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Chapter 29: Problem 59

A radar system produces pulses consisting of 100 full cycles of a sinusoidal \(70-\) GHz electromagnetic wave. The average power while the transmitter is on is \(45 \mathrm{MW}\), and the waves are confined to a beam \(20 \mathrm{cm}\) in diameter. Find (a) the peak electric field. (b) the wavelength, (c) the total energy in a pulse, and (d) the total momentum in a pulse. (e) If the transmitter produces 1000 pulses per second, what's its average power output?

Short Answer

Expert verified
The peak electric field, wavelength, total energy in a pulse, total momentum in a pulse, and average power output will be calculated using the formulas detailed above, with data given in the problem and standard constants. The values may vary based on the precise constants used, but the general process will remain the same. The calculated values will be the solutions to parts (a), (b), (c), (d), and (e) of the exercise.

Step by step solution

01

Calculate Peak Electric Field

Use the formula for power \( P = 0.5 \cdot A^2 \cdot c \cdot \epsilon_0 \cdot S \), where power \(P\) is given as \(45MW\) but for calculation convert it to watts \(45 x 10^6 W\), \(c\) is speed of light \(3 x 10^8 m/s\), \(\epsilon_0\) is permittivity of free space \(8.85 x 10^{-12} C^{2}/NM^{2}\), and \(S\) is the cross-sectional area of the beam computed as \( \pi (0.1m)^2\) because diameter is given as \(20cm\), on conversion becomes \(0.2m\), thus radius is \(0.1m\). \(A\) is the amplitude or the peak electric field we are trying to find. Solving for \(A\) gives us the peak electric field.
02

Determine the Wavelength

Use the formula for wavelength which is calculated as \( \lambda = c/f \), where \(c\) is the speed of light \(3 x 10^8 m/s\) and \(f\) is the frequency \(70 GHz\), but convert it to Hz for the calculation \(70 x 10^9 Hz\). Solving the formula for \( \lambda \) gives the wavelength of the electromagnetic wave.
03

Calculate Total Energy in a Pulse

The total energy in a pulse is given by \(E = n \cdot h \cdot f\), where \(n\) is the number of cycles (100), \(h\) is Planck's constant \((6.63 x 10^{-34} Js)\), and \(f\) is the frequency \(70 x 10^9 Hz\). Solve this formula for \(E\) to get the total energy in a pulse.
04

Determine Total Momentum in a Pulse

The total momentum in a pulse is given by \(p = E / c\), where \(E\) is the total energy computed in the previous step, and \(c\) is the speed of light \(3 x 10^8 m/s\). Solve this formula for \(p\) to get the total momentum in a pulse.
05

Calculate Average Power Output

The average power output is \( E \cdot p \) where \(E\) is the total energy per pulse computed in Step 3, and \(p\) is the number of pulses per second (1000). Solve this product for average power output.

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