1.Units, Dimensions and Measurement
hard

An expression of energy density is given by $u=\frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)$, where $\alpha, \beta$ are constants, $x$ is displacement, $k$ is Boltzmann constant and $t$ is the temperature. The dimensions of $\beta$ will be.

A

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

B

$\left[ M ^{0} L ^{2} T ^{-2}\right]$

C

$\left[ M ^{0} L ^{0} T ^{0}\right]$

D

$\left[ M ^{0} L ^{2} T ^{0}\right]$

(JEE MAIN-2022)

Solution

$\frac{\alpha[ L ]}{\left[ ML ^{2} T ^{-2}\right]}=\left[ M ^{0} L ^{0} T ^{0}\right]$

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

$\frac{\alpha}{\beta}=\frac{\left[ ML ^{2} T ^{-2}\right]}{\left[ L ^{3}\right]} \Rightarrow \beta=\frac{\left[ ML ^{1} T ^{-2}\right]\left[ L ^{3}\right]}{ ML ^{2} T ^{-2}}$

Standard 11
Physics

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