Question

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1.Units, Dimensions and Measurement

normal

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0^\beta h^7 c^5$ and $n$ is a non-zero integer, then $(\alpha, \beta, \gamma, \delta)$ is given by

$(2 n,-n,-n,-n)$

$(n,-n,-2 n,-n)$

$(n,-n,-n,-2 n)$

$(2 n,-n,-2 n,-2 n)$

(IIT-2024)

Solution

For the quantity to be dimensionless

$e ^\alpha \varepsilon_0^\beta h ^\gamma c ^{ d }= M ^0 L ^0 T ^0 A ^0$

$\Rightarrow( AT )^\alpha\left( M ^{-1} L ^{-3} T ^4 A ^2\right)^\beta\left( ML ^2 T ^{-1}\right)^\gamma\left( LT ^{-1}\right)^\delta= A ^0 M ^0 L ^0 T ^0$

$\therefore \alpha+2 \beta=0, \alpha+4 \beta-\gamma-\delta=0,-\beta+\gamma=0 \&-3 \beta+2 \gamma+\delta=0$

$\therefore \alpha=-2 \beta, \beta=\gamma \& \gamma=\delta$

$\therefore$ Option $(A)$ satisfies the given condition

Std 11

Physics

If $\varepsilon_0$ is permittivity of free space, $e$ is charge of proton, $G$ is universal gravitational constant and $m_p$ is mass of a proton then the dimensional formula for $\frac{e^2}{4 \pi \varepsilon_0 G m_p{ }^2}$ is

medium

If velocity$(V)$, force$(F)$ and time$(T)$ are chosen as fundamental quantities then dimensions of energy are

hard

if Energy is given by $U = \frac{{A\sqrt x }}{{{x^2} + B}},\,$, then dimensions of $AB$ is

medium

A physical quantity $\vec{S}$ is defined as $\vec{S}=(\vec{E} \times \vec{B}) / \mu_0$, where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $\mu_0$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity (ies)?

$(A)$ $\frac{\text { Energy }}{\text { charge } \times \text { current }}$

$(B)$ $\frac{\text { Force }}{\text { Length } \times \text { Time }}$

$(C)$ $\frac{\text { Energy }}{\text { Volume }}$

$(D)$ $\frac{\text { Power }}{\text { Area }}$

easy

A dimensionally consistent relation for the volume $V$ of a liquid of coefficient of viscosity $\eta $ flowing per second through a tube of radius $r$ and length $l$ and having a pressure difference $p$ across its end, is

medium

Solution

Similar Questions

If $\varepsilon_0$ is permittivity of free space, $e$ is charge of proton, $G$ is universal gravitational constant and $m_p$ is mass of a proton then the dimensional formula for $\frac{e^2}{4 \pi \varepsilon_0 G m_p{ }^2}$ is

If velocity$(V)$, force$(F)$ and time$(T)$ are chosen as fundamental quantities then dimensions of energy are

if Energy is given by $U = \frac{{A\sqrt x }}{{{x^2} + B}},\,$, then dimensions of $AB$ is

A dimensionally consistent relation for the volume $V$ of a liquid of coefficient of viscosity $\eta $ flowing per second through a tube of radius $r$ and length $l$ and having a pressure difference $p$ across its end, is