A narrow beam of monochromatic light of wavelength `lamda` emitted from a source of power P is propogating in the positive x-direction. After being reflected from a perfectly reflecting plane mirror of area vector `vecA=A(-hati-hatj)`, the beam falls on a metal plate of surface area A[`vecA` is along outward normal at mirror]. the force exerted by light beam on the mirror is `vecF` and `Deltavecp` is change in momentum of each photon then [cis speed of light in vacuum]

Options

A. `DeltavecF=(P)/(c)(hati+hatj),Deltavecp=(h)/(lamda)(-hati-hatj)`
B. `DeltavecF=(P)/(c)(hati+hatj),Deltavecp=(h)/(lamda)(hati+hatj)`
C. `DeltavecF=(P)/(c)(-hati-hatj),Deltavecp=(h)/(lamda)(-hati-hatj)`
D. `DeltavecF=(P)/(c)(-hati-hatj),Deltavecp=(h)/(lamda)(+hati+hatj)`

Explanation

To solve the problem step by step, we will analyze the situation involving the monochromatic light beam, its interaction with the mirror, and the resulting force and momentum change.

### Step 1: Understanding the Problem
We have a monochromatic light beam of wavelength \( \lambda \) emitted from a source with power \( P \). The beam travels in the positive x-direction and reflects off a perfectly reflecting mirror. The area vector of the mirror is given as \( \vec{A} = A(-\hat{i} - \hat{j}) \).

### Step 2: Momentum of a Photon
According to the de Broglie hypothesis, the momentum \( p \) of a single photon is given by:
\[ p = \frac{h}{\lambda} \]
where \( h \) is Planck's constant.

### Step 3: Initial and Final Momentum of the Photon
- The initial momentum of the photon traveling in the positive x-direction is:
\[ \vec{p}_{\text{initial}} = \frac{h}{\lambda} \hat{i} \]
- Upon reflection, the photon will change direction but maintain the same magnitude of momentum. The final momentum will be:
\[ \vec{p}_{\text{final}} = -\frac{h}{\lambda} \hat{j} \]

### Step 4: Change in Momentum of a Single Photon
The change in momentum \( \Delta \vec{p} \) for a single photon is calculated as:
\[ \Delta \vec{p} = \vec{p}_{\text{final}} - \vec{p}_{\text{initial}} = -\frac{h}{\lambda} \hat{j} - \frac{h}{\lambda} \hat{i} = -\frac{h}{\lambda} (\hat{i} + \hat{j}) \]

### Step 5: Total Change in Momentum
If \( n \) is the total number of photons incident on the mirror in a time interval \( \Delta t \), the total change in momentum \( \Delta \vec{P} \) is:
\[ \Delta \vec{P} = n \Delta \vec{p} = n \left(-\frac{h}{\lambda} (\hat{i} + \hat{j})\right) \]

### Step 6: Number of Photons Incident
The energy emitted in time \( \Delta t \) is:
\[ E = P \Delta t \]
The energy of a single photon is:
\[ E_{\text{photon}} = \frac{hc}{\lambda} \]
Thus, the total number of photons \( n \) can be expressed as:
\[ n = \frac{P \Delta t}{\frac{hc}{\lambda}} = \frac{P \lambda \Delta t}{hc} \]

### Step 7: Force Exerted by the Light Beam
The force \( \vec{F} \) exerted on the mirror can be calculated using the rate of change of momentum:
\[ \vec{F} = \frac{\Delta \vec{P}}{\Delta t} = \frac{n \Delta \vec{p}}{\Delta t} \]
Substituting for \( n \) and \( \Delta \vec{p} \):
\[ \vec{F} = \frac{P \lambda \Delta t}{hc} \left(-\frac{h}{\lambda} (\hat{i} + \hat{j})\right) \frac{1}{\Delta t} \]
This simplifies to:
\[ \vec{F} = \frac{P}{c} (-\hat{i} - \hat{j}) \]

### Final Result
Thus, the force exerted by the light beam on the mirror is:
\[ \vec{F} = \frac{P}{c} (-\hat{i} - \hat{j}) \]
And the change in momentum of each photon is:
\[ \Delta \vec{p} = -\frac{h}{\lambda} (\hat{i} + \hat{j}) \]