Magnetic Field Vector Function Verification

To determine which vector function represents a magnetic field \(\vec{B}\), we need to use a fundamental principle from electromagnetism derived from Maxwell's equations. Specifically, Gauss's law for magnetism states that the magnetic field is solenoidal, meaning its divergence is always zero.

The condition is:

$$ \nabla \cdot \vec{B} = 0 $$

Let's check this condition for each of the given options.

Analyzing Vector Function Options

The divergence of a vector field \(\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}\) is calculated as:

$$ \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} $$

Option 1 Analysis: \( \vec{B}_1 = 10x \hat{i} - 30z \hat{j} + 20y \hat{k} \)

• Components: \( F_x = 10x \), \( F_y = -30z \), \( F_z = 20y \)
• Calculate partial derivatives:
  • \( \frac{\partial F_x}{\partial x} = \frac{\partial (10x)}{\partial x} = 10 \)
  • \( \frac{\partial F_y}{\partial y} = \frac{\partial (-30z)}{\partial y} = 0 \)
  • \( \frac{\partial F_z}{\partial z} = \frac{\partial (20y)}{\partial z} = 0 \)
• Calculate divergence: \( \nabla \cdot \vec{B}_1 = 10 + 0 + 0 = 10 \)
• Since \( \nabla \cdot \vec{B}_1 \neq 0 \), this vector function cannot represent a magnetic field.

Option 2 Analysis: \( \vec{B}_2 = 10y \hat{i} + 20x \hat{j} - 10z \hat{k} \)

• Components: \( F_x = 10y \), \( F_y = 20x \), \( F_z = -10z \)
• Calculate partial derivatives:
  • \( \frac{\partial F_x}{\partial x} = \frac{\partial (10y)}{\partial x} = 0 \)
  • \( \frac{\partial F_y}{\partial y} = \frac{\partial (20x)}{\partial y} = 0 \)
  • \( \frac{\partial F_z}{\partial z} = \frac{\partial (-10z)}{\partial z} = -10 \)
• Calculate divergence: \( \nabla \cdot \vec{B}_2 = 0 + 0 + (-10) = -10 \)
• Since \( \nabla \cdot \vec{B}_2 \neq 0 \), this vector function cannot represent a magnetic field.

Option 3 Analysis: \( \vec{B}_3 = 10x \hat{i} + 20y \hat{j} - 30z \hat{k} \)

• Components: \( F_x = 10x \), \( F_y = 20y \), \( F_z = -30z \)
• Calculate partial derivatives:
  • \( \frac{\partial F_x}{\partial x} = \frac{\partial (10x)}{\partial x} = 10 \)
  • \( \frac{\partial F_y}{\partial y} = \frac{\partial (20y)}{\partial y} = 20 \)
  • \( \frac{\partial F_z}{\partial z} = \frac{\partial (-30z)}{\partial z} = -30 \)
• Calculate divergence: \( \nabla \cdot \vec{B}_3 = 10 + 20 + (-30) = 10 + 20 - 30 = 0 \)
• Since \( \nabla \cdot \vec{B}_3 = 0 \), this vector function satisfies the condition for a magnetic field.

Option 4 Analysis: \( \vec{B}_4 = 10z \hat{i} + 20y \hat{j} - 30x \hat{k} \)

• Components: \( F_x = 10z \), \( F_y = 20y \), \( F_z = -30x \)
• Calculate partial derivatives:
  • \( \frac{\partial F_x}{\partial x} = \frac{\partial (10z)}{\partial x} = 0 \)
  • \( \frac{\partial F_y}{\partial y} = \frac{\partial (20y)}{\\partial y} = 20 \)
  • \( \frac{\partial F_z}{\partial z} = \frac{\partial (-30x)}{\partial z} = 0 \)
• Calculate divergence: \( \nabla \cdot \vec{B}_4 = 0 + 20 + 0 = 20 \)
• Since \( \nabla \cdot \vec{B}_4 \neq 0 \), this vector function cannot represent a magnetic field.

Conclusion on Magnetic Field Representation

By applying the divergence test (\( \nabla \cdot \vec{B} = 0 \)), we found that only the vector function in Option 3 satisfies this necessary condition for a magnetic field. Therefore, the vector function representing a magnetic field is \( 10x \hat{i} + 20y \hat{j} - 30z \hat{k} \).