A charged particle enters a magnetic field at right angles to the field. The field exists for a length equal to `1.5` times the radius of circular path of particle. The particle will be deviated from its path by

Options

A. `90^(@)`
B. `sin^(-1)(2//3)`
C. `30^(@)`
D. `180^(@)`

Answer

To solve the problem of how much a charged particle is deviated from its path when it enters a magnetic field at right angles, we can follow these steps:

### Step 1: Understand the Motion of the Charged Particle
When a charged particle enters a magnetic field at right angles, it experiences a magnetic force that causes it to move in a circular path. The radius of this circular path can be determined using the formula:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

### Step 2: Determine the Length of the Magnetic Field
The problem states that the magnetic field exists for a length equal to \( 1.5 \) times the radius of the circular path. Hence, the length of the magnetic field \( L \) can be expressed as:
\[ L = 1.5r \]

### Step 3: Calculate the Maximum Displacement
The maximum displacement \( x_{\text{max}} \) of the particle while it is within the magnetic field can be calculated. Since the particle moves in a circular path, the maximum deviation from the original path will be equal to the radius \( r \) when it exits the magnetic field:
\[ x_{\text{max}} = r \]

### Step 4: Relate the Length of the Magnetic Field to the Circular Path
Since the magnetic field length is \( 1.5r \), and the maximum displacement \( x_{\text{max}} \) is equal to \( r \), we can see that the particle will not complete a full circular path before exiting the magnetic field. Instead, it will deviate by a distance equal to the radius \( r \).

### Step 5: Determine the Angle of Deviation
When the particle exits the magnetic field, it will reverse its direction. The initial direction of the particle is perpendicular to the magnetic field, and upon exiting, it will be directed opposite to its initial path. This results in a deviation of:
\[ \Phi = 180^\circ \]

### Conclusion
Thus, the charged particle will be deviated from its path by an angle of \( 180^\circ \).

### Final Answer
The particle will be deviated from its path by \( 180^\circ \).