GATE PH 2016 • Q12
MCQ • 1 Mark
Section 3: Electromagnetic Theory
Electrostatics and Magnetostatics
Subtopic: Magnetic Vector Potential & Boundary Conditions
Question Statement:
Which of the following magnetic vector potentials gives rise to a uniform magnetic field $B_0\hat{k}$?
Options:
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(A) $B_0 z \hat{k}$
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(B) $-B_0 x \hat{j}$
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(C) $\frac{B_0}{2}(-y\hat{i} + x\hat{j})$
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(D) $\frac{B_0}{2}(y\hat{i} + x\hat{j})$
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2016 Q.12 / PH Q.2)**
**1. Magnetic Vector Potential & Curl Formulation**:
- The magnetic field is related to the vector potential by $\vec{B} = \nabla \times \vec{A}$.
- We require $\vec{B} = B_0 \hat{k}$.
**2. Evaluating the Options**:
- For $\vec{A} = \frac{B_0}{2} (-y \hat{i} + x \hat{j})$:
$\nabla \times \vec{A} = \hat{k} \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) = \hat{k} \left(\frac{B_0}{2} - \left(-\frac{B_0}{2}\right)\right) = B_0 \hat{k}$
- This identically yields the uniform magnetic field $B_0 \hat{k}$.
**3. Conclusion**:
- **Official Master Answer**: **C**
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