Question

Analyze We begin by determining the magnitude of the beam's Poynting vector. Divide the time-averaged power delivered via the electromagnetic wave by the cross-sectional area of the beam: Savg = Pavg / A = Pavg / (?r^2) = (2.7 × 10^-3 W) / (?((1.9 × 10^-3 m)/2)^2) = 952 W/m^2 Analyze Now let's determine the radiation pressure from the laser beam. The equation for radiation pressure exerted on a perfectly reflecting surface indicates that a completely reflected beam would apply an average pressure of Pavg = 2Savg/c. We can model the actual reflection as follows. Imagine that the surface absorbs the beam, resulting in pressure Pavg = Savg/c. Then the surface emits the beam, resulting in additional pressure Pavg = Savg/c. If the surface emits only a fraction f of the beam (so that f is the amount of the incident beam reflected), the pressure due to the emitted beam is Pavg = fSavg/c. Use this model to find the total pressure on the surface due to absorption and re-emission (reflection): Pavg = Savg/c + fSavg/c = (1 + f)Savg/c Evaluate this pressure for a beam that is 70% reflected: Pavg = (1 + 0.70) (952 W/m^2) / (3.0 × 10^8 m/s) Pavg = .00000539 N/m^2 Finalize Consider the magnitude of the Poynting vector, Savg = 952 W/m^2. It is about the same as the intensity of sunlight at the Earth's surface. For this reason, it is not safe to shine the beam of a laser pointer into a person's eyes, which may be more dangerous than looking directly at the Sun. The pressure has an extremely small value, as expected. (Recall from a previous chapter that atmospheric pressure is approximately 10^5 N/m^2.) A 10.0 mW helium-neon laser (? = 632.8 nm) emits a beam of circular cross section with a diameter of 2.00 mm. (a) Find the maximum electric field in the beam. Emax = V/m (b) What total energy is contained in a 7.50 m length of the beam? U = J (c) Find the momentum carried by a 7.50 m length of the beam. p = kg·m/s

          Analyze We begin by determining the magnitude of the beam's Poynting vector.

Divide the time-averaged power delivered via the electromagnetic wave by the cross-sectional area of the beam:
Savg = Pavg / A = Pavg / (?r^2)
= (2.7 × 10^-3 W) / (?((1.9 × 10^-3 m)/2)^2) = 952 W/m^2

Analyze Now let's determine the radiation pressure from the laser beam. The equation for radiation pressure exerted on a perfectly reflecting surface indicates that a completely reflected beam would apply an average pressure of Pavg = 2Savg/c. We can model the actual reflection as follows. Imagine that the surface absorbs the beam, resulting in pressure Pavg = Savg/c. Then the surface emits the beam, resulting in additional pressure Pavg = Savg/c. If the surface emits only a fraction f of the beam (so that f is the amount of the incident beam reflected), the pressure due to the emitted beam is Pavg = fSavg/c.

Use this model to find the total pressure on the surface due to absorption and re-emission (reflection):
Pavg = Savg/c + fSavg/c = (1 + f)Savg/c

Evaluate this pressure for a beam that is 70% reflected:
Pavg = (1 + 0.70) (952 W/m^2) / (3.0 × 10^8 m/s)
Pavg = .00000539 N/m^2

Finalize Consider the magnitude of the Poynting vector, Savg = 952 W/m^2. It is about the same as the intensity of sunlight at the Earth's surface. For this reason, it is not safe to shine the beam of a laser pointer into a person's eyes, which may be more dangerous than looking directly at the Sun. The pressure has an extremely small value, as expected. (Recall from a previous chapter that atmospheric pressure is approximately 10^5 N/m^2.)

A 10.0 mW helium-neon laser (? = 632.8 nm) emits a beam of circular cross section with a diameter of 2.00 mm.

(a) Find the maximum electric field in the beam.
Emax = V/m

(b) What total energy is contained in a 7.50 m length of the beam?
U = J

(c) Find the momentum carried by a 7.50 m length of the beam.
p = kg·m/s

$Analyze We begin by determining the magnitude of the beam's Poynting vector. Divide the time-averaged power delivered via the electromagnetic wave by the cross-sectional area of the beam: Savg = Pavg / A = Pavg / (?r^2) = (2.7 × 10^-3 W) / (?((1.9 × 10^-3 m)/2)^2) = 952 W/m^2 Analyze Now let's determine the radiation pressure from the laser beam. The equation for radiation pressure exerted on a perfectly reflecting surface indicates that a completely reflected beam would apply an average pressure of Pavg = 2Savg/c. We can model the actual reflection as follows. Imagine that the surface absorbs the beam, resulting in pressure Pavg = Savg/c. Then the surface emits the beam, resulting in additional pressure Pavg = Savg/c. If the surface emits only a fraction f of the beam (so that f is the amount of the incident beam reflected), the pressure due to the emitted beam is Pavg = fSavg/c. Use this model to find the total pressure on the surface due to absorption and re-emission (reflection): Pavg = Savg/c + fSavg/c = (1 + f)Savg/c Evaluate this pressure for a beam that is 70% reflected: Pavg = (1 + 0.70) (952 W/m^2) / (3.0 × 10^8 m/s) Pavg = .00000539 N/m^2 Finalize Consider the magnitude of the Poynting vector, Savg = 952 W/m^2. It is about the same as the intensity of sunlight at the Earth's surface. For this reason, it is not safe to shine the beam of a laser pointer into a person's eyes, which may be more dangerous than looking directly at the Sun. The pressure has an extremely small value, as expected. (Recall from a previous chapter that atmospheric pressure is approximately 10^5 N/m^2.) A 10.0 mW helium-neon laser (? = 632.8 nm) emits a beam of circular cross section with a diameter of 2.00 mm. (a) Find the maximum electric field in the beam. Emax = V/m (b) What total energy is contained in a 7.50 m length of the beam? U = J (c) Find the momentum carried by a 7.50 m length of the beam. p = kg·m/s$

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