Question

advanced wave calculations
wavelength equation
$c = \lambda v$
planck equation
$e = hv$
$c = 3.00\times10^{8}\\ m/s$
$h = 6.626\times10^{-34}$
instructions: use the wavelength equations to answer the questions below:

1. calculate the frequency and wavelength of waves with an energy of $3.45\times10^{-21}\\ j$.
2. at hospitals, patients are exposed to x - rays with a wavelength of $4.22\times10^{-10}\\ m$. how much energy (in joules) are patients exposed to from these x - rays?
3. ultraviolet rays from the sun have a wavelength of approximately $3.10\times10^{-8}\\ m$. what is the energy (in joules) of these ultraviolet rays?

Explanation:

Step1: Solve for frequency in question 1

Given $E = h
u$, we can find $
u=\frac{E}{h}$. Substitute $E = 3.45\times 10^{-21}\text{ J}$ and $h = 6.626\times 10^{-34}\text{ J}\cdot\text{s}$ into the formula.
$
u=\frac{3.45\times 10^{-21}\text{ J}}{6.626\times 10^{-34}\text{ J}\cdot\text{s}}\approx5.21\times 10^{12}\text{ Hz}$
Then, using $c = \lambda
u$, we can find $\lambda=\frac{c}{
u}$. Substitute $c = 3.00\times 10^{8}\text{ m/s}$ and $
u = 5.21\times 10^{12}\text{ Hz}$ into the formula.
$\lambda=\frac{3.00\times 10^{8}\text{ m/s}}{5.21\times 10^{12}\text{ Hz}}\approx5.76\times 10^{-5}\text{ m}$

Step2: Solve for energy in question 2

First, from $c = \lambda
u$, we get $
u=\frac{c}{\lambda}$. Substitute $\lambda = 4.22\times 10^{-10}\text{ m}$ and $c = 3.00\times 10^{8}\text{ m/s}$ into the formula.
$
u=\frac{3.00\times 10^{8}\text{ m/s}}{4.22\times 10^{-10}\text{ m}}\approx7.11\times 10^{17}\text{ Hz}$
Then, using $E = h
u$, substitute $
u = 7.11\times 10^{17}\text{ Hz}$ and $h = 6.626\times 10^{-34}\text{ J}\cdot\text{s}$ into the formula.
$E=(6.626\times 10^{-34}\text{ J}\cdot\text{s})\times(7.11\times 10^{17}\text{ Hz})\approx4.71\times 10^{-16}\text{ J}$

Step3: Solve for energy in question 3

First, from $c = \lambda
u$, we get $
u=\frac{c}{\lambda}$. Substitute $\lambda = 3.10\times 10^{-8}\text{ m}$ and $c = 3.00\times 10^{8}\text{ m/s}$ into the formula.
$
u=\frac{3.00\times 10^{8}\text{ m/s}}{3.10\times 10^{-8}\text{ m}}\approx9.68\times 10^{15}\text{ Hz}$
Then, using $E = h
u$, substitute $
u = 9.68\times 10^{15}\text{ Hz}$ and $h = 6.626\times 10^{-34}\text{ J}\cdot\text{s}$ into the formula.
$E=(6.626\times 10^{-34}\text{ J}\cdot\text{s})\times(9.68\times 10^{15}\text{ Hz})\approx6.41\times 10^{-18}\text{ J}$

Answer:

1. Frequency: $5.21\times 10^{12}\text{ Hz}$, Wavelength: $5.76\times 10^{-5}\text{ m}$
2. Energy: $4.71\times 10^{-16}\text{ J}$
3. Energy: $6.41\times 10^{-18}\text{ J}$