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Consider Three Quantities X = E / B , Y = √ 1 / μ 0 ϵ 0 and Z = L C R . Here, L is the Length of a Wire, C is a Capacitance and R is a Resistance. All Other Symbols Have Standard Meanings. - Physics

Short Note

Consider three quantities  \[x = E/B,   y = \sqrt{1/ \mu_0 \epsilon_0}\] and \[z = \frac{l}{CR}\] . Here, l is the length of a wire, C is a capacitance and R is a resistance. All other symbols have standard meanings.

(a) xy have the same dimensions.
(b) yz have the same dimensions.
(c) zx have the same dimensions.
(d) None of the three pairs have the same dimensions.

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(a) x, y have the same dimensions.
(b) y, z have the same dimensions.
(c) z, x have the same dimensions. 

Lorentz Force:

\[qvB = qE\]
\[ \Rightarrow \text{ Dimensions of x }= [v] = \left[ \frac{E}{B} \right] = [ {LT}^{- 1} ]\]

\[y = \frac{1}{\sqrt{\mu_o \epsilon_o}} = \sqrt{\frac{4\pi}{\mu_o} \times \frac{1}{4 \pi\epsilon_o}} = \sqrt{\frac{9 \times {10}^9}{{10}^{- 7}}} = 3 \times {10}^8 = c\]
\[ \Rightarrow \text{ Dimensions of y }= [c] = [ {LT}^{- 1} ]\]

Time constant of RC circuit = RC so dimensionally [RC] = [T]

\[\Rightarrow z = \left[ \frac{l}{RC} \right] \Rightarrow [z] = [ {LT}^{- 1} ]\]

Therefore, x, y and z have the same dimensions.

Concept: Force on a Moving Charge in Uniform Magnetic and Electric Fields
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HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 13 Magnetic Field due to a Current
MCQ | Q 2 | Page 249
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