Question

What is the dimension of magnetic induction?

Answer 1

Step 1: Definition

Magnetic induction would be the occurrence of generating electromotive force (e.m.f) in a conductor due to a change throughout the magnetic flux.

Step 2: Dimensional formula

We know the formula,

$\mathrm{Lorentz}\mathrm{Force}\left(\mathrm{F}\right)=\mathrm{Charge}\times \mathrm{Magnetic}\mathrm{Field}\left(\mathrm{B}\right)\times \mathrm{Velocity}$

Therefore,

$\mathrm{Magnetic}\mathrm{Field}\left(\mathrm{B}\right)=\mathrm{Lorentz}\mathrm{Force}\times {[\mathrm{Charge}\times \mathrm{Velocity}]}^{-1}$ …(equation $1$)

We know that

$\mathrm{Velocity}=\mathrm{Distance}\times {\mathrm{Time}}^{-1}$

∴ The dimensional formula of velocity is:

$\left[{\mathrm{LT}}^{-1}\right]$ …(equation $2$)

Also,

$\mathrm{Electric}\mathrm{charge}=\mathrm{electric}\mathrm{current}\times \mathrm{time}$

∴ The dimensional formula of electric charge is:

$\left[{\mathrm{A}}^1{\mathrm{T}}^1\right]$…(equation $3$)

And,

$$\begin{gathered} \mathrm{Lorentz}\mathrm{Force}=\mathrm{Mass}\times \mathrm{Acceleration} \\ =\mathrm{M}\times \left[{\mathrm{LT}}^{-2}\right] \end{gathered}$$

∴ The dimensional formula of Lorentz Force is:

$\left[{\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-2}\right]$ …(equation $4$)

Now substituting the equation $2,3,$ and $4$ in the equation $1$ we get,

$\mathrm{B}=\left[{\mathrm{M}}^1{\mathrm{L}}^1{\mathrm{T}}^{-2}\right]\times {\left[{\mathrm{A}}^1{\mathrm{T}}^1\right]}^{-1}\times {\left[{\mathrm{LT}}^{-1}\right]}^{-1}=\left[{\mathrm{M}}^1{\mathrm{L}}^0{\mathrm{T}}^{-2}{\mathrm{A}}^{-1}\right]$

Therefore, the dimensional formula of magnetic induction will be $\left[{\mathrm{M}}^1{\mathrm{T}}^{-2}{\mathrm{A}}^{-1}\right]$.