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The physical quantities not having same dimensions are

Option 1)
torque and work
Option 2)
momentum and Planck's constant

Option 3)
stress and Young's modulus
Option 4)
speed and $\left ( \mu _{0}\varepsilon _{0} \right )^{-1/2}$ .

Answers (2)

best_answer

As we learnt in the concept:

Permeability of free space -

Dimension of permeablity of free space ( $\mu_{o}$ )- $M^{1}L^{1}T^{-2}A^{-2}$

- wherein

$\frac{newton}{ampere^{2}},\frac{henry}{metre}$

and

Plank's Constant (h) -

- wherein

Joule-sec

Option 1)

torque and work

Torque = $\vec{r}\times \vec{F}$

$\left [ Torque \right ]= \left [ ML^{2} T^{-2}\right ]$

$work = \vec{F}\cdot \vec{r}$

$\Rightarrow$ Dimension of work = $\left [ ML^{2}T^{-2} \right ]$

Both Torque and work have same dimension.

Option 2)

momentum and Planck's constant

Dimension of momentum = $\left [ MLT^{-1} \right ]$

Dimension of Plank Constant $= \left [ P \right ]\left [ \lambda \right ]$

$= \left [ ML^{2} T^{-1}\right ]$

Hence these two have different dimensions.

Option 3)

stress and Young's modulus

Stress = $\frac{F}{A}$

Young modulus = $\frac{Stress}{Strain}$

Since strain is dimension less.

hence young modulus have dimension same as stress.

Option 4)

speed and $\left ( \mu _{0}\varepsilon _{0} \right )^{-1/2}$ .

Speed of light = $\frac{1}{\sqrt {\mu \epsilon _{0}}}$

Hence two have same dimensions.

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