Question

1. what is the frequency in hz (s⁻¹) of a 486 nm light? 6. a local radio station broadcasts at a frequency of 98.7 mhz. what is the wavelength of this radiation in m? 7. what is the energy (in j) of a wave with a frequency of 4.25×10⁹ hz? 8. what is the energy (in j) of a wave with a wavelength of 652 μm?

Explanation:

Step1: Recall the speed - wavelength - frequency relation

The speed of light $c = \lambda
u$, where $c = 3.0\times10^{8}\ m/s$, $\lambda$ is the wavelength and $
u$ is the frequency. For question 5, $\lambda=486\ nm = 486\times10^{- 9}\ m$. Then $
u=\frac{c}{\lambda}$.
\[

u=\frac{3.0\times 10^{8}\ m/s}{486\times 10^{-9}\ m}\approx6.17\times 10^{14}\ Hz
\]

Step2: For question 6

Given $
u = 98.7\ MHz=98.7\times10^{6}\ Hz$. Using $\lambda=\frac{c}{
u}$, we have
\[
\lambda=\frac{3.0\times 10^{8}\ m/s}{98.7\times 10^{6}\ Hz}\approx3.04\ m
\]

Step3: Recall the energy - frequency relation

The energy of a photon is given by $E = h
u$, where $h = 6.63\times10^{-34}\ J\cdot s$. For question 7, $
u = 4.25\times10^{9}\ Hz$. Then $E=h
u$.
\[
E=(6.63\times 10^{-34}\ J\cdot s)\times(4.25\times 10^{9}\ Hz)=2.82\times 10^{-24}\ J
\]

Step4: For question 8

First, convert $\lambda = 652\ \mu m=652\times10^{-6}\ m$. Using $
u=\frac{c}{\lambda}$ and then $E = h
u$.
\[

u=\frac{3.0\times 10^{8}\ m/s}{652\times 10^{-6}\ m}\approx4.60\times 10^{11}\ Hz
\]
\[
E=(6.63\times 10^{-34}\ J\cdot s)\times(4.60\times 10^{11}\ Hz)\approx3.05\times 10^{-22}\ J
\]

Answer:

1. $6.17\times 10^{14}\ Hz$
2. $3.04\ m$
3. $2.82\times 10^{-24}\ J$
4. $3.05\times 10^{-22}\ J$