An infinitely long cylinder of radius R carries a frozen -in magnetization
M⃗⃗ = ke-sẑ, where k is a constant and s is the distance from the axis of cylinder.
The magnetic permeability of free space is \mu0. There is no free current present
anywhere. The magnetic flux density (B⃗ ) inside the cylinder is
Options:
(A) 0
(B) \mu0ke-Rẑ
(C) \mu0ke-sẑ
(D) \mu0ke-s (R
s ) ẑ
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.20)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 3: Electromagnetic Theory* | **Topic**: *Electrostatics and Magnetostatics* (`Method of Images & Multipole Expansion`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Electrostatics and Magnetostatics** (`Method of Images & Multipole Expansion`).
**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): `𝜇0ke-Rẑ`**.
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