A proton is moving in a uniform magnetic field B in a circular path of radius a in a direction perpendicular to Z- axis along which field B exists. Calculate the angular momentum. If the radius is a and charge on proton is e.

Options

A. `(Be)/(a^(2))`
B. `eB^(2)a`
C. `a^(2) eB`
D. `aeB`

Answer

To solve the problem of calculating the angular momentum of a proton moving in a uniform magnetic field, we can follow these steps:

### Step 1: Understand the situation
We have a proton moving in a circular path of radius \( a \) in a uniform magnetic field \( B \). The motion is perpendicular to the z-axis, which means the magnetic field is aligned along the z-axis.

### Step 2: Write the formula for angular momentum
The angular momentum \( L \) of an object moving in a circular path can be expressed as:
\[ L = m v r \]
where \( m \) is the mass of the proton, \( v \) is the velocity of the proton, and \( r \) is the radius of the circular path, which is given as \( a \).

### Step 3: Identify the centripetal force
The magnetic force acting on the proton provides the necessary centripetal force to keep it moving in a circular path. The magnetic force \( F \) on a charged particle moving in a magnetic field is given by:
\[ F = q v B \sin(\theta) \]
Since the velocity is perpendicular to the magnetic field, \( \theta = 90^\circ \) and \( \sin(90^\circ) = 1\). Thus, the magnetic force becomes:
\[ F = q v B \]

### Step 4: Set the magnetic force equal to centripetal force
The centripetal force required to keep the proton in circular motion is given by:
\[ F_c = \frac{m v^2}{r} \]
Setting the magnetic force equal to the centripetal force gives us:
\[ q v B = \frac{m v^2}{a} \]

### Step 5: Solve for velocity \( v \)
We can rearrange the equation to solve for \( v \):
\[ q v B = \frac{m v^2}{a} \]
\[ q B = \frac{m v}{a} \]
\[ v = \frac{q B a}{m} \]

### Step 6: Substitute \( v \) into the angular momentum formula
Now, we substitute \( v \) back into the angular momentum formula:
\[ L = m v a \]
Substituting for \( v \):
\[ L = m \left(\frac{q B a}{m}\right) a \]
\[ L = q B a^2 \]

### Step 7: Substitute the charge of the proton
The charge of the proton \( q \) is denoted as \( e \). Therefore, we can write:
\[ L = e B a^2 \]

### Final Answer
The angular momentum of the proton moving in the magnetic field is:
\[ L = e B a^2 \]