Table of contents  
The Lorentz Force Law: 
1 Crore+ students have signed up on EduRev. Have you? 
The magnetic force on a charge Q, moving with velocity v in a magnetic field
is
This is known as Lorentz force law.
In the presence of both electric and magnetic fields, the net force on Q would be:
Charged Particle enters in the direction of field (Linear motion)
The force on the charge Q in electric field
Acceleration of the charge particle in the direction of the electric field is
If is the position vector at any time t then
where C is a constant.
Since
where C_{1} is a constant
Let at
If initially
The energy acquired by the charged particle in moving from point 1 to 2 is
If the potential difference between points 1 to 2 is V then
If the particle starts from rest i.e (v_{1} = 0) and final velocity is v then
Kinetic energy of the particle
Charged Particle enters in the direction perpendicular to field (Parabolic motion)
Let us consider a charge particle enters in an electric field region with velocity v_{x} at t = 0 . The electric field is in the ydirection and the field region has length l. After traversing a distance l it strikes a point P on a screen which is placed at a distance L from the field region.
Since electric field is in the ydirection, charge particle will experience force
In time t, charge particle will traverse a distance in ydirection and a distance x = v_{x}t in xdirection.
Thus and which represents parabolic path.
Thus distance of point P from the center of the screen is,
Angle of deviation in the field region
Angle of deviation in the field free region,
he magnetic force on a charge Q, moving with velocity in a magnetic field is,
This is known as Lorentz force law.
Charged Particle enters in the direction perpendicular to field (Circular motion)
If a charge particle enters in a magnetic field at angle of 90^{o}, then motion will be circular with the magnetic force providing the centripetal acceleration. As shown in figure, a uniform magnetic field points into the page; if the charge Q moves counter clockwise, with speed v, around a circle of radius R, the magnetic force points inward, and has a fixed magnitude QvB, just right to sustain uniform circular motion:
where R is the radius of the circle and m is the mass of the charge particle.
Momentum of the charged particle p = QBR
Charged Particle enters in the direction making an angle with the field
(Helical motion)
If the charge particle enters in a magnetic field making an angle θ, then motion will be helical.
and the radius of helix is
Charged Particle in Uniform Electric and Magnetic Field (Cycloid motion)
If points in xdirection and points in zdirection, and a particle at rest is released from origin, then particle will follow cycloid motion. Initially, the particle is at rest, so the magnetic force is zero, and the electric field accelerates the charge in zdirection. As it speeds up, a magnetic force develops which pulls the charge to the right. The faster it goes stronger the magnetic force becomes and it curves the particle back around towards the yaxis. At this point the charge is moving against the electric force, so it begins to slow downthe magnetic force then decreases, and the electrical force takes over, bringing the charge to rest at point a and then process repeats.
Let us solve the above differential equations,
For C.F
For P.I.
Initial Condition:
Thus
This is the formula for a circle, of radius R, whose center is ( 0, Rω t ,R ) travels in the ydirection at constant speed,
The curve generated in this way is called a cycloid.
Magnetic forces do not work because is perpendicular to
so,
Magnetic forces may alter the direction in which a particle moves, but they can not speed up or slow down it.
Example 1: A neutron, a proton, an electron and an α – particle enter a region of constant magnetic field with equal velocities. The magnetic field is along the inward normal to the plane of paper. Label the tracks of the particles.
A → Proton, B → α−particle, C → neutron (undeflected), D → electron.
Example 2: A uniform magnetic field with a slit system, as shown in figure, is to be used as a momentum filter for high energy charged particles. With a field B tesla, it is found that the filter transmits α–particles each of energy 5.3 MeV.
The magnetic field is increased to 2.3 B tesla and deuterons are passed into the filter. Find the energy of each deuteron transmitted by the filter.
Example 3: A beam of protons with velocity 4 ×10^{5} m / sec enters a uniform magnetic field of 0.3 Tesla at an angle of 60^{0} to the magnetic field. Find the radius of the helical path taken by the proton beam. Also find the pitch of the helix.
v = 4×10^{5}m / sec , v_{⊥} =v sin 60 and v_{} =v cos 60
86 videos29 docs22 tests
