A parallel beam of monochromatic light of wavelength 663 nm is incident on a totally reflection plane mirror. The angle of incidence is `60^@` and the number of photons striking the mirror per second is `1.0xx10^19`. Calculate the force exerted by the light beam on the mirror.

Explanation

To solve the problem of calculating the force exerted by a beam of monochromatic light on a totally reflecting plane mirror, we can follow these steps:

### Step 1: Understand the Problem
We have a beam of light with a wavelength of 663 nm, incident on a mirror at an angle of incidence of 60 degrees. The number of photons striking the mirror per second is \(1.0 \times 10^{19}\). We need to calculate the force exerted by this beam of light on the mirror.

### Step 2: Convert Wavelength to Meters
The wavelength is given in nanometers, so we convert it to meters:
\[ \lambda = 663 \, \text{nm} = 663 \times 10^{-9} \, \text{m} \]

### Step 3: Calculate the Momentum of a Single Photon
The momentum \(p\) of a single photon can be calculated using the formula:
\[ p = \frac{h}{\lambda} \]
where \(h\) is Planck's constant, \(h = 6.63 \times 10^{-34} \, \text{Js}\).

Substituting the values:
\[ p = \frac{6.63 \times 10^{-34}}{663 \times 10^{-9}} = 1.00 \times 10^{-25} \, \text{kg m/s} \]

### Step 4: Calculate the Total Momentum Change per Second
When photons strike the mirror, they are reflected, which means their momentum changes direction. The change in momentum for each photon is \(2p\) (since it reverses direction).

The total change in momentum per second (\(\Delta p\)) due to \(n\) photons is:
\[ \Delta p = n \cdot (2p) \]
Substituting \(n = 1.0 \times 10^{19}\):
\[ \Delta p = 1.0 \times 10^{19} \cdot (2 \cdot 1.00 \times 10^{-25}) = 2.0 \times 10^{-6} \, \text{kg m/s} \]

### Step 5: Calculate the Force
The force \(F\) exerted by the light beam on the mirror is given by the rate of change of momentum:
\[ F = \frac{\Delta p}{\Delta t} \]
Since we are considering the change in momentum per second, \(\Delta t = 1 \, \text{s}\):
\[ F = 2.0 \times 10^{-6} \, \text{N} \]

### Final Answer
The force exerted by the light beam on the mirror is:
\[ F = 2.0 \times 10^{-6} \, \text{N} \]