Dimensional Analysis of Formula Relating Magnetic and Electric Fields

To find the values of $p, q, r,$ and $s$, we perform dimensional analysis on the given formula:

$H = \frac{x^p \epsilon^q E^r}{t^s}$

First, let us write the dimensional formulas for each physical quantity in terms of mass ($\text{M}$), length ($\text{L}$), time ($\text{T}$), and electric current ($\text{I}$):

1. Magnetic field strength ($H$): Measured in amperes per meter ($\text{A/m}$) $[H] = \text{M}^0 \text{L}^{-1} \text{T}^0 \text{I}^1$

2. Distance ($x$): $[x] = \text{M}^0 \text{L}^1 \text{T}^0 \text{I}^0$

3. Permittivity ($\epsilon$): $[\epsilon] = \text{M}^{-1} \text{L}^{-3} \text{T}^4 \text{I}^2$

4. Electric field ($E$): $[E] = \text{M}^1 \text{L}^1 \text{T}^{-3} \text{I}^{-1}$

5. Time ($t$): $[t] = \text{M}^0 \text{L}^0 \text{T}^1 \text{I}^0$

Substituting these dimensional formulas into the given equation:

$[H] = [x]^p [\epsilon]^q [E]^r [t]^{-s}$

$\text{M}^0 \text{L}^{-1} \text{T}^0 \text{I}^1 = (\text{L})^p (\text{M}^{-1} \text{L}^{-3} \text{T}^4 \text{I}^2)^q (\text{M}^1 \text{L}^1 \text{T}^{-3} \text{I}^{-1})^r (\text{T})^{-s}$

Combining the exponents for each base dimension on the right-hand side:

$\text{M}^0 \text{L}^{-1} \text{T}^0 \text{I}^1 = \text{M}^{-q + r} \cdot \text{L}^{p - 3q + r} \cdot \text{T}^{4q - 3r - s} \cdot \text{I}^{2q - r}$

By comparing the exponents of fundamental dimensions on both sides of the equation, we obtain the following system of linear equations:

1. For Mass ($\text{M}$): $-q + r = 0 \implies r = q$

2. For Current ($\text{I}$): $2q - r = 1$ Substituting $r = q$: $2q - q = 1 \implies q = 1$ Thus, $r = 1$.

3. For Length ($\text{L}$): $p - 3q + r = -1$ Substituting $q = 1$ and $r = 1$: $p - 3(1) + 1 = -1 \implies p - 2 = -1 \implies p = 1$

4. For Time ($\text{T}$): $4q - 3r - s = 0$ Substituting $q = 1$ and $r = 1$: $4(1) - 3(1) - s = 0 \implies 1 - s = 0 \implies s = 1$

Therefore, the values of $p, q, r,$ and $s$ are respectively: $(p, q, r, s) = (1, 1, 1, 1)$

This corresponds to Option A.