Question

The dimension of electrical conductivity is___________
A). \[[{M^{ - 2}}{L^{ - 4}}{T^3}{A^2}]\]
B). \[[{M^{ - 1}}{L^{ - 2}}{T^3}{A^2}]\]
C). \[[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}]\]
D). \[[{M^{ - 2}}{L^{ - 3}}{T^6}{A^2}]\]

Answer 1

Hint: As we all know that the electrical conductivity is the reciprocal of the electrical resistance. So, the dimension for the electrical conductivity can be derived from the dimension of the reciprocal of the electrical resistance.

Complete step-by-step solution:
We can write the electrical conductivity as follows,
Electrical conductivity $ = C = \dfrac{1}{R}$, where $R = $electrical resistance and $C = $electrical conductivity.
Also, from the ohm’s law as we know that the voltage and the current are directly proportional to each other when the resistance is constant.
i.e. $V = IR$
where $V = $voltage
$I = $current and $R = $electrical resistance.
So from the ohm’s law we can write the value of electrical resistance $R$ in terms of current $I$and voltage $V$, which is as follows,
$R = \dfrac{V}{I}$
So now the formula for the electrical conductivity will be,
$C = \dfrac{1}{{\left[ {\dfrac{V}{I}} \right]}}$
$ \Rightarrow C = \dfrac{I}{V}$-------equation (1)
In the above equation we have voltage and current, and we also know that,
The voltage can be written as work done per unit charge,
$V = \dfrac{W}{Q}$, where $W = $work done to move the charge and $Q = $charge which is being moved by the work $W$.
So now putting this value of the voltage in equation (1)
$C = \dfrac{{IQ}}{W}$---equation (2)
Also, we know that the charge is the product of the current and time so we can write like,
$Q = It$
And also, the work done is the product of force and distance, that is
$W = Fs$
Now we will write the dimensions of each quantity separately.
Dimension of the work done = dimension of the force multiplied by dimension of the distance= $\left[ {ML{T^{ - 2}}.L} \right] = \left[ {M{L^2}{T^{ - 2}}} \right]$
Dimension of the current $ = \left[ A \right]$
Dimension of the charge $ = \left[ {A.T} \right]$
Now putting all these values in the equation (2), we get,
Dimension of the electrical conductivity, $C = \dfrac{{\left[ A \right]\left[ {A.T} \right]}}{{\left[ {M{L^2}{T^{ - 2}}} \right]}}$
$ = [{M^{ - 1}}{L^{ - 2}}{A^2}{T^{1 - ( - 2)}}]$
$ = [{M^{ - 1}}{L^{ - 2}}{A^2}{T^3}]$
$ = [{M^{ - 1}}{L^{ - 2}}{T^3}{A^2}]$
So, the dimension of the electrical conductivity is $[{M^{ - 1}}{L^{ - 2}}{T^3}{A^2}]$
Hence option (B) is the correct answer.

Note: Here is the mention of the electrical conductivity so we should have a clear idea about the electrical conductivity other than the formula for the electrical conductivity. Since we know the formula for the electrical conductivity that it is reciprocal of the electrical resistance. But also, in the other words, the electrical conductivity is the measurement of the ability of the particular material to allow the transport of an electric charge. Its SI unit is siemens per meter (S/m). It is also the ratio of current density and electric field strength.