GATE PH 2023 • Q12
MCQ • 1 Mark
Section 1: Mathematical Physics
Differential equations
Subtopic: First and Second Order ODEs
Question Statement:
Which one of the following is a dimensionless constant?
Options:
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(A) Permittivity of free space
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(B) Permeability of free space
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(C) Bohr magneton
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(D) Fine structure constant
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.12)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 1: Mathematical Physics* | **Topic**: *Linear Algebra & Vector Calculus* (`Matrices, Eigenvalues & Eigenvectors`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Linear Algebra & Vector Calculus** (`Matrices, Eigenvalues & Eigenvectors`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `Permeability of free space`**.
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