Step 1: Understanding the Question: We are tasked with calculating the energy of a single photon given its wavelength (\(\lambda = 500\ \text{nm}\)) and Planck's constant (\(h\)).
Step 2: Key Formula and Approach:
The energy (\(E\)) of a photon can be calculated using the Planck-Einstein relation:
\[ E = \frac{hc}{\lambda} \]
where \(h\) is Planck's constant (\(6.6 \times 10^{-34}\ \text{Js}\)), \(c\) is the speed of light (\(3 \times 10^8\ \text{m/s}\)), and \(\lambda\) is the wavelength in standard SI units (meters).
Step 3: Detailed Explanation:
First, we must convert the given wavelength from nanometers to meters to ensure unit consistency:
\[ \lambda = 500\ \text{nm} = 500 \times 10^{-9}\ \text{m} = 5 \times 10^{-7}\ \text{m} \]
Next, we substitute the known values into the energy formula:
\[ E = \frac{(6.6 \times 10^{-34}) \times (3 \times 10^8)}{5 \times 10^{-7}} \]
We simplify the numerator by multiplying the constants and their respective powers of ten:
\[ 6.6 \times 3 = 19.8 \]
\[ 10^{-34} \times 10^8 = 10^{-26} \]
So, the numerator evaluates to \(19.8 \times 10^{-26}\ \text{J}\cdot\text{m}\).
Now, we divide this result by the denominator:
\[ E = \frac{19.8 \times 10^{-26}}{5 \times 10^{-7}} \]
\[ E = \left(\frac{19.8}{5}\right) \times 10^{-26 - (-7)} \]
\[ E = 3.96 \times 10^{-19}\ \text{J} \]
Step 4: Final Answer: The calculated energy of the photon is \(3.96 \times 10^{-19}\ \text{J}\).