Question

In terms of potential difference V, electric current I, permittivity ${\varepsilon _{\text{o}}}$, permeability ${\mu _{\text{o}}}$ and speed of light c, the dimensionally correct equation(s) is/are:
(This question has multiple correct options)
$
  {\text{A}}{\text{. }}{\mu _{\text{o}}}{{\text{I}}^2} = {\varepsilon _{\text{o}}}{{\text{V}}^2} \\
  {\text{B}}{\text{. }}{\mu _{\text{o}}}{\text{I}} = {\varepsilon _{\text{o}}}{\text{V}} \\
  {\text{C}}{\text{. I}} = {\varepsilon _{\text{o}}}{\text{cV}} \\
  {\text{D}}{\text{. }}{\mu _{\text{o}}}{\text{cI}} = {\varepsilon _{\text{o}}}{\text{V}} \\
$

Answer 1

- Hint: In order to find all the dimensionally correct options we check each individual option separately by using the formulae of speed of light c, resistance R in terms of permittivity and permeability and Ohm’s law.
Ohm’s law: V = IR

Formula Used,
${\text{C = }}\dfrac{1}{{\sqrt {{\mu _{\text{o}}}{\varepsilon _o}} }}$
${\text{R = }}\sqrt {\dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}}} $
Ohm’s Law – V = IR

Complete step-by-step solution:
We could check if all the options are dimensionally correct or not by two methods. We could use their formulae to verify or we could write down the units of each quantity and verify.
We use the formulae of speed of light C and resistance R in terms of ${\mu _{\text{o}}}{\text{ and }}{\varepsilon _{\text{o}}}$, to find the answer.

The speed of light C is given as ${\text{C = }}\dfrac{1}{{\sqrt {{\mu _{\text{o}}}{\varepsilon _o}} }}$
The resistance can be expressed as ${\text{R = }}\sqrt {\dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}}} $
And we know, V = IR, where V, I, R are the voltage, current and resistance respectively.
${\mu _{\text{o}}}{{\text{I}}^2} = {\varepsilon _{\text{o}}}{{\text{V}}^2}$
$ \Rightarrow \dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}} = {\text{ }}{\left( {\dfrac{{\text{V}}}{{\text{I}}}} \right)^2}$
$ \Rightarrow \dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}} = {\text{ }}{\left( {\text{R}} \right)^2}$
$ \Rightarrow \dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}} = {\text{ }}\left( {\dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}}} \right){\text{ - - - - - Since R = }}\sqrt {\dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}}} $
Option A is correct.

${\mu _{\text{o}}}{\text{I}} = {\varepsilon _{\text{o}}}{\text{V}}$
$
   \Rightarrow \dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}} = \dfrac{{\text{V}}}{{\text{I}}} \\
   \Rightarrow \dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}} = {\text{R}} \\
   \Rightarrow \dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}} = \sqrt {\dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}}} \\
$
Option B is not correct.

${\text{I}} = {\varepsilon _{\text{o}}}{\text{cV}}$
$
   \Rightarrow \dfrac{{\text{I}}}{{\text{V}}} = {\varepsilon _{\text{o}}}\dfrac{1}{{\sqrt {{\varepsilon _{\text{o}}}{\mu _{\text{o}}}} }} \\
   \Rightarrow \dfrac{1}{{\text{R}}} = \sqrt {\dfrac{{{\varepsilon _{\text{o}}}}}{{{\mu _{\text{o}}}}}} \\
   \Rightarrow \sqrt {\dfrac{{{\varepsilon _{\text{o}}}}}{{{\mu _{\text{o}}}}}} = \sqrt {\dfrac{{{\varepsilon _{\text{o}}}}}{{{\mu _{\text{o}}}}}} \\
$
Option C is correct.

${\mu _{\text{o}}}{\text{cI}} = {\varepsilon _{\text{o}}}{\text{V}}$
$
   \Rightarrow \dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}}\dfrac{1}{{\sqrt {{\varepsilon _{\text{o}}}{\mu _{\text{o}}}} }} = \dfrac{{\text{V}}}{{\text{I}}} \\
   \Rightarrow \sqrt {\dfrac{{{\mu _{\text{o}}}}}{{{\varepsilon _{\text{o}}}}}} = {\text{R}}{\varepsilon _{\text{o}}} \\
$
Option D is not correct.
Options A and C are the correct options.

Note – In order to answer this type of question the key is to know to express the given equation in terms of one another. We can also solve this question by only verifying the options using the units of given variables in the question.
The dimensions of the terms given are –
$
  [{\text{V] = [}}{{\text{M}}^{ - 1}}{{\text{L}}^2}{{\text{T}}^{ - 3}}{{\text{A}}^{ - 1}}] \\
  [{\text{I] = [A]}} \\
  {\text{[c] = [}}{{\text{L}}^1}{{\text{T}}^{ - 1}}] \\
  [{\varepsilon _{\text{o}}}]{\text{ = [}}{{\text{M}}^{ - 1}}{{\text{L}}^{ - 3}}{{\text{T}}^4}{{\text{A}}^2}] \\
  [{\mu _{\text{o}}}]{\text{ = [}}{{\text{M}}^1}{{\text{L}}^1}{{\text{T}}^{ - 2}}{{\text{A}}^{ - 2}}] \\
$
These dimensions can be performed in each option to verify them, we should still get the same answer.