the same?

h

$h$ : Planck's constant,

K

$K$ : kinetic energy,

ω

$\omega$ : angular speed/frequency,

F

$F$ : force,

L

$L$ : inductance,

i

$i$ : current,

q

$q$ : charge,

t

$t$ : time,

x

$x$ : distance
1.

h, F t x, L i q

$h, ~Ftx,~ Liq$
2.

K, h ω, ω L i

$K , h\omega , \omega Li$
3.

F x, L i^{2}, K ω

$Fx, ~Li^2,~K \omega$
4.

\frac{F x}{t}, K x, h t

$\dfrac{Fx}{t} ,~ Kx, ~ ht$

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Q8:

Which of the following quantities have the same dimension?

(A)	$\dfrac{\text{(Magnetic flux)}^2}{\text{Electrical resistance}}$	(B)	$\text{Torque}\times\text{time}$
(C)	$\text{Momentum}\times\text{length}$	(D)	$\dfrac{\text{Power}}{\text{time}}$

1. B, C
2. A, B, C
3. B, D
4. A, B, C, D

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Q9:

What is the unit of the quantity represented by:

\frac{(Angular momentum)}{{(electric charge)}^{2}}

$\dfrac{\text{(Angular momentum)}}{\text{(electric charge)}^2}$

1.

Ω

$\Omega$ (ohm)
2. s/m
3.

H

$\text{H }$ (henry)
4.

F

$\text{F }$ (farad)

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Q10:

The quantity

η

$\eta$ is defined by:

η = \frac{1}{μ_{0} σ}

$\eta=\dfrac{1}{\mu_0\sigma}$
where

μ_{0}

$\mu_0$ is the permeability of free space and

σ

$\sigma$ is the electrical conductivity (of a plasma).

η

$\eta$ is referred to as the magnetic diffusivity. Its SI unit is:
1. m²/s
2. m/s²
3. C-m²/s
4. T-m/s

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Q1:

Let the two quantities be defined as follows:
$\int \vec{B}.\vec{dl} = W$ and $\int \vec{E}.\vec{dS} = U$
where $\vec{ B}$ is the magnetic field, $d\vec{l}$ is a small length, $\vec{ E}$ is the electric field and $d\vec{S}$ is a small area. The ratio $\dfrac{U}{W}$ has the dimensions of:
1. $\left[L^2T^{-2}\right]$
2. $\left[LT^{-2}\right]$
3. $\left[L^2T^{-1}\right]$
4. $\left[LT^{-1}\right]$

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1.	$I~ \text{and}~II$
2.	$I~ \text{and}~III$
3.	$II~ \text{and}~III$
4.	$I,~II~\text{and}~III$

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