Chapter 9, Problem 20P

(a)

To determine

Write Maxwell’s Equations for a linear, homogeneous medium in terms of $E_s$ and $H_s$.

Answer to Problem 20P

The Maxwell’s Equations for a linear, homogeneous medium are $\underset{\_}{\nabla \times E_s=j\omega \mu H_s}$ and $\underset{\_}{\nabla \times H_s=\left(\sigma -j\omega \epsilon \right)E_s}$.

Explanation of Solution

Calculation:

Write Maxwell’s Equations for a linear, homogeneous medium in terms of $E_s$ and $H_s$.

$\nabla \times E_s=j\omega \mu H_s$        (1)

Here,

$E_s$ is the phasor form of electric field,

$\omega$ is the wave frequency,

$\mu$ is the permeability of the medium, and

$H_s$ is the phasor form of magnetic field.

$\nabla \times H_s=\left(\sigma -j\omega \epsilon \right)E_s$        (2)

Here,

$\sigma$ is the conductivity and

$\epsilon$ is the permittivity of the medium.

Equations (1) and (2) represent the Maxwell’s Equations for a linear, homogeneous medium.

Conclusion:

Thus, the Maxwell’s Equations for a linear, homogeneous medium are $\underset{\_}{\nabla \times E_s=j\omega \mu H_s}$ and $\underset{\_}{\nabla \times H_s=\left(\sigma -j\omega \epsilon \right)E_s}$.

(b)

To determine

Write the point form of Maxwell’s Equations as eight scalar equations.

Explanation of Solution

Calculation:

Write the generalized forms of Maxwell’s four Equations.

$\nabla \cdot D={\rho }_v$        (3)

$\nabla \cdot B=0$        (4)

$\nabla \times E=-\frac{\partial B}{\partial t}$        (5)

$\nabla \times H=J+\frac{\partial D}{\partial t}$        (6)

Rewrite Equation (3) in point form.

$\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\cdot \left(D_x{a}_x+D_y{a}_y+D_z{a}_z\right)={\rho }_v$

$\frac{\partial D_x}{\partial x}+\frac{\partial D_y}{\partial y}+\frac{\partial D_z}{\partial z}={\rho }_v$        (7)

Rewrite Equation (4) in point form.

$\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\cdot \left(B_x{a}_x+B_y{a}_y+B_z{a}_z\right)=0$

$\frac{\partial B_x}{\partial x}+\frac{\partial B_y}{\partial y}+\frac{\partial B_z}{\partial z}=0$        (8)

Rewrite Equation (5) in point form.

$\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(E_x{a}_x+E_y{a}_y+E_z{a}_z\right)=-\left(\frac{\partial B_x}{\partial t}{a}_x+\frac{\partial B_y}{\partial t}{a}_y+\frac{\partial B_z}{\partial t}{a}_z\right)$        (9)

Find $\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(E_x{a}_x+E_y{a}_y+E_z{a}_z\right)$.

$\begin{array}{c}\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(E_x{a}_x+E_y{a}_y+E_z{a}_z\right)=\left|\begin{array}{ccc}{a}_x& a_y& a_z\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ E_x& E_y& E_z\end{array}\right|\\ =\left[\begin{array}{l}\left(\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right)a_x-\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right)a_y\\ +\left(\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right)a_z\end{array}\right]\\ =\left[\begin{array}{l}\left(\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right)a_x+\left(\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}\right)a_y\\ +\left(\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right)a_z\end{array}\right]\end{array}$

Substitute $\left[\left(\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right)a_x+\left(\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}\right)a_y+\left(\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right)a_z\right]$ for $\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(E_x{a}_x+E_y{a}_y+E_z{a}_z\right)$ in Equation (9).

$\left(\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right)a_x-\left(\frac{\partial E_z}{\partial x}-\frac{\partial E_x}{\partial z}\right)a_y+\left(\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right)a_z=-\left(\frac{\partial B_x}{\partial t}{a}_x+\frac{\partial B_y}{\partial t}{a}_y+\frac{\partial B_z}{\partial t}{a}_z\right)$

Write the scalar equations from the obtained expression.

$\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}=-\frac{\partial B_x}{\partial t}$        (10)

$\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}=-\frac{\partial B_y}{\partial t}$        (11)

$\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}=-\frac{\partial B_z}{\partial t}$        (12)

Rewrite Equation (6) in point form.

$\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(H_x{a}_x+H_y{a}_y+H_z{a}_z\right)=\left[\begin{array}{l}\left(J_x+\frac{\partial D_x}{\partial t}\right)a_x\\ +\left(J_y+\frac{\partial D_y}{\partial t}\right)\\ +\left(J_z+\frac{\partial D_z}{\partial t}\right)\end{array}\right]$        (13)

Find $\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(H_x{a}_x+H_y{a}_y+H_z{a}_z\right)$.

$\begin{array}{c}\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(H_x{a}_x+H_y{a}_y+H_z{a}_z\right)=\left|\begin{array}{ccc}{a}_x& a_y& a_z\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ H_x& H_y& H_z\end{array}\right|\\ =\left[\begin{array}{l}\left(\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}\right)a_x-\left(\frac{\partial H_z}{\partial x}-\frac{\partial H_x}{\partial z}\right)a_y\\ +\left(\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}\right)a_z\end{array}\right]\\ =\left[\begin{array}{l}\left(\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}\right)a_x+\left(\frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}\right)a_y\\ +\left(\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}\right)a_z\end{array}\right]\end{array}$

Substitute $\left[\left(\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}\right)a_x+\left(\frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}\right)a_y+\left(\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}\right)a_z\right]$ for $\left(\frac{\partial }{\partial x}{a}_x+\frac{\partial }{\partial y}{a}_y+\frac{\partial }{\partial z}{a}_z\right)\times \left(H_x{a}_x+H_y{a}_y+H_z{a}_z\right)$ in Equation (13).

$\left(\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}\right)a_x+\left(\frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}\right)a_y+\left(\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}\right)a_z=\left[\begin{array}{l}\left(J_x+\frac{\partial D_x}{\partial t}\right)a_x\\ +\left(J_y+\frac{\partial D_y}{\partial t}\right)+\left(J_z+\frac{\partial D_z}{\partial t}\right)\end{array}\right]$

Write the scalar equations from the obtained expression.

$\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}=J_x+\frac{\partial D_x}{\partial t}$        (14)

$\frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}=J_y+\frac{\partial D_y}{\partial t}$        (15)

$\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}=J_z+\frac{\partial D_z}{\partial t}$        (16)

Equations (7), (8), (10), (11), (12), (14), (15), and (16) represent the point of Maxwell’s Equations as eight scalar equations.

Conclusion:

Thus, the point form of Maxwell’s Equations is written.