Light of wavelength 221 nm falls on a totally reflecting plane mirror at an angle `30^@` with the mirror . If `10^(20)` photons are striking the mirror per second, then the force exerted on the mirror is `F = alpha xx 10^(-7)N` , Find the value of `alpha`
`{:(0,1,2,3,4,5,6,7,8,9):}`

Explanation

To solve the problem, we need to calculate the force exerted on a totally reflecting plane mirror by light of a given wavelength and number of photons striking the mirror per second. Here are the steps:

### Step 1: Convert Wavelength to Meters
The wavelength of light is given as 221 nm. We need to convert this to meters.
\[ \lambda = 221 \, \text{nm} = 221 \times 10^{-9} \, \text{m} \]

### Step 2: Calculate the Linear Momentum of One Photon
The linear 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:
\[ p = \frac{6.63 \times 10^{-34}}{221 \times 10^{-9}} \approx 3.00 \times 10^{-27} \, \text{kg m/s} \]

### Step 3: Calculate the Change in Linear Momentum
Since the light is totally reflected, the change in momentum \( \Delta p \) for each photon is given by:
\[ \Delta p = 2p \cos \theta \]
where \( \theta \) is the angle of incidence with respect to the normal. Given that the angle with the mirror is \( 30^\circ \), the angle with the normal is \( 60^\circ \).
\[ \Delta p = 2p \cos(60^\circ) = 2 \times 3.00 \times 10^{-27} \times \frac{1}{2} = 3.00 \times 10^{-27} \, \text{kg m/s} \]

### Step 4: Calculate the Total Change in Momentum per Second
The total number of photons striking the mirror per second is given as \( n = 10^{20} \). Therefore, the total change in momentum per second (which is the force) can be calculated as:
\[ F = n \Delta p = 10^{20} \times 3.00 \times 10^{-27} = 3.00 \times 10^{-7} \, \text{N} \]

### Step 5: Express the Force in the Given Format
The force can be expressed as:
\[ F = \alpha \times 10^{-7} \, \text{N} \]
From our calculation, we have:
\[ 3.00 \times 10^{-7} = \alpha \times 10^{-7} \]
Thus, we find:
\[ \alpha = 3 \]

### Final Answer
The value of \( \alpha \) is \( 3 \).

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