Question:
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:
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:
Updated On: Apr 13, 2026
- 1, 1, 1, 1
- -1, 1, 2, 1
- 1, -1, -2, 1
- -1, -2, -2, 1
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
We use dimensional analysis to find the powers $p, q, r,$ and $s$. We equate the dimensions of the Left Hand Side (LHS) with the Right Hand Side (RHS).
Step 2: Key Formula or Approach:
Dimensions: - Magnetic field intensity (\(H\)): \([L^{-1} A]\) - Electric field (\(E\)): \([M L T^{-3} A^{-1}]\) - Permittivity (\(\varepsilon\)): \([M^{-1} L^{-3} T^4 A^2]\) - Distance (\(x\)): \([L]\), Time (\(t\)): \([T]\)
Step 3: Detailed Explanation:
Equating dimensions: \[ [L^{-1} A] = \frac{[L]^p [M^{-1} L^{-3} T^4 A^2]^q [M L T^{-3} A^{-1}]^r}{[T]^s} \] \[ [L^{-1} A] = M^{-q+r} L^{p-3q+r} T^{4q-3r-s} A^{2q-r} \] Comparing powers of $A$: \(2q - r = 1\) Comparing powers of $M$: \(-q + r = 0 \implies q = r\) Substituting \(q = r\) into the $A$ equation: \(2r - r = 1 \implies r = 1, q = 1\). Comparing powers of $L$: \(p - 3(1) + 1 = -1 \implies p - 2 = -1 \implies p = 1\). Comparing powers of $T$: \(4(1) - 3(1) - s = 0 \implies 1 - s = 0 \implies s = 1\).
Step 4: Final Answer:
The values are \(p=1, q=1, r=1, s=1\).
We use dimensional analysis to find the powers $p, q, r,$ and $s$. We equate the dimensions of the Left Hand Side (LHS) with the Right Hand Side (RHS).
Step 2: Key Formula or Approach:
Dimensions: - Magnetic field intensity (\(H\)): \([L^{-1} A]\) - Electric field (\(E\)): \([M L T^{-3} A^{-1}]\) - Permittivity (\(\varepsilon\)): \([M^{-1} L^{-3} T^4 A^2]\) - Distance (\(x\)): \([L]\), Time (\(t\)): \([T]\)
Step 3: Detailed Explanation:
Equating dimensions: \[ [L^{-1} A] = \frac{[L]^p [M^{-1} L^{-3} T^4 A^2]^q [M L T^{-3} A^{-1}]^r}{[T]^s} \] \[ [L^{-1} A] = M^{-q+r} L^{p-3q+r} T^{4q-3r-s} A^{2q-r} \] Comparing powers of $A$: \(2q - r = 1\) Comparing powers of $M$: \(-q + r = 0 \implies q = r\) Substituting \(q = r\) into the $A$ equation: \(2r - r = 1 \implies r = 1, q = 1\). Comparing powers of $L$: \(p - 3(1) + 1 = -1 \implies p - 2 = -1 \implies p = 1\). Comparing powers of $T$: \(4(1) - 3(1) - s = 0 \implies 1 - s = 0 \implies s = 1\).
Step 4: Final Answer:
The values are \(p=1, q=1, r=1, s=1\).
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