Question

A quantity X is given by ${{{\varepsilon }}_{{o}}}{{L}}\dfrac{{{{\Delta V}}}}{{{{\Delta T}}}}$where ${\varepsilon _ \circ }$is the permittivity of the free space, L is length. $\Delta V$ is the potential difference and $\Delta T$ is the time interval. The dimensional formula for X is the same as that of
A) Resistance
B) Charge
C) Voltage
D) Current

Answer 1

Hint: Reduce the given formula of X to the maximum limit where we may get definite quantity and then find dimensional formula of that quantity to compare with the formula given by
${{X = }}{{{\varepsilon }}_{{o}}}{{L}}\dfrac{{{{\Delta V}}}}{{{{\Delta T}}}}$
Where,
${\varepsilon _ \circ }$=permittivity of the free space
L= length,
$\Delta V$ = potential difference
$\Delta T$= time interval
After converting we can easily compare the dimensional formula of X with the given options.

Complete step by step solution:
According to question
Now, firstly we will try to reduce the formula of X into the formula of a dimensionally valid quantity.
${{X = }}{{{\varepsilon }}_{{o}}}{{L}}\dfrac{{{{\Delta V}}}}{{{{\Delta T}}}}$
In the above equation the term ${\varepsilon _ \circ }L$ represent the capacitance because we know that
\[{\varepsilon _ \circ }L{{ }} = C\;\]
Now, on putting value of ${\varepsilon _ \circ }L$ in formula of X, we get
 ${{X = C}}\dfrac{{{{\Delta V}}}}{{{{\Delta T}}}}$
The term $C\Delta V$ can be replaced by charge $\Delta Q$because we know that
${{C\Delta V = \Delta Q}}$
Now, putting value of$C\Delta V$ in formula of X we get
\[{{X = }}\dfrac{{{{\Delta Q}}}}{{{{\Delta T}}}}\]
We know that $\dfrac{{{{\Delta Q}}}}{{{{\Delta T}}}}$ is equal to current.
 $\therefore {{\;X = }}\dfrac{{{{\Delta Q}}}}{{{{\Delta T}}}}{{ = }} \Rightarrow X{{ = I}}$
Where ${{I}}$ stands for current


$\therefore $ Quantity $X$ is dimensionally equal to the current. Option (D) is correct.

Note:
We can also solve this question by using the dimensions of each quantity and then reducing it by putting in the formula of X. In this way we get the dimension of X. then also we need to find the dimension of each quantity given in the option. After all this we finally compare the dimension of X with any other physical quantity.