The charged particle entering a magnetic field describes an arc which is at most a semicircle. If the span of magnetic field is limited, then there is no further bending of path due to magnetic force. Let us consider a case in which a particle traveling in the plane of drawing enters a region of magnetic field at angle α.

We should realize here that even though the charged particle enters magnetic region obliquely (i.e at an angle) in the plane of motion, the directions of velocity and magnetic field are still perpendicular to each other. The particle, in turn, follows a circular path. However, the particle needs to move in the region behind the boundary YY’ in order to complete the circular path. But, there is no magnetic field behind the boundary. Therefore, the charged particle is unable to complete the circular path. From geometry, it is clear that point of entry and point of exit are points on the circle which is intersected by the boundary YY’. By symmetry, the angle that the velocity vector makes with the boundary YY’ at the point of entry is same as the angle that velocity vector makes with the boundary YY' at the point of exit.

By geometry, the angle between pair of lines is same as the angle between the lines perpendicular to them. Hence,

∠
O
A
D
=
∠
C
O
D
=
α
∠
O
A
D
=
∠
C
O
D
=
α

and

⇒
∠
A
O
C
=
2
α
⇒
∠
A
O
C
=
2
α

The length of arc, AEC is :

l
=
A
E
C
=
2
α
R
l
=
A
E
C
=
2
α
R

Substituting for R, we have :

⇒
l
=
2
α
m
v
q
B
⇒
l
=
2
α
m
v
q
B

The time of travel in the magnetic field is :

⇒
t
=
l
v
=
2
α
m
q
B
⇒
t
=
l
v
=
2
α
m
q
B

When charged particle enters magnetic field at right angle, velocity vector is perpendicular to the boundary of magnetic field. We know that a tangent can be drawn on a circle in this direction only at the points obtained by the intersection of the circle by the boundary line which divides the circle in two equal sections. A charged particle can, therefore, travel a semicircular path when it enters into the region magnetic field at right angle, provided of course the span of magnetic is sufficient.

We should understand that circular arc path as obtained by the analysis above can be subject to availability of magnetic field till the charged particle begins to move backwards. For a smaller extent of the magnetic field, we find that the particle emerges out of the magnetic field without being further deviated. If the extent of magnetic field is greater than or equal to R, then charged particle describes up to a semicircle depending on the angle at which it enters magnetic region. However, if the extent of magnetic field is less than R, then particle emerges out of the magnetic field without being further deviated.