A physical quantity obtained from the ratio of the coefficient of thermal conductivity to the universal…

A physical quantity obtained from the ratio of the coefficient of thermal conductivity to the universal gravitational constant has a dimensional formula $\left[\mathrm{M}^{2 a} \mathrm{~L}^{4 b} \mathrm{~T}^{2 c} \mathrm{~K}^d\right]$, then the value of $\frac{a+b}{c+b}-d$ is
  1. $+\frac{3}{2}$
  2. $-\frac{1}{2}$
  3. $-\frac{3}{2}$
  4. $+\frac{1}{2}$

Solution

Dimensional formula of thermal conductivity $[k]=\left[\mathrm{M}^1 \mathrm{~L}^1 \mathrm{~T}^{-3} \mathrm{~K}^{-1}\right].$
Dimensional formula of universal gravitational constant, $[G]=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$
Now, $\frac{[k]}{[G]}=\left[\mathrm{M}^2 \mathrm{~L}^{-2} \mathrm{~T}^{-1} \mathrm{~K}^{-1}\right]$
Compare above equation with $\left[\mathrm{M}^{2 a} \mathrm{~L}^{4 b} \mathrm{~T}^{2 c} \mathrm{~K}^d\right]$
This will give us, $a=1, b=-\frac{1}{2}, c=-\frac{1}{2}$ and $d=-1$
Now, $\frac{a+b}{c+b}-d=\frac{1-\frac{1}{2}}{-\frac{1}{2}-\frac{1}{2}}-(-1)$ or $\quad \frac{a+b}{c+b}-d=\frac{1}{2}$

Asked in: AP EAMCET 2019 (20 Apr Shift 2)

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