Answer :
To find the frequency of an x-ray wave with an energy of [tex]\( E = 2.0 \times 10^{-17} \)[/tex] Joules and Planck's constant [tex]\( h = 6.626 \times 10^{-34} \)[/tex] Joules [tex]\(\cdot\)[/tex] second, we can use the relationship between energy and frequency, which is given by the formula:
[tex]\[ E = h \cdot f \][/tex]
where [tex]\( f \)[/tex] is the frequency. Rearranging this formula to solve for [tex]\( f \)[/tex], we get:
[tex]\[ f = \frac{E}{h} \][/tex]
Substitute the given values into the formula:
[tex]\[ f = \frac{2.0 \times 10^{-17} \, \text{Joules}}{6.626 \times 10^{-34} \, \text{Joules} \cdot \text{seconds}} \][/tex]
Perform the division:
[tex]\[ f = \frac{2.0}{6.626} \times 10^{-17 - (-34)} \][/tex]
Simplify the exponent part:
[tex]\[ f = \frac{2.0}{6.626} \times 10^{17} \times 10^{34} \][/tex]
[tex]\[ f = \frac{2.0}{6.626} \times 10^{17 + 34} \][/tex]
[tex]\[ f = \frac{2.0}{6.626} \times 10^{51} \][/tex]
Now, calculate the numerical coefficient:
[tex]\[ f \approx \frac{2.0}{6.626} = 0.3018412315122246 \][/tex]
So,
[tex]\[ f \approx 3.018 \times 10^{51 - 34} \][/tex]
[tex]\[ f \approx 3.018 \times 10^{16} \][/tex]
Thus, the frequency of the x-ray wave is:
[tex]\[ f \approx 3.018 \times 10^{16} \, \text{Hz} \][/tex]
[tex]\[ E = h \cdot f \][/tex]
where [tex]\( f \)[/tex] is the frequency. Rearranging this formula to solve for [tex]\( f \)[/tex], we get:
[tex]\[ f = \frac{E}{h} \][/tex]
Substitute the given values into the formula:
[tex]\[ f = \frac{2.0 \times 10^{-17} \, \text{Joules}}{6.626 \times 10^{-34} \, \text{Joules} \cdot \text{seconds}} \][/tex]
Perform the division:
[tex]\[ f = \frac{2.0}{6.626} \times 10^{-17 - (-34)} \][/tex]
Simplify the exponent part:
[tex]\[ f = \frac{2.0}{6.626} \times 10^{17} \times 10^{34} \][/tex]
[tex]\[ f = \frac{2.0}{6.626} \times 10^{17 + 34} \][/tex]
[tex]\[ f = \frac{2.0}{6.626} \times 10^{51} \][/tex]
Now, calculate the numerical coefficient:
[tex]\[ f \approx \frac{2.0}{6.626} = 0.3018412315122246 \][/tex]
So,
[tex]\[ f \approx 3.018 \times 10^{51 - 34} \][/tex]
[tex]\[ f \approx 3.018 \times 10^{16} \][/tex]
Thus, the frequency of the x-ray wave is:
[tex]\[ f \approx 3.018 \times 10^{16} \, \text{Hz} \][/tex]
