Introduction / Context:
Dimensional analysis in electromagnetism relates electrical quantities to base dimensions M (mass), L (length), T (time), and I (current). Matching correct dimensions helps validate formulas and perform unit checks across problems.
Given Data / Assumptions:
- A: Resistance (ohm = V/A).
- B: Inductance (henry = V*s/A).
- C: Capacitance (farad = C/V).
- D: Reluctance (magnetic) is analogous to 1/inductance in dimensional terms.
Concept / Approach:
Use base relations: V = W/Q = (J/s)/A = (M L^2 T^-3)/I; then compute each quantity by substitution and simplification.
Step-by-Step Solution:
Voltage dimension: V → M L^2 T^-3 I^-1.A (Resistance R) = V / I → M L^2 T^-3 I^-2 → match 4.B (Inductance L) = (V * s) / I → (M L^2 T^-3 I^-1 * T) / I = M L^2 T^-2 I^-2 → match 2.C (Capacitance C) = Q / V = (I * T) / (M L^2 T^-3 I^-1) = M^-1 L^-2 T^4 I^2 → match 3.D (Reluctance S) ≈ 1 / inductance → (M L^2 T^-2 I^-2)^-1 = M^-1 L^-2 T^2 I^2 → match 1.
Verification / Alternative check:
Cross-check using energy in inductor W = (1/2) L I^2 → L = 2W / I^2; since W has dimensions M L^2 T^-2, L → M L^2 T^-2 I^-2, consistent with above.
Why Other Options Are Wrong:
- A-1, B-2, C-3, D-4: Assigns reluctance and resistance incorrectly.
- A-2, B-1, C-3, D-4: Misplaces resistance and inductance; reluctance not 4.
- A-2, B-4, C-3, D-1: Gives an impossible dimension (4) to inductance.
Common Pitfalls:
- Forgetting that resistance involves T^-3 (from V) rather than T^-2.
- Confusing magnetic reluctance with electrical resistance; they are analogous but dimensionally inverse to inductance.
Final Answer:
A-4, B-2, C-3, D-1
Discussion & Comments