*||gm law of resultant of two vectors
Resultant of two vectors inclined at an angle θ is given by the diagonal of the parallelogram formed by them. If 
are along adjacent sides of a parallelogram 
, then will be the diagonal of the same parallelogram provided 
all have same order. act at an angle θ have, then
c2 = a2 +b2 + 2 ab cosθ [Law of cosine
Velocity selector
A beam of charged particles is passed in a region of electric & magnetic fields acting at right angles (called cross fields) with such a velocity that the electric & magnetic force balances each other & the beam of charge particles pass undeflected through the region of cross fields. i.e.
Felectric = Fmagnetic ⇒ qE = Bqv ⇒ 
Hall effect
Is the phenomena of production of transverse emf 
in a current carrying strip of metal or a semiconductor when it is placed in a region of uniform magnetic field acting at right angles to current. It can be used to, calculate drift velocity of charge carriers, number density of charge carriers & nature of charge carriers.
Deviation of charge in magnetic field
Consider a positive charge of charge + q & mass m fired horizontally in to a region of uniform magnetic field acting normally inwards to the plane of as shown in the diagram.
Let x < r be the thickness of the magnetic field region. Let δ is the deviation suffered by the charge particle as it comes out of the region of magnetic field. From diagram we can write
Cyclotron
Also called magnetic resonance oscillator) & is the first circular accelerator designed was by American Physicist Ernest O. Lawrence. + vely charged particles like proton, deutron & alpha particles etc. can be suitably accelerated by cyclotron. It can’t be used to accelerate light particles (e.g. electrons) & neutral particles (e.g. neutrons).
Charge particle in magnetic field
A point charge moving in uniform magnetic field experiences a force on 
. Behaviour of charge particle depends on the angle ‘θ ‘ between 
.
Case 1, if θ = 00 or 1800
When the charged particle moves parallel or anti parallel to field then no net force acts on it & its trajectory remains a straight line.
Case 2, If θ is other than 00, 1800 & 900
Charge particle moves in a helical path of radius 
, completes one circle in time 
& travels a distance in one time period in the direction of field called pitch & is given by,
Case 3, If θ is 900
Charge particle moves in a circle at uniform speed.
Radius of circular path is 
KE of particle is 
constant. It completes a cycle in time 
at angular frequency 
.
As the force acting on the particle is only normal & no tangential force is available thus speed & hence KE also of the particle will remain unchanged. As the kinetic energy of the charged particle remains constant. Hence no work is done in moving a charge particle moving at right angles to the magnetic field. However due to change in direction 
& are variables in direction.
Toroid
Magnetic field inside is B = m0 nI & outside is zero.
Solenoid
Mag. field on the axis of a solenoid is
If solenoid is very long, loops are tight, then for a point situated on the axis & well inside, B = μ0 nI & for a point near end on the axis, 
Field pattern of a long solenoid & a bar magnet are similar, with the difference that magnetic field is maximum at the end of poles where as for a long solenoid it is at centre.
Thick solid conductor or rod
Magnetic field for a thick sold conductor or a rod of uniform current density using Ampere’s law is
Also inside a piper carrying hollow conductor magnetic field is zero inside everywhere, maximum at surface. Outside & at surface described by same functions as that of a rod.
Ampere’s Circuital Law
Ampere’s law is useful to find magnetic field in symmetrical situations. It states that the line integral of magnetic field for a closed plane curve is equal to ‘m 0’ times the current crossing the area bounded by the closed curve provided the electric field inside the loop remains the constant. i.e.
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