Animated Solution for Physics - Physics and Measurement: Write the dimensions of the following in terms of mass, time, length and charge. (a) Magnetic flux (b) Rigidity modulus

Dimensional analysis is a powerful tool in physics that allows us to understand the fundamental nature of physical quantities. In this problem, we are tasked with finding the dimensions of magnetic flux and rigidity modulus in terms of mass (M), length (L), time (T), and charge (Q).

Magnetic flux, denoted by $\varphi$, is a measure of the total magnetic field $B$ passing through a given area $s$. Mathematically, it is expressed as:

To find the dimensions of $\varphi$, we first need the dimensions of the magnetic field $B$. We can derive this from the Lorentz force equation for a moving charge:

Rearranging for $B$, we get:

Substituting this back into our flux equation gives us a formula entirely in terms of quantities whose dimensions we know:

Now, let's substitute the dimensional formulas for each of these fundamental quantities. We know that force $[F]=[{\text{MLT}}^{-2}]$, area $[s]=[{\text{L}}^2]$, charge $[q]=[\text{Q}]$, and velocity $[v]=[{\text{LT}}^{-1}]$. Plugging these into our equation yields:

Simplifying the expression by canceling out common terms, we arrive at the final dimension for magnetic flux:

Next, let's determine the dimensions of the rigidity modulus. The rigidity modulus, like any modulus of elasticity, is defined as the ratio of shear stress to shear strain:

Since strain is a ratio of lengths (change in dimension over original dimension), it is a dimensionless quantity. Therefore, the dimensions of the rigidity modulus are identical to those of stress. Stress is defined as force per unit area:

Substituting the dimensions for force $[{\text{MLT}}^{-2}]$ and area $[{\text{L}}^2]$, we get:

And there we have it! By breaking down complex quantities into their fundamental components, dimensional analysis provides a clear and systematic path to the solution.