Using dimensional analysis, the resistivity i | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

$(c)$ Let resistivity depends on given fundamental constants.

$\rho=h^{} m_{e}^{b} c^{c} e^{d} \varepsilon_{0}^{f}$

where, $k=$ a numeric consta

Vedclass · Accepted Answer

$\frac{h^{2}}{m_{e} c e^{2}}$

Vedclass · Answer

$(c)$ Let resistivity depends on given fundamental constants.

$\rho=h^{} m_{e}^{b} c^{c} e^{d} \varepsilon_{0}^{f}$

where, $k=$ a numeric constant.

Now, substituting dimensions of different physical constants, we have

$\left[ ML ^{3} T ^{-3} A ^{-2}\right]=k\left[[ \mathrm { ML } ^ { 2 } \mathrm { T } ^ { - 1 } ] ^ { k } \left[ M ^{b}\left[ LT ^{-1}\right]^{c}\right.\right.$

$\left.[ AT ]^{d}\left[ M ^{-1} L ^{-3} T ^{4} A ^{2}\right]^{f}\right]$

Equating dimensions, we have

$1=a+b-f$

$3=2 a+c-3 f$

$-3=-a-c+d+4 f$

$-2=d+2 f$

Solving these, we get

$a=2$

$b=-1$

$c=-1$

$d=-2$

$f=0$

So, resistivity can be expressed as

$\rho=k\left(\frac{h^{2}}{m_{e} c e^{2}}\right)$

Using dimensional analysis, the resistivity in terms of fundamental constants $h, m_{e}, c, e, \varepsilon_{0}$ can be expressed as

Similar Questions

If energy $(E)$, velocity $(v)$and force $(F)$ be taken as fundamental quantity, then what are the dimensions of mass

The density of a material in $SI\, units$ is $128\, kg\, m^{-3}$. In certain units in which the unit of length is $25\, cm$ and the unit of mass $50\, g$, the numerical value of density of the material is

The frequency $(v)$ of an oscillating liquid drop may depend upon radius $(r)$ of the drop, density $(\rho)$ of liquid and the surface tension $(s)$ of the liquid as : $v=r^{ a } \rho^{ b } s ^{ c }$. The values of $a , b$ and $c$ respectively are

Frequency is the function of density $(\rho )$, length $(a)$ and surface tension $(T)$. Then its value is

The equation of a wave is given by$Y = A\sin \omega \left( {\frac{x}{v} - k} \right)$where $\omega $ is the angular velocity and $v$ is the linear velocity. The dimension of $k$ is