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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# A special kind of lightbulb emits monochromatic light of wavelength 630 nm. Electrical energy is supplied to it at the rate of 60W, and the bulb is 93% efficient at converting that energy to light energy. How many photons are emitted by the bulb during its lifetime of 730h?

The number of photons are $4.6×{10}^{26}\mathrm{photons}.$

See the step by step solution

## Describe the expression of frequency and energy of the photon:

The expression for frequency of photon is given by,

${f}{=}\frac{c}{\lambda }$ ….. (1)

Here, c is the speed of light and ${\lambda }$ is the wavelength.

The energy of a photon is given by,

${E}{}{=}{}{h}{f}$ ….. (2)

Here, h is Planck’s constant and is the frequency.

Combine the above two equations.

$E=\frac{hc}{\lambda }$ ….. (3)

## Determine the number of photons emitted by the bulb during its lifetime of 730h:

Consider the given data as below.

The wavelength, $\lambda =630nm\phantom{\rule{0ex}{0ex}}$

Plank’s constant, $h=6.626×{10}^{-34}\mathrm{J}·\mathrm{s}$

The speed of light, $c=3×{10}^{8}m/s$

Substitute the above values in eq 3.

Rate at which electrical energy is supplied is 60W. But the bulb is only 93% efficient in converting this energy into light.

The power delivered from the light is,

$\begin{array}{c}P=\frac{93}{100}×60\mathrm{W}\\ =55.8\mathrm{W}\end{array}$

$\mathrm{Numberofphotons}=\frac{\mathrm{power}\mathrm{delivered}\mathrm{out}}{\mathrm{energy}\mathrm{of}\mathrm{each}\mathrm{photon}}\phantom{\rule{0ex}{0ex}}=\frac{55.8\mathrm{W}}{3.15×{10}^{-19}\mathrm{J}}\phantom{\rule{0ex}{0ex}}=1.77×{10}^{20}\mathrm{photons}/\mathrm{s}.\phantom{\rule{0ex}{0ex}}=\left(1.77×{10}^{20}\mathrm{photons}/\mathrm{s}\right)\left(\frac{60\mathrm{s}}{\mathrm{min}}\right)\left(\frac{60\mathrm{min}}{\mathrm{h}}\right)$

Simplify further.

$\mathrm{Numberofphotons}=6372×{10}^{20}photons/h\phantom{\rule{0ex}{0ex}}=6372×{10}^{20}photons/h×730h\phantom{\rule{0ex}{0ex}}=4.6×{10}^{26}\mathrm{photons}$

Hence, the number of photons arelocalid="1663135942932" style="width:30%" style="width:30%" style="width:30%" style="width:30%" style="max-width: none; vertical-align: -4px;" $4.6×{10}^{26}\mathrm{photons}$.