Question

$100\\ \text{cm} = 1\\ \text{m}$ $1000\\ \text{mm} = 1\\ \text{m}$

1. a television station has a wavelength of 920 cm. what is the frequency of that station?
2. what is the wavelength of light for a country station with a frequency of 93.1 mhz?
3. how much energy does a photon of light have if the frequency is $6.92 \times 10^{14}\\ \text{hz}$?
4. what is the energy of a form of light with a wavelength of 890 nm?
5. what is the frequency of a photon of light with $9.30 \times 10^{-16}\\ \text{j}$ of energy?

Explanation:

Step1: Convert wavelength to meters

Given $\lambda = 920\ \text{cm}$, convert to meters:
$\lambda = 920\ \text{cm} \times \frac{1\ \text{m}}{100\ \text{cm}} = 9.20\ \text{m}$

Step2: Use wave speed formula for frequency

Electromagnetic wave speed $c = 3.00 \times 10^8\ \text{m/s}$. Use $c = \lambda f$, rearrange to $f = \frac{c}{\lambda}$:
$f = \frac{3.00 \times 10^8\ \text{m/s}}{9.20\ \text{m}} \approx 3.26 \times 10^7\ \text{Hz}$

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Step1: Convert frequency to Hz

Given $f = 93.1\ \text{MHz}$, convert to Hz:
$f = 93.1\ \text{MHz} \times \frac{10^6\ \text{Hz}}{1\ \text{MHz}} = 9.31 \times 10^7\ \text{Hz}$

Step2: Use wave speed formula for wavelength

Rearrange $c = \lambda f$ to $\lambda = \frac{c}{f}$:
$\lambda = \frac{3.00 \times 10^8\ \text{m/s}}{9.31 \times 10^7\ \text{Hz}} \approx 3.22\ \text{m}$

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Step1: Use photon energy formula

Use $E = hf$, where Planck's constant $h = 6.626 \times 10^{-34}\ \text{J·s}$:
$E = (6.626 \times 10^{-34}\ \text{J·s})(6.92 \times 10^{14}\ \text{Hz}) \approx 4.58 \times 10^{-19}\ \text{J}$

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Step1: Convert wavelength to meters

Given $\lambda = 890\ \text{nm}$, convert to meters:
$\lambda = 890\ \text{nm} \times \frac{1\ \text{m}}{10^9\ \text{nm}} = 8.90 \times 10^{-7}\ \text{m}$

Step2: Calculate frequency first

Use $f = \frac{c}{\lambda}$:
$f = \frac{3.00 \times 10^8\ \text{m/s}}{8.90 \times 10^{-7}\ \text{m}} \approx 3.37 \times 10^{14}\ \text{Hz}$

Step3: Calculate photon energy

Use $E = hf$:
$E = (6.626 \times 10^{-34}\ \text{J·s})(3.37 \times 10^{14}\ \text{Hz}) \approx 2.23 \times 10^{-19}\ \text{J}$

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Step1: Rearrange photon energy formula for frequency

Rearrange $E = hf$ to $f = \frac{E}{h}$:
$f = \frac{9.30 \times 10^{-16}\ \text{J}}{6.626 \times 10^{-34}\ \text{J·s}} \approx 1.40 \times 10^{18}\ \text{Hz}$

Answer:

1. $3.26 \times 10^7\ \text{Hz}$
2. $3.22\ \text{m}$
3. $4.58 \times 10^{-19}\ \text{J}$
4. $2.23 \times 10^{-19}\ \text{J}$
5. $1.40 \times 10^{18}\ \text{Hz}$