A homogeneous ball (mass=m) of ideal black material at rest is illuminated with a radiation having a set of photons (wavelength`=lamda`), each with the same momentum and the same energy. The rate at whoch photons fall on the ball is n. the linear acceleration of the ball is

Options

A. `(mlamda)/(nh)`
B. `(nh)/(mlamda)`
C. `(nh)/((2pi)(mlamda))`
D. `(2lamda)/(nh)`

Explanation

To find the linear acceleration of the ball when illuminated by photons, we can follow these steps:

### Step 1: Understand the Problem
We have a homogeneous ball of mass \( m \) that is illuminated by photons with a wavelength \( \lambda \). The rate at which photons strike the ball is \( n \). Each photon has a certain momentum and energy.

### Step 2: Determine the Momentum of a Single Photon
The momentum \( p \) of a single photon can be expressed using the formula:
\[ p = \frac{h}{\lambda} \]
where \( h \) is Planck's constant.

### Step 3: Calculate the Total Momentum Imparted to the Ball
If \( n \) photons are falling on the ball per second, the total momentum imparted to the ball per unit time (which is the force \( F \)) can be calculated as:
\[ F = n \cdot p = n \cdot \frac{h}{\lambda} \]

### Step 4: Apply Newton's Second Law
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration \( a \):
\[ F = m \cdot a \]
Substituting the expression for force from Step 3, we have:
\[ n \cdot \frac{h}{\lambda} = m \cdot a \]

### Step 5: Solve for Linear Acceleration \( a \)
Rearranging the equation to solve for acceleration \( a \):
\[ a = \frac{n \cdot \frac{h}{\lambda}}{m} \]
This simplifies to:
\[ a = \frac{n \cdot h}{m \cdot \lambda} \]

### Final Answer
Thus, the linear acceleration of the ball is:
\[ a = \frac{n \cdot h}{m \cdot \lambda} \]