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

The correct option is: 10x x̂ + 20y ŷ - 30z ẑ

To determine which vector function represents a physically possible magnetic field $\overrightarrow{B}$, we must apply Maxwell's second equation (also known as Gauss's law for magnetism). This fundamental law of electromagnetism states that magnetic monopoles do not exist, which mathematically translates to the divergence of any magnetic field being zero everywhere:

$\nabla ·\overrightarrow{B}=0$

For a vector field $\overrightarrow{B}=B_x\hat{x}+B_y\hat{y}+B_z\hat{z}$, the divergence in Cartesian coordinates is given by:

$\nabla ·\overrightarrow{B}=\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}$

Let us evaluate the divergence for the correct option, $\overrightarrow{B}=10x\hat{x}+20y\hat{y}-30z\hat{z}$:

Here, the components of the field are:
$B_x=10x$
$B_y=20y$
$B_z=-30z$

Now, we compute the partial derivatives:
$\frac{\partial B_x}{\partial x}=\frac{\partial }{\partial x}\left(10x\right)=10$
$\frac{\partial B_y}{\partial y}=\frac{\partial }{\partial y}\left(20y\right)=20$
$\frac{\partial B_z}{\partial z}=\frac{\partial }{\partial z}(-30z)=-30$

Substituting these values back into the divergence formula:
$\nabla ·\overrightarrow{B}=10+20-30=0$

Since the divergence of this vector field is zero, it represents a physically possible magnetic field.