The workdone by a gas molecule in an isolated system is given by, $W =\alpha \beta^{2} e ^{-\frac{ x ^{2}}{\alpha kT }},$ where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature, $\alpha$ and $\beta$ are constants. Then the dimension of $\beta$ will be

  • [JEE MAIN 2021]
  • A

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

  • B

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

  • C

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

  • D

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

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In a particular system of units, a physical quantity can be expressed in terms of the electric charge $c$, electron mass $m_c$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4 \pi \epsilon_0}$, where $\epsilon_0$ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is $[B]=[c]^\alpha\left[m_c\right]^\beta[h]^\gamma[k]^\delta$. The value of $\alpha+\beta+\gamma+\delta$ is. . . . .

  • [IIT 2022]

Match the following two coloumns

  Column $-I$   Column $-II$
$(A)$ Electrical resistance $(p)$ $M{L^3}{T^{ - 3}}{A^{ - 2}}$
$(B)$ Electrical potential $(q)$ $M{L^2}{T^{ - 3}}{A^{ - 2}}$
$(C)$ Specific resistance $(r)$ $M{L^2}{T^{ - 3}}{A^{ - 1}}$
$(D)$ Specific conductance $(s)$ None of these