Deriving Magnetic Flux Density from Vector Potential

The magnetic flux density, denoted by $\vec{B}$, is related to the magnetic vector potential, $\vec{A}$, through the curl operation. The fundamental relationship is given by:

$ \vec{B} = \nabla \times \vec{A} $

In Cartesian coordinates, the curl operator $\nabla \times$ is defined as:

$ \nabla \times \vec{A} = \begin{vmatrix} \vec{a_x} & \vec{a_y} & \vec{a_z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix} $

Calculating the Magnetic Flux Density ($\vec{B}$)

The given magnetic vector potential is:

$ \vec{A} = (y \cos ax) \vec{a_x} + 0 \vec{a_y} + (y+e^x) \vec{a_z} $

From this, we identify the components:

• $ A_x = y \cos(ax) $
• $ A_y = 0 $
• $ A_z = y + e^x $

Now, let's calculate the necessary partial derivatives:

• $ \frac{\partial A_z}{\partial y} = \frac{\partial}{\partial y}(y + e^x) = 1 $
• $ \frac{\partial A_y}{\partial z} = \frac{\partial}{\partial z}(0) = 0 $
• $ \frac{\partial A_x}{\partial z} = \frac{\partial}{\partial z}(y \cos ax) = 0 $
• $ \frac{\partial A_z}{\partial x} = \frac{\partial}{\partial x}(y + e^x) = e^x $
• $ \frac{\partial A_y}{\partial x} = \frac{\partial}{\partial x}(0) = 0 $
• $ \frac{\partial A_x}{\partial y} = \frac{\partial}{\partial y}(y \cos ax) = \cos ax $

Substitute these into the curl formula:

$ \vec{B} = \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) \vec{a_x} + \left( \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} \right) \vec{a_y} + \left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) \vec{a_z} $

$ \vec{B} = (1 - 0) \vec{a_x} + (0 - e^x) \vec{a_y} + (0 - \cos ax) \vec{a_z} $

$ \vec{B} = 1 \vec{a_x} - e^x \vec{a_y} - \cos(ax) \vec{a_z} $

Evaluating $\vec{B}$ at the Origin

The origin is the point where $ x=0 $, $ y=0 $, and $ z=0 $. We substitute these values into the expression for $\vec{B}$:

• $ B_x = 1 $
• $ B_y = -e^0 = -1 $
• $ B_z = -\cos(a \cdot 0) = -\cos(0) = -1 $

Therefore, the magnetic flux density vector at the origin is:

$ \vec{B}_{origin} = 1 \vec{a_x} - 1 \vec{a_y} - 1 \vec{a_z} $

Calculating the Magnitude of $\vec{B}$

The magnitude of the magnetic flux density vector $\vec{B}$ is calculated using the formula:

$ |\vec{B}| = \sqrt{B_x^2 + B_y^2 + B_z^2} $

Substituting the components at the origin:

$ |\vec{B}_{origin}| = \sqrt{(1)^2 + (-1)^2 + (-1)^2} $

$ |\vec{B}_{origin}| = \sqrt{1 + 1 + 1} $

$ |\vec{B}_{origin}| = \sqrt{3} \, \text{Wb/m}^2 $