(a) Deduce an expression for the frequency of revolution of a charged particle in a magnetic field and show that it is independent of velocity or energy of the particle.
(b) Draw a schematic sketch of a cyclotron. Explain, giving the essential details of its construction, how it is used to accelerate the charged particles.
Answers
Derived the expression for frequency of revolution of a charged particle in magnetic field
(a)
• When a charged particle with charge q & velocity v enters the region of magnetic field B it experiences magnetic Force
| F | = qvB
This F behaves as centripetal force
mv²/R = qvB
R = mv/qB _______(1)
Now , Time period of revolution will be
T = Dist./speed = 2πR/v
T = 2π(mv/qB)/v
T = 2πm/qB
•Now , frequency = 1/T
frequency = qB/2πm
Angular frequency = 2π/T = qB/m
•Hence , frequency of revolution of a charged particle in a magnetic field and show that it is independent of velocity or energy of the particle.
(b) Explained construction & principle of cyclotron with diagram
•Cyclotron : It is an experimental device which is used to accelerate heavy charged particles to a very high speed .
•Principle :
A cyclotron uses a uniform magnetic field & a uniform Electric feild. It is based on the principle that a charged particle can be accelerated to a very high energy with a small alternating Electric feild. By making it to cross the same feild , time again & again with the help of magnetic field.
•Construction:
1) Cyclotron consists of two D -shaped hollow semicircular metal chambers D1 & D2 called as D's
2) Two D's are placed horizontally with a small gap , where the modest potential difference which alternates in polarity with a certain frequency is maintained.
3) This system of two D's are enclosed in a chamber where there exists a uniform magnetic field within the poles of strong electromagnet . The magnetic feild so produced is perpendicular to the plane of the D's
Cyclotron
Explanation:
(a) Magnetic force on charged particle in magnetic field is:
F = qvB
This is equal to the centripetal force acting on the particle.
F = mv²/r = qvB
v/r = qB/m
Frequency f v/2πr = qB/2πm
So frequency is independent of velocity of the particle.
(b) Please find the attached picture for a schematic diagram of a cyclotron.
Cyclotron is used to accelerate particles to very high speeds. Protons are injected from a source in a region consisting of orthogonal electric and magnetic fields. The particle feels a centripetal force due to the magnetic field and an accelerating force due to the electric field.
Radius of a particle in the cyclotron r = mv/qB
We can find the speed of the particle if we know the radius of the particle, which can be used to accelerate the particle to a required value.
