Consider three quantities x= E B , y= 1 _0 _0 and z=1 CR . Here, … - Vidyadip

MCQ

Consider three quantities $\text{x}=\frac{\text{E}}{\text{B}},\ \text{y}=\sqrt{\frac{1}{\mu_0\in_0}}$ and $\text{z}=\frac{1}{\text{CR}}.$ Here, $l$ is the length of a wire, $C$ is a capacitance and $R$ is a resistance. All other symbols have standard meanings.

A
$x, y$ have the same dimensions.
B
$y, z$ have the same dimensions.
C
$z, x$ have the same dimensions.
✓
All of the above

✓

Answer

Correct option: D.

All of the above

Lorentz Force:
$\text{qvB}=\text{qE}$
$\Rightarrow$ Dimensions of $\text{x}=[\text{v}]=\Big[\frac{\text{E}}{\text{B}}\Big]=\big[\text{LT}^{-1}\big]$
$\text{y}=\frac{1}{\sqrt{\mu_0\in_0}}$
$=\sqrt{\frac{4\pi}{\mu_0}\times\frac{1}{4\pi\in_0}}$
$=\sqrt{\frac{9\times10^9}{10^{-7}}}$
$=3\times10^8$
$=\text{c}$
$\Rightarrow$ Dimensions of $\text{y}=[\text{c}]=\big[\text{LT}^{-1}\big]$
Time constant of $RC$ circuit $= RC$ so dimensionally $[RC] = [T]$
$\Rightarrow\text{z}=\Big[\frac{\text{l}}{\text{RC}}$
$\Rightarrow[\text{z}]=[\text{LT}^{-1}]$
Therefore, $x, y$ and $z$ have the same dimensions.

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