Question

Which of the following units denotes the dimensions ${\rm{M}}{{\rm{L}}^{\rm{2}}}{\rm{/}}{{\rm{Q}}^{\rm{2}}}$, where Q denotes the electric charge.
A. Weber (Wb)
B. Wb/m$^2$
C. Henry (H)
D. H/m²

Answer 1

Hint: We know that mainly seven fundamental dimensions represent all the physical quantities. The primary three fundamental dimensions are M, L, T. Apart from its charge it has a fundamental dimension called Q.

Complete step by step solution:
We know that weber is the unit of magnetic flux and it has dimensions $\left[ {M{L^2}{T^{ - 1}}{Q^{ - 1}}} \right]$
We can derive weber per meter square as Wb/m2,
\[\begin{array}{l}
\dfrac{{Wb}}{{{m^2}}} = \left[ {\dfrac{{M{L^2}{T^{ - 1}}{Q^{ - 1}}}}{{{A^2}}}} \right]\\
 = M{L^2}{T^{ - 1}}{Q^{ - 1}}{A^{ - 2}}
\end{array}\]

We know that Henry is the unit of inductance, and it can be written as,
$H = \dfrac{{BA}}{I}$
Here, H is Henry, B is the magnetic field, A is the area, and I current.
We know that the dimensional formula for B is $M{T^{ - 1}}{Q^{ - 1}}$ , and A is ${L^2}$ , and I is $Q{T^{ - 1}}$. We will substitute all these in the above equation to find the value of H.
$\begin{array}{l}
H = \dfrac{{\left( {M{T^{ - 1}}{Q^{ - 1}}} \right)\left( {{L^2}} \right)}}{{Q{T^{ - 1}}}}\\
 = \left[ {M{L^2}{Q^{ - 2}}} \right]
\end{array}$
We can find the dimensional formula for H/m2 as,
$\begin{array}{l}
\dfrac{H}{{{m^2}}} = \dfrac{{\left[ {M{L^2}{Q^{ - 2}}} \right]}}{{{L^2}}}\\
\dfrac{H}{{{m^2}}} = M{L^2}{Q^{ - 2}}{L^{ - 2}}
\end{array}$

Therefore, the dimensions $\dfrac{{M{L^2}}}{{{Q^2}}}$ are for Henry, and the correct option is C.

Note: There are various fundamental units in which there are no dimensions and are dimensionless. Therefore, the dimensions of those physical quantities are defined by ${M^0}{L^0}{T^0}$ it means that it has no mass, no length, and no time as a physical quantity. Apart from fundamental physical quantities, there are some derived physical quantities also for ex-speed, work, force, etc. and they all have the terms M, L, and T. For example, work is defined as distance multiplied by force. It has the dimensions of $M{L^2}{T^{ - 2}}$. Since work and energy are analogous, and hence energy also has the same dimensions.