The Voyager 1 spacecraft is at the edge of the solar system, about 11 billion km away. (a) How

long does a radio signal from Voyager 1 take to get to the Earth? (b) Its radio works at 8 GHz.
What is the wavelength? (c) What type of light is this? (d) What is the energy of 1 photon? (e)
It has a 23 watt radio wave transmitter. How many photons per second is this? (f) This radio
wave is aimed towards the Earth by a dish; by the time the signal gets to the Earth, the area of
the radio beam is 4x10^15 km^2
 What is the photon flux density, in photons per second per square
km? (g) How about in photons per second per square meter?

(a) To calculate the time it takes for a radio signal from Voyager 1 to reach Earth, we can use the speed of light. The speed of light is approximately 299,792 kilometers per second. Therefore, we can divide the distance between Voyager 1 and Earth (11 billion km) by the speed of light:

Time = Distance / Speed
Time = 11,000,000,000 km / 299,792 km/s

Calculating this, the radio signal from Voyager 1 takes approximately 36.7 hours to reach Earth.

(b) To find the wavelength of the radio wave, we can use the formula:

Wavelength = Speed of Light / Frequency
Wavelength = 299,792 km/s / 8 GHz

Converting the speed of light to meters per second to match the frequency:

299,792 km/s = 299,792,000 m/s

Wavelength = 299,792,000 m/s / 8 × 10^9 Hz

Simplifying the calculation:

Wavelength = 37.474 m

The wavelength of the radio wave is approximately 37.474 meters.

(c) Radio waves occupy the radio frequency range in the electromagnetic spectrum.

(d) The energy of a single photon can be calculated using the formula:

Energy = Planck's Constant × Frequency
Energy = 6.62607015 × 10^-34 J·s × 8 × 10^9 Hz

Calculating this:

Energy = 5.30085612 × 10^-24 J

The energy of one photon is approximately 5.301 × 10^-24 Joules.

(e) To calculate the number of photons per second from a 23 watt transmitter, we can use the formula:

Number of photons/s = Power / Energy of one photon
Number of photons/s = 23 W / 5.301 × 10^-24 J

Calculating this:

Number of photons/s ≈ 4.342 × 10^23 photons/s

There are approximately 4.342 × 10^23 photons per second.

(f) The photon flux density is the number of photons passing through a given area per unit time. We can calculate the photon flux density using the formula:

Photon flux density = Number of photons/s / Area

Given that the area of the radio beam is 4 × 10^15 km², we need to convert it to square meters:

Area = 4 × 10^15 km² × (1 × 10^6 m/km)^2

Calculating this:

Area = 4 × 10^27 m²

Photon flux density = (4.342 × 10^23 photons/s) / (4 × 10^27 m²)

Photon flux density ≈ 1.085 x 10^-4 photons/s/m²

The photon flux density is approximately 1.085 × 10^-4 photons per second per square meter.

(g) To convert the photon flux density to photons per second per square kilometer, we need to multiply by 10^6:

Photon flux density = 1.085 × 10^-4 photons/s/m² × (1 × 10^6 m²/km²)

Photon flux density ≈ 108.5 photons/s/km²

The photon flux density is approximately 108.5 photons per second per square kilometer.

(a) To find out how long a radio signal from Voyager 1 takes to get to Earth, we can use the speed of light. The speed of light is approximately 299,792,458 meters per second. Since the distance between Voyager 1 and Earth is given in kilometers, we need to convert it to meters first: 11 billion km = 11,000,000,000,000 meters.

To calculate the time it takes for the radio signal to reach Earth, we divide the distance by the speed of light:

Time = Distance / Speed of Light
= 11,000,000,000,000 meters / 299,792,458 meters per second
≈ 36,712 seconds

Therefore, a radio signal from Voyager 1 takes approximately 36,712 seconds to reach Earth.

(b) The wavelength can be determined using the formula:

Wavelength = Speed of Light / Frequency

Given that the radio works at 8 GHz, we need to convert this to Hz by multiplying by 1,000,000,000:

Frequency = 8 GHz * 1,000,000,000
= 8,000,000,000 Hz

Now we can calculate the wavelength:

Wavelength = 299,792,458 meters per second / 8,000,000,000 Hz
≈ 0.0375 meters

Therefore, the wavelength of the radio signal is approximately 0.0375 meters.

(c) To determine the type of light, we can use the electromagnetic spectrum. Radio waves fall in the long-wavelength region of the spectrum, which includes frequencies between 3 kHz and 300 GHz. Since the frequency of Voyager 1's radio signal is 8 GHz, it falls within this range, indicating that it is a radio wave.

(d) The energy of a single photon can be calculated using the formula:

Energy of Photon = Planck's Constant * Frequency

Planck's Constant (h) is approximately 6.62607015 x 10^(-34) Joule-seconds.

Let's plug in the values:

Energy of Photon = 6.62607015 x 10^(-34) J·s * 8,000,000,000 Hz
≈ 5.30085612 x 10^(-24) Joules

Therefore, the energy of one photon in Voyager 1's radio wave is approximately 5.30085612 x 10^(-24) Joules.

(e) To calculate the number of photons per second emitted by a 23-watt radio wave transmitter, we can use the equation:

Number of Photons per Second = Power / Energy of One Photon

Given that the power of the transmitter is 23 watts, we need to convert it to Joules per second:

Power = 23 watts

Number of Photons per Second = 23 J/s / (5.30085612 x 10^(-24) J)
≈ 4.3399 x 10^24 photons per second

Therefore, the transmitter emits approximately 4.3399 x 10^24 photons per second.

(f) To find the photon flux density, we need to divide the number of photons per second by the area of the radio beam.

Given that the area is 4 x 10^15 km^2, we need to convert it to square meters by multiplying by 1,000,000,000:

Area = 4 x 10^15 km^2 * 1,000,000,000 m^2/km^2
= 4 x 10^24 m^2

Photon Flux Density = Number of Photons per Second / Area

Photon Flux Density = (4.3399 x 10^24 photons per second) / (4 x 10^24 m^2)
≈ 1.08498 photons per second per square meter

Therefore, the photon flux density is approximately 1.08498 photons per second per square meter.

(g) To find the photon flux density in photons per second per square kilometer, we need to convert the area to square kilometers:

Area = 4 x 10^24 m^2 * (1 km^2 / 1,000,000 m^2)
≈ 4 x 10^18 km^2

Photon Flux Density = Number of Photons per Second / Area

Photon Flux Density = (4.3399 x 10^24 photons per second) / (4 x 10^18 km^2)
≈ 1.08498 x 10^6 photons per second per square kilometer

Therefore, the photon flux density is approximately 1.08498 x 10^6 photons per second per square kilometer.