Question

Consider three quantities x=E/B, y=√1/μ0ε0 and z=lCR. 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.
(d) None of the three pairs have the same dimensions.

Answer 1

The correct answer is Option a)x, y have the same dimensions , b) y, z have the same dimensions and c)z, x have the same dimensions.

Explanation:

According to Lorentz Force:

We know that  

qvB = qE

The Dimensions of   $x=[v]=\frac{E}{B}=\left[L T^{-1}\right]$

  $y=\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}$

    $=\sqrt{\frac{4 \pi}{\mu_{0}} \times \frac{1}{4 \pi \varepsilon_{0}}}$

    $=\sqrt{\frac{9 \times 10^{9}}{10^{-7}}}$  

     $=3 \times 10^{8}=c$

Actual dimensions $y=[c]=\left[L T^{-1}\right]$

RC Circuit Time Constant= RC so dimensionally [RC] = [T]

  $z=\frac{l}{R C}$

$[z]=\left[L T^{-1}\right]$

The same dimensions thus apply to x, y and z.