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.

According to Gauss's law for magnetism (one of Maxwell's equations), the divergence of any magnetic field vector $\overrightarrow{B}$ must always be zero:
$\nabla ·\overrightarrow{B}=0$
This physical property indicates that magnetic monopoles do not exist, and magnetic field lines always form closed loops.

For a general vector field in Cartesian coordinates represented as $\overrightarrow{B}=B_x\hat{x}+B_y\hat{y}+B_z\hat{z}$, the divergence is calculated as:
$\nabla ·\overrightarrow{B}=\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}$
Let us evaluate the divergence for the correct option, $\overrightarrow{B}=10x\hat{x}+20y\hat{y}-30z\hat{z}$:

First, identify the components:
$B_x=10x$
$B_y=20y$
$B_z=-30z$

Now, calculate the partial derivatives with respect to their corresponding variables:
$\frac{\partial B_x}{\partial x}=\frac{\partial }{\partial x}\left(10x\right)=10$
$\frac{\partial B_y}{\partial y}=\frac{\partial }{\partial y}\left(20y\right)=20$
$\frac{\partial B_z}{\partial z}=\frac{\partial }{\partial z}(-30z)=-30$

Adding these together gives the divergence:
$\nabla ·\overrightarrow{B}=10+20-30=0$
Since the divergence of this vector function is zero, it represents a physically possible magnetic field.