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Chapter 27: Problem 59

A surgeon is attempting to correct a detached retina by using a pulsed laser. (a) If the pulses last for \(20.0 \mathrm{ms}\) and if the output power of the laser is \(0.500 \mathrm{W},\) how much energy is in each pulse? (b) If the wavelength of the laser light is \(643 \mathrm{nm}\), how many photons are present in each pulse?

Short Answer

Expert verified
Solution: (a) The energy in each pulse is 0.010 J. (b) The number of photons in each pulse can be calculated using the formula \(n = \frac{0.010 \, \text{J}}{E_{photon}}\) after calculating the energy per photon.

Step by step solution

01

Calculate the total energy used during the pulse

Use the formula for power: \(P = \frac{E}{t}\). We know the power (P) and the time duration of the pulse (t). We have to find the energy (E) in each pulse. So, rearrange the equation to get the energy: \(E = P \times t\). Now substitute the given values: \(E = 0.500 \, \text{W} \times 20.0 \, \text{ms}\) #Step 2: Convert ms to s#
02

Convert time duration from milliseconds to seconds

Since we need the time duration in seconds, convert \(20.0 \, \text{ms}\) into seconds by dividing by 1000: \(\frac{20.0 \, \text{ms}}{1000} = 0.020 \, \text{s}\) #Step 3: Calculate the energy in Joules#
03

Calculate the energy in Joules (J)

Now, multiply the power by the time duration in seconds: \(E = 0.500 \, \text{W} \times 0.020 \, \text{s} = 0.010 \, \text{J}\) The energy in each pulse is \(0.010 \, \text{J}\). #Step 4: Calculate the energy per photon#
04

Calculate the energy per photon using the given wavelength

From the given wavelength, calculate the energy per photon using the formula: \(E_{photon} = \frac{hc}{\lambda}\), where h is the Planck's constant (\(6.63 \times 10^{-34} \, \text{Js}\)), c is the speed of light (\(3.00 \times 10^8 \, \text{m/s}\)), and \(\lambda\) is the wavelength. Convert the given wavelength from nanometers to meters by dividing by \(10^9\): \(\frac{643 \, \text{nm}}{10^9} = 6.43 \times 10^{-7} \, \text{m}\). Then, calculate the energy per photon: \(E_{photon} = \frac{(6.63 \times 10^{-34} \, \text{Js})(3.00 \times 10^8 \, \text{m/s})}{6.43 \times 10^{-7} \, \text{m}}\) #Step 5: Calculate the number of photons in one pulse#
05

Calculate the number of photons in each pulse

Divide the total energy of the pulse (0.010 J) by the energy per photon to find the number of photons in each pulse: \(n = \frac{0.010 \, \text{J}}{E_{photon}}\) (a) The energy in each pulse is \(0.010 \, \text{J}\). (b) The number of photons in each pulse is given by the formula \(n = \frac{0.010 \, \text{J}}{E_{photon}}\) after calculating the energy per photon.

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