Q. 385.0( 1 Vote )
A particle of mass m and positive charge q, moving with a uniform velocity v, enters a magnetic field B as shown in figure.
(a) Find the radius of the circular arc it describes in the magnetic field.
(b) Find the angle subtended by the arc at the centre.
(c) How long does the particle stay inside the magnetic field?
(d) Solve the three parts on the above problem if the charge q on the particle is negative.
Answer :
Given-
Mass of the particle = m
Positive charge on the particle = q
velocity of the particle = v
Magnetic field = B
(a)The radius of the circular arc described by the particle in the magnetic field-
We know,
The radius of the circle is given by,
where,
m is the mass of a proton
v= velocity of the particle
B = magnetic force
q= charge on the particle =
C
(b) The angle subtended by the arc at the centre
Line MAB is the tangent to arc ABC,
When the particle enters into the magnetic field, it follows a path in the form of arc as shown in fig.
Now, the angle described by the charged particle is nothing but the angle ∠ MAO is ,
∠MAO = 90°
Also from fig , ∠NAC = 90°
∠OAC = ∠ OCA = θ
Then, by angle-sum property of a triangle, sum of all angles of a triangle is 1800
∠AOC =180°− (θ + θ)
= 
(c)The time for which the particle stay inside the magnetic field is given by-
Distance covered by the particle inside the magnetic field,is the length of arc subtended by angle θ and the radius
l = rθ
from (1)
l= r
time taken for a complete cycle will be its circumference 
and
the velocity is V -
Also The radius of the circle is given by,
where,
m is the mass of a proton
v= velocity of the particle
B = magnetic force
q= charge on the particle =
C
(d) If the charge q on the particle is negative, then
(i) Radius of circular arc is given by
(ii) The centre of the arc lies within the magnetic field
Therefore, the angle subtended by the arc = 
(iii) The time taken by the particle to cover the path inside the magnetic field
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