(a) f = 5.00 × 10²⁰ Hz, E = 3.32 × 10⁻¹³ J;
(b) f = 1.20 × 10¹⁰ Hz, E = 7.96 × 10⁻²⁴J.
Explanation
What's the similarity between a gamma ray and a microwave?
Both gamma rays and microwave rays are electromagnetic radiations. Both travel at the speed of light at [tex]3.00 \times 10^{8}\;\text{m}\cdot\text{s}^{-1}[/tex] in vacuum.
[tex]f = \dfrac{c}{\lambda}[/tex]
where
- f is the frequency of the electromagnetic radiation,
- c is the speed of light, and
- [tex]\lambda[/tex] is the wavelength of the radiation.
(a)
Convert all units to standard ones.
[tex]\lambda = 0.600\;\text{pm} = 0.600 \times 10^{-12} \;\text{m}[/tex].
The unit of [tex]f[/tex] shall also be standard.
[tex]f = \dfrac{c}{\lambda} = \dfrac{3.00\times 10^{8}\;\text{m}\cdot\text{s}^{-1}}{0.600\times 10^{12}\;\text{m}} = 5.00 \times 10^{20}\;\text{s}^{-1}= 5.00\times 10^{20}\;\text{Hz}[/tex].
For each particle,
[tex]E = h\cdot f[/tex],
where
- [tex]E[/tex] is the energy of the particle,
- [tex]h[/tex] is the planck's constant where [tex]h = 6.63\times 10^{-34}\;\text{J}\cdot\text{s}^{-1}[/tex], and
- [tex]f[/tex] is the frequency of the particle.
[tex]E = h \cdot f = 6.63\times10^{-34}\;\text{J}\cdot\text{s}\times 5.00\times 10^{20}\;\text{s}^{-1} = 3.32\times10^{-13}\;\text{J}[/tex].
(b)
Try the steps in (a) for this beam of microwave with
- [tex]\lambda = 2.50 \;\text{cm} = 2.50\times 10^{-2}\;\text{m}[/tex].
Expect the following results:
- [tex]f = 1.20\times 10^{10}\;\text{Hz}[/tex], and
- [tex]E = 7.96\times 10^{-24}\;\text{J}[/tex].
