Answer :

Solution :

Concept:

The magnetic field forms a closed loop, i.e. the amount of field leaving a point equals the amount entering. i.e. Magnetic monopoles do not exist.

Since the divergence of a field gives the net outflow of a field and is calculated as

So, ∇ ⋅ B = 0

B = Magnetic flux density

Since B is related to H (magnetic field intensity) as:

B = μH

so, ∇ ⋅ H = 0

Application:

Option: 1

B = 10x x̂ - 30z ŷ + 20y ẑ

$\nabla \cdot B=\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=\frac{\partial 10x}{\partial x}+\frac{\partial (-30z)}{\partial y}+\frac{\partial 20y}{\partial z}=10$

∇ ⋅ B ≠ 0, so function does not represent magnetic field.

Option: 2

B = 10y x̂ + 20x ŷ - 10z ẑ

$\mathrm{\nabla }\cdot B=\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=\frac{\partial 10y}{\partial x}+\frac{\partial 20x}{\partial y}+\frac{\partial (-10z)}{\partial z}=-10$

∇ ⋅ B ≠ 0, so function does not represent magnetic field.

Option: 3

B = 10x x̂ + 20y ŷ - 30z ẑ

$\mathrm{\nabla }\cdot B=\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=\frac{\partial 10x}{\partial x}+\frac{\partial 20y}{\partial y}+\frac{\partial (-30z)}{\partial z}=10+20-30=0$

∇ ⋅ B = 0, so function represent magnetic field.

Option: 4

B = 10z x̂ + 20y ŷ - 30x ẑ

$\mathrm{\nabla }\cdot B=\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=\frac{\partial 10z}{\partial x}+\frac{\partial 20y}{\partial y}+\frac{\partial (-30x)}{\partial z}=20$

∇ ⋅ B ≠ 0, so function does not represent magnetic field.