Question

A sodium lamp emits light at the power P = 70.0 W and at the wavelength λ = 600 nm, and the emission is uniformly in all directions.

(a) At what rate are photons emitted by the lamp?

(b) At what distance from the lamp will a totally absorbing screen absorb photons at the rate of 1.00 photon /cm2s?

(c) What is the rate per square meter at which photons are intercepted by a screen at a distance of 1.90 m from the lamp?

Answer 1

a) # = 21,116 10¹⁹ photons , b) R = 4,099 10⁷ m , c) I = 4.66 10¹⁸ photons / m²

Step-by-step explanation:

a) In this problem we need to know the energy of each photon and the total emitted power

Let's start with the Planck equation

E = h f

c =λ f

E = h c / λ

E = 6.63 10⁻³⁴ 3 10⁸/600 10⁻⁹

E = 3,315 10⁻¹⁹ J

This is the energy of a photon

The definition of power is

P = W / t

Work is equal to energy change

P = E / t

E = 70.0 1

`E_(total)` = 70 J

This is the total energy emitted

Let's use a rule of proportions to find the number of photons

# =
`E_(total)` / E

# = 70 / 3,315 10⁻¹⁹

# = 21,116 10¹⁹ photons

b) The definition of intensity is

I = P / A

P = I A

We want an intensity

I = 1 photon / cm²

The area of ​​a sphere is

A = 4π r²

P = I 4π R²

R = √ P / 4π I

The power is equal to the number of photons per energy of each photon emitted per unit of time

R = √ (#
`E_(total)` / 4π I)

Intensity is

I = 1 photon / cm² (100 cm / 1m) 2 = 1 10⁴ photon / m²

R = √ (21,116 10¹⁹ / 4π 10⁴)

R = √ (16.80 10¹⁴)

R = 4,099 10⁷ m

c) we calculate the intensity

A = 4π R²

A = 4π 1.9²

A = 45.36 m²

I = P / A

I = 21,116 10¹⁹ / 45.36

I = 4.66 10¹⁸ photons / m²