now let us study the situation of projection
of a charge particle at some angle to magnetic
field. say these are the magnetic line of
forces of uniform magnetic induction b. and
in this region say we are given with a charge
plus q, which is projected into the magnetic
fielded with a velocity vector v. at an angle
theta to the direction of magnetic induction.
in this situation we can directly state, a
component of velocity which is v coz theta.
that will be parallel to the direction of
magnetic induction and another component v
sine theta will be, perpendicular to the direction
of magnetic induction. here if we talk about
the direction of motion of charge parallel
to magnetic field it does not experience any
magnetic force. and it has a tendency to move
straight. and the component of velocity with
which the charge is moving perpendicular to
the direction of magnetic field. it experiences
a magnetic force. and here we can write. magnetic
force on the charge particle will be q v-b
sine theta as theta is the angle between the
2, so here we can write it is due to, the
component v sine theta, it is experiencing
the magnetic force. if we just have a look
on the direction of magnetic force by righthand
palm rule we can see magnetic induction is
toward right. so if i point my fingers to
right and thumb along the direction of positive
charge. so my palm is facing in inward direction.
so it’ll experience the magnetic force,
in the direction into the plane of the surface.
and due to this v sine theta it’ll have
a tendency to circulate the charge. with its
central line in the direction of an axis passing
through this point and perpendicular to the
surface. and it’ll have a circulation tendency.
and v coz theta will be having a tendency
to pull it in forward direction. so resulting
motion of this charge particle will be bounded
in this cylindrical zone. and due to this
component v-sine theta it’ll be circulating
and due to v-coz theta it’ll be continuously
moving forward. so the resulting path of particle
will be, a helical path. if we can write this
will be a helical path. which the particle
will follow. and the radius of this helical
path r-h will be due to the circulation of
this particle. by the speed v-sine theta.
so we can directly write here radius of. this
helical path or radius of this hellix which
is. traced by the particle we can write as,
r-h is equal to. the formula is m v by q b
so here we write m-v sine theta by q-b. that
is the expression use for the radius of helical
path. similarly we can find out the pitch
of this hellix, or the helical path that is
the distance covered by the particle during
1 complete revolution. so here it is following
. the path along the, direction of magnetic
field with the velocity v coz theta. so here
we can write directly, pitch of hellix. can
be given as p-h this is. v-coz theta multiplied
by time period of revolution. and the time
period of revolution we can separately find
out this 2 pie by omega. and we already studied.
that, the value of angular speed is independent
of the velocity of particle in a magnetic
field which is given as q-b by m. so the time
period of revolution here is 2 pie m over
q-b. if we substitute it here pitch of hellix
we are getting as 2 pie. m v coz theta by
q-b this is the expression for calculation
of. pitch of the helical path that is the
distance. covered by the particle along the
direction of magnetic field during one revolution.
and this actual situation of this helical
path. or we can also explain with a realistic
situation, lets continue on the next sheet
to understand, how this helical path is being
traced by the particle once again. in continuation
lets discuss a realistic situation where.
we can practically understand how the helical
path is being followed. here you can see this
is the uniform magnetic field in the region
which exist from left to right. and from.
left-side a charge plus q enters into it,
with a velocity v vector. for reference we
can consider the point where the charge is
getting into the magnetic field as origin.
horizontal direction as x axis and vertical
direction as y axis and. in outward direction
positive z axis. now here you can see. the
horizontal component of velocity v coz theta
particle will not experience any force. so
here we can say. force acting on the particle
due to v coz theta here is equal to zero.
and force experienced due to the velocity
v sine theta here will be. q v b sine theta.
and here you can see by, righthand palm rule
here fingers are pointing in, magnetic induction.
thumb along the direction of v-sine theta
so palm is pointing in the direction of minus
z axis. so we can put the unit vector minus
k cap here. now if we just rotate this situation
and have a look on this situation. from the
side of. x axis. then we can see. the magnetic
force acting on, the particle is due to v-sine
theta is toward left. and when the particle
is starts it moving. in a circular path due
to v sine theta about the centre c which we
can see here in this situation. so the radius
of this helical path, r-h, or you can simply
say radius of this circle is. m-v sine theta
by q-b. and if we just repeat the experiment
and have a look on the situation once again.
in 3 dimensional situation here you can see
when the particle is thrown, it follows the
helical path. in, the direction of x axis
and if look on it from side of minus x axis,
it looks like a circle, as. the projection
of helical path on x-y plane here is the circle.
so with this, realistic situation you can
understand. how the helical path is being
followed by the charge particle.
