A parallel beam of particles each of mass `m` moving with velocity `v` impinges on a wall at an angle `theta` to its normal. The number of particles per unit volume in the beam is `n`. If the collision of particles with the wall is elastic, then the pressure exerted by this beam on the wall is

Options

A. `2mnv^(2)costheta`
B. `2mnv^(2)cos^(2)theta`
C. `2mn v costheta`
D. `2mn v cos^(2)theta`

Explanation

To find the pressure exerted by a parallel beam of particles on a wall, we will follow these steps:

### Step 1: Understand the problem
We have a beam of particles, each of mass \( m \), moving with a velocity \( v \) at an angle \( \theta \) to the normal of a wall. The number of particles per unit volume in the beam is \( n \). The collisions with the wall are elastic, meaning the particles will bounce back with the same speed but in the opposite direction.

### Step 2: Resolve the velocity into components
The velocity \( v \) can be resolved into two components:
- The component perpendicular to the wall (normal component): \( v \cos \theta \)
- The component parallel to the wall (tangential component): \( v \sin \theta \)

### Step 3: Calculate the change in momentum
When a particle strikes the wall, it reverses its normal component of velocity. The change in momentum \( \Delta p \) for one particle can be calculated as:
\[ \Delta p = m(v \cos \theta - (-v \cos \theta)) = 2m v \cos \theta \]

### Step 4: Determine the number of particles striking the wall
The volume \( dV \) of the beam of particles striking the wall can be expressed as:
\[ dV = dA \cdot dx \cdot \cos \theta \]
where \( dA \) is the area of the wall and \( dx \) is the thickness of the beam.

The number of particles \( dN \) in this volume is:
\[ dN = n \cdot dV = n \cdot dA \cdot dx \cdot \cos \theta \]

### Step 5: Calculate the total change in momentum for all particles
The total change in momentum for all particles striking the wall is:
\[ \Delta P = dN \cdot \Delta p = (n \cdot dA \cdot dx \cdot \cos \theta) \cdot (2m v \cos \theta) \]
Substituting the expression for \( dN \):
\[ \Delta P = 2m n v \cos^2 \theta \cdot dA \cdot dx \]

### Step 6: Calculate the force exerted on the wall
The force \( F \) exerted on the wall is the rate of change of momentum. If \( dt \) is the time taken for the particles to strike the wall, we can express the force as:
\[ F = \frac{\Delta P}{dt} \]
The distance \( dx \) can be expressed as \( v \cdot dt \), hence:
\[ F = \frac{2m n v \cos^2 \theta \cdot dA \cdot v \cdot dt}{dt} = 2m n v^2 \cos^2 \theta \cdot dA \]

### Step 7: Calculate the pressure
Pressure \( P \) is defined as force per unit area:
\[ P = \frac{F}{dA} = \frac{2m n v^2 \cos^2 \theta \cdot dA}{dA} = 2m n v^2 \cos^2 \theta \]

### Final Answer
The pressure exerted by the beam of particles on the wall is:
\[ P = 2m n v^2 \cos^2 \theta \]

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