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Step-by-Step Solution
Step 1: Identify the Physical Quantities and Known Constants
• Power of the light bulb, P = 50 W (which is 50 J of energy emitted per second)
• Wavelength of the emitted red light, \lambda = 795 \times 10^{-9} \text{ m}
• Planck’s constant, h = 6.63 \times 10^{-34} \text{ J·s}
• Speed of light, c = 3.0 \times 10^{8} \text{ m/s}
Step 2: Write the Formula for Energy of One Photon
The energy of a single photon of wavelength \lambda is given by the relation:
E_{\text{photon}} = \frac{h \, c}{\lambda}
Step 3: Express the Total Energy Emitted per Second
The total energy emitted per second by the bulb is 50 J (because 50 W means 50 J/s). If n is the number of photons emitted per second, then the total energy per second is:
E_{\text{total}} = n \times E_{\text{photon}} = n \times \frac{h \, c}{\lambda}
Given that E_{\text{total}} = 50 \, \text{J} , we have:
50 = n \times \frac{(6.63 \times 10^{-34}) \times (3.0 \times 10^8)}{795 \times 10^{-9}}
Step 4: Solve for the Number of Photons (n)
Rearranging to find n:
n = \frac{50 \times 795 \times 10^{-9}}{6.63 \times 10^{-34} \times 3.0 \times 10^8}
Performing the calculation:
n \approx 1.998 \times 10^{20} \approx 2.0 \times 10^{20}
Step 5: Identify the Value of x
The problem states that the number of photons per second is expressed as x \times \, 10^{20} . Since we have found n \approx 2.0 \times 10^{20} , it follows that:
x = 2.0
Answer
The value of x is 2.
