Chapter 1, Problem 1.18P

To determine

(a)

The value of $S=E\times H$ and convert the value of $S$ in rectangular coordinates.

Answer to Problem 1.18P

The value of $S=E\times H$ is $\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}$ and the rectangular expression for $S$ is

   $\left(\frac{AB}{r^2}\right){\sin }^2\theta \left(\sin \theta \cos \varphi a_x+\sin \theta \sin \varphi a_y+\cos \theta a_z\right)$.

Explanation of Solution

Given:

The value of $E$ is $E=\left(\frac{A}{r}\right)\sin \theta a_{\theta }$.

The value of $H$ is $H=\left(\frac{B}{r}\right)\sin \theta a_{\varphi }$

The expression for $S$

is

Concept used:

Write the expression for cross product of two vector.

   $\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\widehat{r}& \widehat{\theta }& \widehat{\varphi }\\ A_r& A_{\theta }& A_{\varphi }\\ B_r& B_{\theta }& B_{\varphi }\end{array}\right|$ ...... (1)

Here, $r$ is the radius, $\theta$ is angle between $z$ axis and line drawn from origin, $\varphi$ is the angle between $x$ axis and line drawn from origin, $A_r$ is the component of vector $\overrightarrow{A}$ along $\widehat{r}$ , $A_{\theta }$

is the component of vector along m:math display='block'>Aϕ

is the component of vector along m:math display='block'>Br is the component of vector $\overrightarrow{B}$ along $\widehat{r}$ , $B_{\theta }$ is the component of vector $\overrightarrow{B}$

along m:math display='block'>Bϕ is the component of vector $\overrightarrow{B}$

along

Calculation:

Substitute $0$

for m:math display='block'>(Ar)sinθ

for m:math display='block'>0

form:math display='block'>0

for for m:math display='block'>(Br)sinθ

forfor and for $\overrightarrow{B}$ in equation (1).

   $\begin{array}{c}E\times H=\left|\begin{array}{ccc}\widehat{r}& \widehat{\theta }& \widehat{\varphi }\\ 0& \left(\frac{A}{r}\right)\sin \theta & 0\\ 0& 0& \left(\frac{B}{r}\right)\sin \theta \end{array}\right|\\ =\widehat{r}\left[\left(\left(\frac{A}{r}\right)\sin \theta \right)\left(\left(\frac{B}{r}\right)\sin \theta \right)-0\right]-\widehat{\theta }\left[\left(0\right)\left(\left(\frac{B}{r}\right)\sin \theta \right)-0\right]+\widehat{\varphi }\left[0-\left(0\right)\left(\left(\frac{A}{r}\right)\sin \theta \right)\right]\\ =\left(\frac{AB}{r^2}\right){\sin }^2\theta \widehat{r}-\widehat{\theta }\left(0\right)+\widehat{\varphi }\left(0\right)\\ =\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}\end{array}$ Substitute $\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}$ for $E\times H$ in given equation.

   $S=\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}$

Write the expression of $x$ component for $S$.

$S_x=S.a_x$ ...... (2)

Here, $S_x$

is component for $S$.

Substitute $\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}$ for $S$ in equation (2).

   $\begin{array}{c}{S}_x=\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}.a_x\\ =\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]a_r.a_x\\ =\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\left(\sin \theta \cos \varphi \right)\left\{\because a_r.a_x=\sin \theta \cos \varphi \right\}\end{array}$

Simplify further.

$S_x=\left(\frac{AB}{r^2}\right){\sin }^3\theta \cos \varphi$ ...... (3)

Write the expression of $y$ component for $S$.

$S_y=S.a_y$ ...... (4)

Here, $S_y$ is $y$

component for

Substitute $\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}$ for $S$ in equation (4).

   $\begin{array}{c}{S}_y=\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}.a_y\\ =\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]a_r.a_y\\ =\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\left(\sin \theta \sin \varphi \right)\left\{\because a_r.a_y=\sin \theta \sin \varphi \right\}\end{array}$

Simplify further.

$S_y=\left(\frac{AB}{r^2}\right){\sin }^3\theta \sin \varphi$ ...... (5)

Write the expression of $z$ component for $S$.

$S_z=S.a_z$ ...... (6)

Here, $S_z$ is $z$

component for

Substitute $\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}$ for $S$ in equation (6).

   $\begin{array}{c}{S}_z=\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}.a_z\\ =\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]a_r.a_z\\ =\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\left(\cos \theta \right)\left\{\because a_r.a_z=\cos \theta \right\}\end{array}$

Simplify further.

$S_z=\left(\frac{AB}{r^2}\right){\sin }^2\theta \cos \theta$ ...... (7)

So the rectangular expression for $S$.

   $\begin{array}{c}S=\left(\left(\frac{AB}{r^2}\right){\sin }^3\theta \cos \varphi \right)a_x+\left(\left(\frac{AB}{r^2}\right){\sin }^3\theta \sin \varphi \right)a_y+\left(\left(\frac{AB}{r^2}\right){\sin }^2\theta \cos \theta \right)a_z\\ =\left(\frac{AB}{r^2}\right){\sin }^2\theta \left(\sin \theta \cos \varphi a_x+\sin \theta \sin \varphi a_y+\cos \theta a_z\right)\end{array}$

Conclusion:

Thus,the value of $S=E\times H$ is $\left[\left(\frac{AB}{r^2}\right){\sin }^2\theta \right]\widehat{r}$ and the rectangular expression for $S$ is

   $\left(\frac{AB}{r^2}\right){\sin }^2\theta \left(\sin \theta \cos \varphi a_x+\sin \theta \sin \varphi a_y+\cos \theta a_z\right)$.

To determine

(b)

The value of $S$ along $x$ , $y$ and $z$ axes.

Answer to Problem 1.18P

Thevalue of $S$ along $x$ axis is $\left(\frac{AB}{r^2}\right){\sin }^3\theta \cos \varphi$ , the value of $S$ along $y$ axis is $\left(\frac{AB}{r^2}\right){\sin }^3\theta \sin \varphi$ and the value of $S$ along $z$ axis is $\left(\frac{AB}{r^2}\right){\sin }^2\theta \cos \theta$.

Explanation of Solution

Calculation:

From equation (3) the value of $S_x$ is

   $S_x=\left(\frac{AB}{r^2}\right){\sin }^3\theta \cos \varphi$

So the value of $S$ along $x$ axis is $\left(\frac{AB}{r^2}\right){\sin }^3\theta \cos \varphi$.

From equation (5) the value of $S_y$ is

   $S_y=\left(\frac{AB}{r^2}\right){\sin }^3\theta \sin \varphi$

So the value of $S$ along $y$ axis is $\left(\frac{AB}{r^2}\right){\sin }^3\theta \sin \varphi$.

From equation (7) the value of $S_z$ is

   $S_z=\left(\frac{AB}{r^2}\right){\sin }^2\theta \cos \theta$

So the value of $S$ along $z$ axis is $\left(\frac{AB}{r^2}\right){\sin }^2\theta \cos \theta$.

Conclusion:

Thus, the value of $S$ along $x$ axis is $\left(\frac{AB}{r^2}\right){\sin }^3\theta \cos \varphi$ , the value of $S$ along $y$ axis is $\left(\frac{AB}{r^2}\right){\sin }^3\theta \sin \varphi$ and the value of $S$ along $z$ axis is $\left(\frac{AB}{r^2}\right){\sin }^2\theta \cos \theta$.