Question:

Planck's constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is(are)

Updated On: Apr 2, 2024

$M \propto \sqrt{c}$
$M \propto \sqrt{G}$
$L \propto \sqrt{h}$
$L \propto \sqrt{G}$

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The Correct Option is D

Solution and Explanation

$E = \frac{hc}{\lambda}$ $E = \frac{-Gm_{1}m_{2}}{r}$ $\frac{hc}{\lambda} = \frac{Gm^{2}}{r}$ unit of $?$ and r is L So $M = \left(\frac{hc}{G}\right)^{1/2}\quad\quad\dots\left(1\right)$ $M \propto \sqrt{c}\quad\quad M \propto \frac{1}{\sqrt{G}}\quad\quad Ans. \left(A\right)$ $E = mc^{2}$ $E = \frac{hc}{\lambda}$ $\frac{hc}{\lambda} = mc^{2}$ $\lambda = \frac{h}{m}$ $\therefore\,L = \frac{h}{M}$ $L = \frac{h}{\left(hc\right)^{1/2}}G^{1/2}\quad\quad$ from $\left(1\right)$ $\therefore\,L \propto \sqrt{G} \quad L \propto \sqrt{h}$

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Concepts Used:

Dimensional Analysis

Dimensional Analysis is a process which helps verify any formula by the using the principle of homogeneity. Basically dimensions of each term of a dimensional equation on both sides should be the same.

Limitation of Dimensional Analysis: Dimensional analysis does not check for the correctness of value of constants in an equation.

Using Dimensional Analysis to check the correctness of the equation

Let us understand this with an example:

Suppose we don’t know the correct formula relation between speed, distance and time,

We don’t know whether

(i) Speed = Distance/Time is correct or

(ii) Speed =Time/Distance.

Now, we can use dimensional analysis to check whether this equation is correct or not.

By reducing both sides of the equation in its fundamental units form, we get

(i) [L][T]^-¹ = [L] / [T] (Right)

(ii) [L][T]^-¹ = [T] / [L] (Wrong)

From the above example it is evident that the dimensional formula establishes the correctness of an equation.