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

Step 1: Write dimensional formulae
The dimensional formula of magnetic field intensity is: \[ [H] = [M L^{0} T^{-2} A^{-1}] \] The dimensional formulae of other quantities are: \[ [\epsilon] = [M^{-1} L^{-3} T^{4} A^{2}] \] \[ [E] = [M L T^{-3} A^{-1}] \] \[ [x] = [L] \] \[ [t] = [T] \] Step 2: Apply dimensional equation
Given, \[ H = \frac{\epsilon^r E^p x^q}{t^s} \] Taking dimensions on both sides: \[ [H] = \frac{[\epsilon]^r [E]^p [x]^q}{[t]^s} \] Substituting dimensions: \[ [M L^{0} T^{-2} A^{-1}] = \frac{ [M^{-1} L^{-3} T^{4} A^{2}]^r [M L T^{-3} A^{-1}]^p [L]^q }{ [T]^s } \] \[ = M^{-r+p} L^{-3r+p+q} T^{4r-3p-s} A^{2r-p} \] Step 3: Compare powers
Comparing powers of $M, L, T,$ and $A$: For $M$: \[ -r+p=1 \] \[ p-r=1 \tag{1} \] For $A$: \[ 2r-p=-1 \tag{2} \] For $L$: \[ -3r+p+q=0 \tag{3} \] For $T$: \[ 4r-3p-s=-2 \tag{4} \] Step 4: Solve equations
Adding (1) and (2): \[ (p-r)+(2r-p)=1+(-1) \] \[ r=0 \] Substitute in (1): \[ p-0=1 \] \[ p=1 \] Substitute in (3): \[ -3(0)+1+q=0 \] \[ q=-1 \] Substitute in (4): \[ 4(0)-3(1)-s=-2 \] \[ -3-s=-2 \] \[ s=1 \] Step 5: Final Answer
\[ \boxed{r=0,\; p=1,\; q=-1,\; s=1} \] Hence, option (1) is correct.