To find the energy of an orange lamp with a given frequency, we can use the formula derived from Planck's equation:
[tex]\[ E = h \cdot f \][/tex]
where:
- [tex]\( E \)[/tex] is the energy in joules (J),
- [tex]\( h \)[/tex] is Planck's constant, approximately [tex]\( 6.626 \times 10^{-34} \)[/tex] joules per second (J·s),
- [tex]\( f \)[/tex] is the frequency, given as [tex]\( 5.10 \times 10^{14} \)[/tex] hertz (Hz).
Now, let’s break down the solution step-by-step:
1. Identify the given frequency [tex]\( f \)[/tex]:
[tex]\[ f = 5.10 \times 10^{14} \, \text{Hz} \][/tex]
2. Find Planck's constant [tex]\( h \)[/tex]:
[tex]\[ h = 6.626 \times 10^{-34} \, \text{J·s} \][/tex]
3. Apply the formula [tex]\( E = h \cdot f \)[/tex]:
[tex]\[ E = (6.626 \times 10^{-34} \, \text{J·s}) \cdot (5.10 \times 10^{14} \, \text{Hz}) \][/tex]
4. Calculate the energy [tex]\( E \)[/tex]:
[tex]\[ E = 3.3792599999999997 \times 10^{-19} \, \text{J} \][/tex]
Thus, the energy of an orange lamp with a frequency of [tex]\( 5.10 \times 10^{14} \)[/tex] Hz is:
[tex]\[ E \approx 3.38 \times 10^{-19} \, \text{J} \][/tex]
This gives us a numerical solution for the energy, accurately representing the energy associated with the given frequency of the orange lamp.
