Consider the equation $H = \dfrac{x^p \epsilon^q E^r}{t^s}$
Where $H =$ magnetic field; $E =$ electric field, $\epsilon =$ permittivity, $x =$ distance, $t =$ time. The values of $p, q, r$ and $s$ respectively are :
Answer: (A) $1, 1, 1, 1$
Concept. Equate the dimensions of both sides. With four unknown powers we need four base dimensions: $M$, $L$, $T$ and $A$.
Step 1. Dimensions of each symbol.
$$[H]=\frac{[B]}{[\mu_0]}=L^{-1}A,\quad[\varepsilon]=M^{-1}L^{-3}T^4A^2,\quad[E]=MLT^{-3}A^{-1},\quad[x]=L,\quad[t]=T$$
Step 2. Equate powers in $L^{-1}A=L^p\,(M^{-1}L^{-3}T^4A^2)^q\,(MLT^{-3}A^{-1})^r\,T^{-s}$:
- $M$: $-q+r=0$
- $A$: $2q-r=1$
- $L$: $p-3q+r=-1$
- $T$: $4q-3r-s=0$
Step 3. Solve. From $M$: $r=q$. Then $A$: $q=1$, so $r=1$. $L$: $p-3+1=-1\Rightarrow p=1$. $T$: $4-3-s=0\Rightarrow s=1$.
Answer. $p=q=r=s=1$ (Option A). Indeed $\varepsilon E/t$ is a displacement current density, and multiplying by a length gives $H$ (Ampère's law).
Solution by Sreeraj P, M.Sc Physics
