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1.Units, Dimensions and Measurement
hard
The quantities $\quad x=\frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}, y=\frac{E}{B}$ and $z=\frac{l}{C R}$ are defined where $C-$ capacitance $R-$Resistance, $l-$length, $E-$Electric field, $B-$magnetic field and $\varepsilon_{0}, \mu_{0},$ -free space permittivity and permeability respectively. Then....
AOnly $x$ and $y$ have the same dimension
B$x, y$ and $z$ have the same dimension
COnly $x$ and $z$ have the same dimension
DOnly $y$ and $z$ have the same dimension
(JEE MAIN-2020)
Solution
$x =\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}=\operatorname{speed} \Rightarrow[ x ]=\left[ L ^{1} T ^{-1}\right]$
$y =\frac{ E }{ B }=\operatorname{speed} \Rightarrow[ y ]=\left[ L ^{1} T ^{-1}\right]$
$z=\frac{\ell}{R C}=\frac{\ell}{\tau} \Rightarrow[z]=\left[L^{1} T^{-1}\right]$
So, $x, y, z$ all have the same dimensions.
$y =\frac{ E }{ B }=\operatorname{speed} \Rightarrow[ y ]=\left[ L ^{1} T ^{-1}\right]$
$z=\frac{\ell}{R C}=\frac{\ell}{\tau} \Rightarrow[z]=\left[L^{1} T^{-1}\right]$
So, $x, y, z$ all have the same dimensions.
Standard 11
Physics
