Chapter 9: 9.37P (page 433)

A microwave antenna radiating at 10GHz is to be protected from the environment by a plastic shield of dielectric constant 2.5. What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9. 199.]

Short Answer

Expert verified

The minimum thickness of the shielding is 9.5 mm.

Step by step solution

Expression for the minimum thicknesof the shielding:

Write the expression for minimum thickness.

$d = \frac{π}{k}$ ........(1)

Determine the minimum thickness of the shielding

Substitute $k = \frac{ω}{v}$ and $ω = 2 πν$ in equation (1).

$d = \frac{π}{(\frac{ω}{ν})} d = \frac{π}{(\frac{2 πν}{v})} d = \frac{v}{2 ν}$

Substitute $ν = \frac{c}{n}$ and $n = \sqrt{\frac{εμ}{ε_{0} μ_{0}}}$ in the above expression.

$d = \frac{(\frac{c}{n})}{2 ν} d = \frac{c}{2 ν \sqrt{\frac{εμ}{ε_{0} μ_{0}}}}$ ....... (2)

Here, $μ \approx μ_{0}$ and $\sqrt{\frac{ε}{ε_{0}}} = \sqrt{ε_{T}}$ .

Substitute $μ \approx μ_{0}$ and $\sqrt{\frac{ε}{ε_{0}}} = \sqrt{ε_{T}}$ in equation (2).

$d = \frac{c}{2 ν \sqrt{ε_{T}}}$ ........(3)

Substitute $c = 3 \times 10^{8} m / s$ , $ν = 10 GHz$ and $ε_{T} = 2.5$ in equation (3).

$d = \frac{3 \times 10^{8} m / s}{2 \times (10 GHZ \times \frac{10^{9} Hz}{1 GHz}) \sqrt{2.5}} d = 9.49 \times 10^{- 3} m \times (\frac{10^{3} mm}{1 m}) d = 9.5 mm$

Therefore, the minimum thickness of the shielding is 9.5 mm.

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A microwave antenna radiating at 10GHz is to be protected from the environment by a plastic shield of dielectric constant 2.5. What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9. 199.]

Short Answer

Step by step solution

Expression for the minimum thicknesof the shielding:

Determine the minimum thickness of the shielding

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