A sensor is exposed for time t to a lamp of power P placed at a distance l. The sensor has an opening that is 4d in diameter. Assuming all energy of the lamp is given off as light, the number of photons entering the sensor if the wavelength of light is `lamda` is

Options

A. `N=(Plamdad^2t)/(hcl^2)`
B. `N=(4lamdad^2t)/(hcl^2)`
C. `N=(Plamdad^2t)/(4hcl^2)`
D. `N=(Plamdad^2t)/(16hcl^2)`

Explanation

To solve the problem of finding the number of photons entering the sensor, we can follow these steps:

### Step 1: Calculate the Power per Unit Area Reaching the Sensor
The lamp emits power \( P \) uniformly in all directions. The power per unit area at a distance \( l \) from the lamp can be calculated using the formula for the surface area of a sphere:

\[ \text{Power per unit area} = \frac{P}{\text{Surface Area of Sphere}} = \frac{P}{4\pi l^2} \]

### Step 2: Calculate the Area of the Sensor
The sensor has a diameter of \( 4d \), so its radius \( r \) is \( 2d \). The area \( A \) of the sensor can be calculated using the formula for the area of a circle:

\[ A = \pi r^2 = \pi (2d)^2 = 4\pi d^2 \]

### Step 3: Calculate the Incident Power on the Sensor
The incident power \( P_{\text{incident}} \) on the sensor can be calculated by multiplying the power per unit area by the area of the sensor:

\[ P_{\text{incident}} = \left(\frac{P}{4\pi l^2}\right) \times (4\pi d^2) = \frac{P d^2}{l^2} \]

### Step 4: Calculate the Total Energy Incident on the Sensor
The total energy \( E \) incident on the sensor over a time \( t \) is given by:

\[ E = P_{\text{incident}} \times t = \left(\frac{P d^2}{l^2}\right) \times t = \frac{P d^2 t}{l^2} \]

### Step 5: Calculate the Energy of a Single Photon
The energy \( E_{\text{photon}} \) of a single photon is given by the equation:

\[ E_{\text{photon}} = \frac{hc}{\lambda} \]

where \( h \) is Planck's constant and \( c \) is the speed of light.

### Step 6: Calculate the Number of Photons
The number of photons \( n \) entering the sensor can be calculated by dividing the total energy incident on the sensor by the energy of a single photon:

\[ n = \frac{E}{E_{\text{photon}}} = \frac{\frac{P d^2 t}{l^2}}{\frac{hc}{\lambda}} = \frac{P d^2 t \lambda}{h c l^2} \]

Thus, the final expression for the number of photons entering the sensor is:

\[ n = \frac{P d^2 t \lambda}{h c l^2} \]

### Final Answer:
The number of photons entering the sensor is given by:

\[ n = \frac{P d^2 t \lambda}{h c l^2} \]

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