Consider the equation H = (x^p ^q E^r)/(t^s) , where H= magnetic...

The dimensional formulas for the given physical quantities are:

Magnetic field intensity $[H] = [M^0 L^{-1} T^0 A^1]$

Distance $[x] = [M^0 L^1 T^0 A^0]$

Permittivity $[\epsilon] = [M^{-1} L^{-3} T^4 A^2]$

Electric field $[E] = [M^1 L^1 T^{-3} A^{-1}]$

Time $[t] = [M^0 L^0 T^1 A^0]$

Given the equation $H = \dfrac{x^p \epsilon^q E^r}{t^s}$, we can write it in terms of dimensions as:

$[M^0 L^{-1} T^0 A^1] = [L]^p [M^{-1} L^{-3} T^4 A^2]^q [M L T^{-3} A^{-1}]^r [T]^{-s}$

Combining the powers of $M, L, T$, and $A$ on the right side:

$[M^0 L^{-1} T^0 A^1] = [M^{-q+r} L^{p-3q+r} T^{4q-3r-s} A^{2q-r}]$

Equating the powers of corresponding fundamental quantities from both sides:

For $M$: $0 = -q + r \implies q = r$

For $A$: $1 = 2q - r$

Substituting $r = q$ into the equation for $A$:

$1 = 2q - q \implies q = 1$

Since $q = r$, we get $r = 1$.

For $L$: $-1 = p - 3q + r$

Substituting $q = 1$ and $r = 1$:

$-1 = p - 3(1) + 1 \implies -1 = p - 2 \implies p = 1$

For $T$: $0 = 4q - 3r - s$

Substituting $q = 1$ and $r = 1$:

$0 = 4(1) - 3(1) - s \implies 0 = 1 - s \implies s = 1$

Thus, the values are $p = 1$, $q = 1$, $r = 1$, and $s = 1$.

Answer: $1, 1, 1, 1$