Police radar detects the speed of a car (Fig. P38.15) as follows. Microwaves of a precisely known frequency are broadcast toward the car. The moving car reflects the microwaves with a Doppler shift. The reflected waves are received and combined with an attenuated version of the transmitted wave. Beats occur between the two microwave signals. The beat frequency is measured. (a) For an electromagnetic wave reflected back to its source from a mirror approaching at speed v , show that the reflected wave has frequency f ′ = c + v c − v f where f is the source frequency. (b) Noting that v is much less than c , show that the beat frequency can be written as f beat = 2 v/λ . (c) What beat frequency is measured for a car speed of 30.0 m/s if the microwaves have frequency 10.0 GHz? (d) If the beat frequency measurement in part (c) is accurate to ±5.0 Hz, how accurate is the speed measurement?
Police radar detects the speed of a car (Fig. P38.15) as follows. Microwaves of a precisely known frequency are broadcast toward the car. The moving car reflects the microwaves with a Doppler shift. The reflected waves are received and combined with an attenuated version of the transmitted wave. Beats occur between the two microwave signals. The beat frequency is measured. (a) For an electromagnetic wave reflected back to its source from a mirror approaching at speed v , show that the reflected wave has frequency f ′ = c + v c − v f where f is the source frequency. (b) Noting that v is much less than c , show that the beat frequency can be written as f beat = 2 v/λ . (c) What beat frequency is measured for a car speed of 30.0 m/s if the microwaves have frequency 10.0 GHz? (d) If the beat frequency measurement in part (c) is accurate to ±5.0 Hz, how accurate is the speed measurement?
Police radar detects the speed of a car (Fig. P38.15) as follows. Microwaves of a precisely known frequency are broadcast toward the car. The moving car reflects the microwaves with a Doppler shift. The reflected waves are received and combined with an attenuated version of the transmitted wave. Beats occur between the two microwave signals. The beat frequency is measured. (a) For an electromagnetic wave reflected back to its source from a mirror approaching at speed v, show that the reflected wave has frequency
f
′
=
c
+
v
c
−
v
f
where f is the source frequency. (b) Noting that v is much less than c, show that the beat frequency can be written as fbeat = 2v/λ. (c) What beat frequency is measured for a car speed of 30.0 m/s if the microwaves have frequency 10.0 GHz? (d) If the beat frequency measurement in part (c) is accurate to ±5.0 Hz, how accurate is the speed measurement?
Interaction between an electric field and a magnetic field.
High-Energy Cancer Treatment. Scientists are working on a new technique to kill cancer cells by zapping them with ultrahighenergy (in the range of 1012 W) pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk 5.0 um in diameter, with the pulse lasting for 4.0 ns with an average power of 2.0 * 1012 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in W/m2) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?
A sinusoidal plane electromagnetic wave of frequency 20 GHz moves in the positive y direction and its electric field is directed along the z-axis. The amplitude of the electric field is 10 V/m. The start of time is chosen so that at t = 0, the electric field has a value +(10 V/m)k at the origin.
a. Write the function that describes the electric field of the wave as a function of position y and time t. Hint: Among other things, you need to decide whether to use a cosine or sine function, and whether a negative sign is needed inside and/or outside the sine/cosine function.
b. Explain why the electric field is not a function of the x or z coordinate.
c. Find the function that describes magnetic field of the wave as a function of position y and time t.
A plane electromagnetic wave of frequency 25 MHz travels in free space along the x-direction. At a particular point in space and time, E = 6.3 ˆj V/m. What is B at this point?
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