TV-reception antennas for VHF (channels 2 through 13) are constructed with cross wires supported at their centers, as shown in Figure 23.24. The ideal length for the cross wires is one-half the wavelength to be received, with the more expensive antennas having one for each channel. Suppose you measure the lengths of the wires for channels 2 and 9 and find them to be 2.71 and 0.801 m long, respectively.

What is the frequency for channel 2?
1 Hz
What is the frequency for channel 9?
2 Hz

L = λ/2, λ = 2•L

f1 = c/λ1 = c/2•L1= 3•10^8/2•2.71 =5.54•10^7 Hz
f2 = c/λ2 = c/2•L2= 3•10^8/2•0.801 =1.87•10^8 Hz

To determine the frequency for channel 2 and channel 9, we need to use the formula relating wavelength, frequency, and the speed of light:

c = λ * f

Where:
c = speed of light (approximately 3 x 10^8 m/s)
λ = wavelength
f = frequency

For the TV reception antennas, the ideal length of the cross wires is one-half the wavelength for the channel to be received. Since the length of the wires for channel 2 is given as 2.71 m, we can calculate the wavelength for channel 2:

λ2 = 2 * 2.71 m

Simplifying, we get:

λ2 = 5.42 m

Now, we can rearrange the equation to solve for the frequency:

f2 = c / λ2

Substituting the known values, we find:

f2 = (3 x 10^8 m/s) / 5.42 m
f2 ≈ 5.53 x 10^7 Hz

Therefore, the frequency for channel 2 is approximately 5.53 x 10^7 Hz.

Similarly, for channel 9, with a wire length of 0.801 m, we can calculate the wavelength:

λ9 = 2 * 0.801 m
λ9 = 1.602 m

Using the same formula as before, we can determine the frequency for channel 9:

f9 = c / λ9
f9 = (3 x 10^8 m/s) / 1.602 m
f9 ≈ 1.87 x 10^8 Hz

Hence, the frequency for channel 9 is approximately 1.87 x 10^8 Hz. Keep in mind that the frequency scale is given in Hz (Hertz), which represents the number of cycles per second.

To find the frequency for channel 2, we can use the formula:

Wavelength = Speed of Light / Frequency

Let's assume the speed of light is approximately 3 x 10^8 m/s.

For channel 2, we have the length of the wire (L) as 2.71 m, which is half the wavelength. So, the wavelength for channel 2 is 2 x 2.71 = 5.42 m.

Using the formula above, we can rearrange it to solve for frequency:

Frequency = Speed of Light / Wavelength

Plugging in the values, we get:

Frequency for channel 2 = 3 x 10^8 m/s / 5.42 m ≈ 5.53 x 10^7 Hz

Therefore, the frequency for channel 2 is approximately 5.53 x 10^7 Hz.

Similarly, we can find the frequency for channel 9.

For channel 9, the length of the wire (L) is 0.801 m, which is half the wavelength. So, the wavelength for channel 9 is 2 x 0.801 = 1.602 m.

Using the same formula, we can find the frequency:

Frequency for channel 9 = 3 x 10^8 m/s / 1.602 m ≈ 1.87 x 10^8 Hz

Therefore, the frequency for channel 9 is approximately 1.87 x 10^8 Hz.