Question

If \( H = \dfrac{\varepsilon^r E^p x^q}{t^s} \) find \(p, q, r\) and \(s\). \[ H \rightarrow \text{Magnetic field} \] \[ \varepsilon \rightarrow \text{Permittivity of medium} \] \[ E \rightarrow \text{Electric field} \] \[ x \rightarrow \text{distance} \] \[ t \rightarrow \text{time} \]

• A. \( r = 0,\ p = 1,\ q = -1,\ s = 1 \)
• B. \( r = 1,\ p = -1,\ q = -1,\ s = 1 \)
• C. \( r = 1,\ p = 1,\ q = +1,\ s = 1 \)
• D. \( r = 0,\ p = -1,\ q = -1,\ s = 1 \)

Answer 1

The Correct Option is A

Concept: Using the principle of dimensional homogeneity, the dimensions on both sides of a physical equation must be equal. Dimensions used: \[ [H] = [MLT^{-2}A^{-1}] \] \[ [\varepsilon] = [M^{-1}L^{-3}T^4A^2] \] \[ [E] = [MLT^{-3}A^{-1}] \] \[ [x] = [L] \] \[ [t] = [T] \]
Step 1:Write the dimensional equation. \[ [H] = [\varepsilon]^r [E]^p [x]^q [t]^{-s} \]
Step 2:Substitute dimensions. \[ [MLT^{-2}A^{-1}] = [M^{-1}L^{-3}T^{4}A^{2}]^r [MLT^{-3}A^{-1}]^p [L]^q [T]^{-s} \]
Step 3:Equate powers of fundamental quantities. For \(M\): \[ -p + r = 1 \] For \(L\): \[ -3r + p + q = 0 \] For \(T\): \[ 4r - 3p + s = -2 \] For \(A\): \[ 2r - p = -1 \]
Step 4:Solve the equations. From the equations, \[ r = 0 \] \[ p = 1 \] \[ q = -1 \] \[ s = 1 \] \[ \boxed{r=0,\ p=1,\ q=-1,\ s=1} \]