Consider the equation \(H = \frac{x^p \varepsilon^q E^r}{t^s}\), where \(H\) = magnetic field, \(E\) = electric field, \(\varepsilon\) = permittivity, \(x\) = distance, \(t\) = time. The values of \(p, q, r\) and \(s\) respectively are:
Step 1: Understanding the Concept:
We must perform a dimensional analysis of the given equation to find the exponents $p, q, r, s$. By substituting the dimensional formulas for each physical quantity and equating the powers of the fundamental dimensions ($M, L, T, A$) on both sides, we can solve for the unknowns. Step 2: Key Formula or Approach:
Identify the dimensions:
Magnetic Field Intensity ($H$): $[M^0 L^{-1} T^0 A^1]$
Permittivity ($\epsilon$): $[M^{-1} L^{-3} T^4 A^2]$
Electric Field ($E$): $[M^1 L^1 T^{-3} A^{-1}]$
Distance ($x$): $[L^1]$
Time ($t$): $[T^1]$
(Note: Often $H$ refers to Magnetic Field Intensity rather than Magnetic Flux Density $B$. The Ampere-Maxwell law directly relates $H$ to $\frac{\epsilon E}{t}$). Step 3: Detailed Explanation:
The given equation is $H = x^p \epsilon^q E^r t^{-s}$.
Substitute the dimensional formulas:
$[M^0 L^{-1} T^0 A^1] = [L]^p [M^{-1} L^{-3} T^4 A^2]^q [M^1 L^1 T^{-3} A^{-1}]^r [T]^{-s}$
Combine the powers of $M, L, T, A$ on the right hand side:
$[M^0 L^{-1} T^0 A^1] = M^{-q+r} L^{p-3q+r} T^{4q-3r-s} A^{2q-r}$.
Now, equate the exponents on both sides:
For M: $0 = -q + r \implies q = r$.
For A: $1 = 2q - r$.
Substitute $r = q$ into the A equation:
$1 = 2q - q \implies q = 1$.
Since $q = r$, we get $r = 1$.
For L: $-1 = p - 3q + r$.
Substitute $q = 1$ and $r = 1$:
$-1 = p - 3(1) + 1 \implies -1 = p - 2 \implies p = 1$.
For T: $0 = 4q - 3r - s$.
Substitute $q = 1$ and $r = 1$:
$0 = 4(1) - 3(1) - s \implies 0 = 1 - s \implies s = 1$.
Therefore, the values are $p=1, q=1, r=1, s=1$. Step 4: Final Answer:
The values of $p, q, r, s$ are $1, 1, 1, 1$.
