Displacement of a Charged Particle in a Magnetic Field:

Introduction:
When a charged particle enters a magnetic field, it experiences a force due to the interaction between its charge and the magnetic field. This force can cause the particle to move in a curved path, resulting in a displacement from its initial position.

Given Information:
- Magnetic induction B exists on the other side of the line.
- A charged particle with charge q and mass m enters the magnetic field at an angle of 45 degrees.

Explanation:

1. Force on the Charged Particle:
When a charged particle with charge q enters a magnetic field with magnetic induction B, it experiences a force given by the equation:

F = q * v * B * sin(θ)

Where:
- F is the force on the charged particle
- q is the charge of the particle
- v is the velocity of the particle
- B is the magnetic induction
- θ is the angle between the velocity vector and the magnetic induction vector

In this case, the angle θ is 45 degrees, so the force on the particle can be written as:

F = q * v * B * sin(45°)

2. Circular Motion:
When a charged particle enters a magnetic field at an angle, it experiences a force perpendicular to its velocity. This force acts as a centripetal force, causing the particle to move in a circular path.

The centripetal force acting on the charged particle is given by the equation:

F_c = (m * v^2) / r

Where:
- F_c is the centripetal force
- m is the mass of the particle
- v is the velocity of the particle
- r is the radius of the circular path

Since the force on the particle due to the magnetic field is acting as the centripetal force, we can equate the two forces:

q * v * B * sin(45°) = (m * v^2) / r

3. Displacement of the Particle:
To find the displacement of the particle when it emerges out of the magnetic field, we need to determine the distance it travels in the circular path.

The distance traveled in a circle is given by the formula:

d = 2 * π * r

To find the radius of the circular path, we can rearrange the equation from step 2:

r = (m * v) / (q * B * sin(45°))

Substituting this value of r in the formula for distance, we get:

d = 2 * π * [(m * v) / (q * B * sin(45°))]

Conclusion:
The displacement of the charged particle when it emerges out of the magnetic field is equal to the distance it travels in the circular path. This distance can be calculated using the formula:

d = 2 * π * [(m * v) / (q * B * sin(45°))]

Note: Please note that this is a general explanation of the displacement of a charged particle in a magnetic field. The specific values and calculations may vary depending on