Correct Answer - Option 3 : 10x x̂ + 20y ŷ - 30z ẑ
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 \(∇ .\vec F\)
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
\(∇ ⋅ B = \frac{{\partial {B_x}}}{{\partial x}} + \frac{{\partial {B_y}}}{{\partial y}} + \frac{{\partial {B_z}}}{{\partial z}} = \frac{{\partial 10x}}{{\partial x}} + \frac{{\partial ( - 30z)}}{{\partial y}} + \frac{{\partial 20y}}{{\partial z}} = 10\)
∇ ⋅ B ≠ 0, so function does not represent magnetic field.
Option: 2
B = 10y x̂ + 20x ŷ - 10z ẑ
\(\nabla ⋅ B = \frac{{\partial {B_x}}}{{\partial x}} + \frac{{\partial {B_y}}}{{\partial y}} + \frac{{\partial {B_z}}}{{\partial z}} = \frac{{\partial 10y}}{{\partial x}} + \frac{{\partial 20x}}{{\partial y}} + \frac{{\partial ( - 10z)}}{{\partial z}} = - 10\)
∇ ⋅ B ≠ 0, so function does not represent magnetic field.
Option: 3
B = 10x x̂ + 20y ŷ - 30z ẑ
\(\nabla ⋅ B = \frac{{\partial {B_x}}}{{\partial x}} + \frac{{\partial {B_y}}}{{\partial y}} + \frac{{\partial {B_z}}}{{\partial z}} = \frac{{\partial 10x}}{{\partial x}} + \frac{{\partial 20y}}{{\partial y}} + \frac{{\partial ( - 30z)}}{{\partial z}} = 10 + 20 - 30 = 0\)
∇ ⋅ B = 0, so function represent magnetic field.
Option: 4
B = 10z x̂ + 20y ŷ - 30x ẑ
\(\nabla ⋅ B = \frac{{\partial {B_x}}}{{\partial x}} + \frac{{\partial {B_y}}}{{\partial y}} + \frac{{\partial {B_z}}}{{\partial z}} = \frac{{\partial 10z}}{{\partial x}} + \frac{{\partial 20y}}{{\partial y}} + \frac{{\partial ( - 30x)}}{{\partial z}} = 20\)
∇ ⋅ B ≠ 0, so function does not represent magnetic field.
