Question

Which one of the following vector functions represents a magnetic field $\vec{B}$?   ($\hat{x}$, $\hat{y}$, and $\hat{z}$ are unit vectors along x-axis, y-axis, and z-axis, respectively)

• A. $10x\hat{x} + 20y\hat{y} - 30z\hat{z}$
• B. $10y\hat{x} + 20x\hat{y} - 10z\hat{z}$
• C. $10z\hat{x} + 20y\hat{y} - 30x\hat{z}$
• D. $10x\hat{x} - 30z\hat{y} + 20y\hat{z}$

Hint:

Always check $\nabla \cdot \vec{B} = 0$ to verify if a vector field can represent a magnetic field.

Answer 1

The Correct Option is A

Step 1: Use $\nabla \cdot \vec{B} = 0$.
For a magnetic field, divergence must be zero. Compute divergence for each option: 
Option (A): $\frac{\partial}{\partial x}(10x) + \frac{\partial}{\partial y}(20y) + \frac{\partial}{\partial z}(-30z)$ = $10 + 20 - 30 = 0$ ✓ 
Option (B): $0 + 0 - 10 \neq 0$ ✗ 
Option (C): $0 + 20 + 0 \neq 0$ ✗ 
Option (D): $10 + 0 + 0 \neq 0$ ✗ 
 

Step 2: Conclusion. 
Only option (A) satisfies the divergence-free condition for magnetic fields.