Chapter 22, Problem 52GP

(a)

To determine

The amount of energy that crosses a transmitting antenna per second.

Answer to Problem 52GP

The answer is $0.40W$ .

Explanation of Solution

Given data:

Power, ${\text{P}}_{\text{0}}=50\times {10}^3\text{W}$

Distance (radius), $r=100\text{m}$

Formula used:

It is known that: $S=\frac{P_0}{4\pi r^2}$

Where,

  $S$ is the power flux.

  $P_0$ is power.

  $r$ is the radius.

Calculation:

Substitute the given data into the formula $S=\frac{P_0}{4\pi r^2}$ ,

  $S=\frac{P_0}{4\pi r^2}=\frac{50\times {10}^3\text{W}}{4\pi \left(100\text{m}\right)}=0.398W/m^2$

Therefore, the power output is:

  $P=SA=\left(0.398W/m^2\right)\times {\left(1.0m\right)}^2=0.40W$

Conclusion: The amount of energy that crosses a transmitting antenna per second will be $0.40W$ .

(b)

To determine

The root mean square value of the electric field.

Answer to Problem 52GP

The answer is $12\text{V/m}$ .

Explanation of Solution

Given data:

Power over unit length, $S=0.398W/m^2$

Speed of the light, $c=3\times {10}^8\text{m/s}$

Permittivity of vacuum, ${\epsilon }_0=8.85\times {10}^{-12}{\text{C}}^{\text{2}}\text{/N}{\text{.m}}^{\text{2}}$

Formula used:

It is known that: $E_{rms}=\sqrt{\frac{S}{c{\epsilon }_0}}$

Where,

  $\text{S}$ is the power flux.

  $c$ is speed of the light.

  $E_{rms}$ is root square means value of the electric field.

  ${\epsilon }_0$ is the permittivity of vacuum.

Calculation:

Substitute the given data into the formula $E_{rms}=\sqrt{\frac{S}{c{\epsilon }_0}}$ ,

  $E_{rms}=\sqrt{\frac{0.398W/m^2}{\left(3\times {10}^8\text{m/s}\times 8.85\times {10}^{-12}{\text{C}}^{\text{2}}\text{/N}{\text{.m}}^{\text{2}}\right)}}=12\text{V/m}$

Conclusion: The root mean square value of the electric field is $12\text{V/m}$ .

(c)

To determine

The voltage induced in antenna.

Answer to Problem 52GP

The answer is $12\text{V}$ .

Explanation of Solution

Given data:

Distance $d=1.0\text{m}$

Root square means value of the electric field, $E_{rms}=12\text{V/m}$

Formula used:

It is known that: $E_{rms}=V_{rms}=E_{rms}d$

Where,

  $E_{rms}$ is root square means value of the electric field.

Calculation:

Substitute the given data into the formula $V_{rms}=E_{rms}d$ ,

  ${\text{V}}_{\text{rms}}={\text{E}}_{\text{rms}}\text{d}=\text{12}\text{V/m}\times \left(\text{1}\text{.0}\text{m}\right)=\text{12}\text{V}$

Conclusion: The voltage induced in antenna is $12\text{V}$ .