5.1 The electromagnetic spectrum — Practise with help

Work each problem first. Open a hint only if you are stuck, and the full working only to check.

Warm-up: remember

Answer these from memory. They use ideas from Section 4.1.

Use $v = f\lambda$ to find $v$ when $f = 200\,\text{Hz}$ and $\lambda = 1.5\,\text{m}$ .
Write 600 nm in metres, using standard form.
Name the two kinds of wave that pass through a vacuum.

Answers

$v = 200 \times 1.5 = 300\,\text{m/s}$ .
$6.0 \times 10^{-7}\,\text{m}$ .
Electromagnetic waves pass through a vacuum. Light is one example.

If question 1 went wrong, look again at the wave equation and keep the units with the numbers. If question 2 went wrong, remember that nano means $10^{-9}$ .

Practise with help

1. An FM radio station broadcasts at 100 MHz. Calculate the wavelength of the radio waves.

Hint

The wave equation is $v = f\lambda$ . You know the speed, and you want the wavelength. Rearrange the equation so that $\lambda$ is on its own.

Next step

Write $\lambda = \dfrac{c}{f}$ and convert 100 MHz to $1.0 \times 10^8\,\text{Hz}$ .

Full working

\begin{aligned} \lambda &= \frac{c}{f} \\ &= \frac{3.0 \times 10^8}{1.0 \times 10^8} \\ &= 3.0\,\text{m} \end{aligned}

2. A radar transmitter sends microwaves at 10 GHz. Calculate their wavelength.

Hint

The speed of a microwave in a vacuum is the same as the speed of light. Convert the frequency to hertz before you divide.

Next step

$10\,\text{GHz} = 1.0 \times 10^{10}\,\text{Hz}$ . Put this into $\lambda = \dfrac{c}{f}$ .

Full working

\begin{aligned} \lambda &= \frac{c}{f} \\ &= \frac{3.0 \times 10^8}{1.0 \times 10^{10}} \\ &= 0.030\,\text{m} \\ &= 3.0\,\text{cm} \end{aligned}

3. A detector measures infrared radiation with a frequency of $6.0 \times 10^{13}\,\text{Hz}$ . Calculate its wavelength, and give your answer in micrometres.

Hint

Find the wavelength in metres first. Then convert metres to micrometres. One micrometre is $1 \times 10^{-6}\,\text{m}$ .

Next step

Write $\lambda = \dfrac{3.0 \times 10^8}{6.0 \times 10^{13}}$ , then divide by $1 \times 10^{-6}$ to change metres into micrometres.

Full working

\begin{aligned} \lambda &= \frac{c}{f} \\ &= \frac{3.0 \times 10^8}{6.0 \times 10^{13}} \\ &= 5.0 \times 10^{-6}\,\text{m} \\ &= 5.0\,\mu\text{m} \end{aligned}

4. Place these four groups in order of increasing wavelength: gamma rays, infrared, radio waves, ultraviolet.

Hint

Picture the spectrum as one long line. Start at the end with the shortest wavelength and move along it.

Next step

Gamma rays have the shortest wavelength. Radio waves have the longest. Decide where the other two go between them.

Full working

Gamma rays, ultraviolet, infrared, radio waves.

Gamma rays have the shortest wavelength and the highest frequency. Radio waves have the longest wavelength and the lowest frequency. Infrared comes after visible light, and ultraviolet comes before it, so ultraviolet is shorter than infrared.

5. A radio wave and a gamma ray travel through a vacuum side by side. Explain how their speeds compare, and how their wavelengths compare.

Hint

They are both electromagnetic waves. Think about what every electromagnetic wave has in common in a vacuum.

Next step

Name the one speed they share. Then use the link between wavelength and frequency to compare the sizes.

Full working

Both waves travel at the same speed in a vacuum, $3.0 \times 10^8\,\text{m/s}$ . They have different frequencies. The radio wave has the lower frequency, so it has the longer wavelength. The gamma ray has the higher frequency, so it has the much shorter wavelength.

This answer is a model answer for you to compare with your own wording.