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Waves: Question 3

Syllabus 7.4

Multiple choice AS 1 mark

Four electromagnetic waves, W, X, Y and Z, travel in a vacuum with the following wavelengths.

W: 6.0×10−7 m6.0 \times 10^{-7}\text{ m}

X: 3.0×10−2 m3.0 \times 10^{-2}\text{ m}

Y: 1.2×10−10 m1.2 \times 10^{-10}\text{ m}

Z: 1.5×103 m1.5 \times 10^{3}\text{ m}

Which list places these four waves in order of increasing frequency (lowest frequency first)?

Choose an answer to check it, then compare with the worked solution below.
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Worked solution

Step 1: Recall the relationship between speed, frequency and wavelength

All electromagnetic waves travel at the same speed in a vacuum, c=3.00×108 m s−1c = 3.00 \times 10^{8}\text{ m s}^{-1}, so c=fλc = f\lambda gives: f=cλf = \frac{c}{\lambda}

Since cc is the same for every wave, frequency and wavelength are inversely related: the wave with the largest wavelength has the smallest frequency, and vice versa.

Step 2: Calculate the frequency of each wave

fW=3.00×1086.0×10−7=5.0×1014 Hzf_\text{W} = \frac{3.00 \times 10^{8}}{6.0 \times 10^{-7}} = 5.0 \times 10^{14}\text{ Hz} fX=3.00×1083.0×10−2=1.0×1010 Hzf_\text{X} = \frac{3.00 \times 10^{8}}{3.0 \times 10^{-2}} = 1.0 \times 10^{10}\text{ Hz} fY=3.00×1081.2×10−10=2.5×1018 Hzf_\text{Y} = \frac{3.00 \times 10^{8}}{1.2 \times 10^{-10}} = 2.5 \times 10^{18}\text{ Hz} fZ=3.00×1081.5×103=2.0×105 Hzf_\text{Z} = \frac{3.00 \times 10^{8}}{1.5 \times 10^{3}} = 2.0 \times 10^{5}\text{ Hz}

Check (independent method (compare wavelengths directly instead): ranking the wavelengths from largest to smallest gives λZ\lambda_\text{Z} (largest) >λX>λW>λY> \lambda_\text{X} > \lambda_\text{W} > \lambda_\text{Y} (smallest). Since frequency and wavelength are inversely related, this must correspond to Z having the smallest frequency and Y the largest) exactly the order found by direct calculation above.

Step 3: Order the waves by increasing frequency

Comparing the four values: fZ (2.0×105)<fX (1.0×1010)<fW (5.0×1014)<fY (2.5×1018)f_\text{Z}\,(2.0\times10^{5}) < f_\text{X}\,(1.0\times10^{10}) < f_\text{W}\,(5.0\times10^{14}) < f_\text{Y}\,(2.5\times10^{18}).

So the order of increasing frequency is Z, X, W, Y. This is a radio wave (Z), a microwave (X), visible light (W), and an X-ray/gamma ray (Y), consistent with their positions in the electromagnetic spectrum.

Why the other options are wrong

  • B (Y, W, X, Z): this is the exact reverse of the correct order, it correctly ranks the wavelengths from smallest to largest, but mistakes that for increasing frequency instead of decreasing frequency.
  • C (W, X, Y, Z): this simply repeats the order the waves were listed in the question, without actually ranking them by frequency (or wavelength) at all.
  • D (Z, Y, X, W): this correctly starts with Z (lowest frequency) but then swaps the positions of Y and W, incorrectly placing the wave with the largest exponent magnitude (Y, at 10−1010^{-10}) as if it were only moderately high in frequency.

Final answer

  • Order of increasing frequency: Z, X, W, Y\boxed{\text{Z, X, W, Y}}, option A.