Cyclic Accelerator

Cyclic Accelerator(Cyclotron)

In cyclic accelerator, the charge particles moves in curved path or circular path. the motion of particles is accelerator by alternating electric field and path of particle is bend alternating electric field and path of particle is bend in circular path by magnetic field.

Theory:

Let V be the potential difference between \(D_1\) and \(D_2\) at any instant of time. The K.E gain by charge particle of mass 'm' and charge 'e' is given by,

\(K.E=ev\)

\(\frac12 mv^2=ev\dotsm(1)\)

\(\Rightarrow v=\sqrt\frac{2ev}{m}\)

Where, v= speed of charge particle.

\(D_1\) & \(D_2\) are two semi-circular plates connected to alternating voltage source. If the charge particle leaves the source 's' when \(D_1\) is at positive potential and \(D_2\) is at negative potential then the magnetic force due to B is

\(Bev=\frac{mv^2}{r}\dotsm(2)\)

where, r=radius of circular path

or \(r= \frac{mv}{Be}\)

\(\Rightarrow r\propto v\)

The radius of circular path is directly proportional to speed of charge particle.

As the charge particle comes in between \(D_1\) & \(D_2\) in half time period (\(\frac T2=t\)) (inside electric field)

The speed of charge particle increases due to electric field. The radius of circular path also increases. the charge particle moves in larger circular path.The time taken by the charge particle to move in semi-circular path is

\(\frac T2=t=\biggl(\frac{2\pi m}{Be}\biggr)\times\frac12\)

=\(\frac{\pi m}{Be}\dotsm(3)\)

The K.E of charge particles in terms of B is

K.E=\frac12 mv^2\)

=\(\frac12 m(\frac{Bev}{m}0^2\)

=\(\frac12 m\frac{B^2 e^2 r^2}{m^2}\)

\(K.E=\frac12\frac{B^2 e^2 r^2}{m}\dotsm(4)\)

The maximum K.E of charge particle is

\((K.E)_max=\frac12\frac{B^2e^2r^2_max}{m}\dotsm(5)\)

Maximum K.E in terms of frequency of applied field:

From equation (2) , \(T=\frac{2\pi m}{Be}\)

\(\therefore f= \frac1T=\frac{Be}{2\pi m}\)

From equation (4) and (5)

(\(K.E)_{max}=\frac12 \frac{4\pi^2 f^2 m^2}{m}\cdot r^2_{max}\)

\(\therefore\;\;\;(K.E)_{max}=2\pi^2f^2m\cdot r^2_{max}\dotsm(7)\)

Limitation of cyclotron :

1. The frequency of oscillating charge particle should be same as frequency of applied A-C to accelerate charge particle in each time when it combine between \(D_1\) & \(D_2\). This condition is known as 'Resonance'.

The frequency of charged particle is

\(f=\frac{Be}{2\pi m}\)

As speed of charge particle increases its mass also increases

\(m=\frac{m_o}{\sqrt{1-\frac{v^2}{c^2}}}\)

The frequency of oscillating charge particle decreases ( times period increases). It takes longer time to move circular path. the charge particles lags behind the frequency of a.c so it couldn't be further accelerated.

To overcome above limitation.

i.e \( f=\frac{Be}{2\pi m_\circ}\sqrt{1-\frac{v^2}{c^2}}\)

i. Variation of magnetic field.

Frequency of oscillating charge particle kept constant by increasing strength of magnetic field in such a way that

\(B\sqrt{1-\frac{v^2}{c^2}}= constant \Rightarrow f= constant \)

such type of cyclotron is known as synchroton.

II. Variation of frequency of source

The frequency of applied is so adjusted that frequency of oscillating charge particle is equal to frequency of a.c. This type of yclotron is known as frequency modulated cyclotron or synchro-cyclotron.

2. Neutron and Neutral particle cannot be accelerated by cyclotron.

3. Electron and light particles can not be accelerated in cyclotron.

Instead of increasing K.E of electron it's mass increases rapidly as given by equation

\(m=\frac{m_\circ}{\sqrt{1-\frac{v^2}{c^2}}}\)

4. Maximum radius of curvature

If the charge particle moves 'n' times the gap between \(D_1\) &\(D_2\) then

\(K.E)_{max}= n.eV\)

\(\frac12 \frac{B^2e^2 v_{max}^2}{m}=nev\)

\(\therefore r_{max}=\sqrt{\frac{2meV}{Be}}\)

\(\Rightarrow r_{max}\propto \sqrt n\)

The maximum value of radius of circula rpath is equal to radius of circular Dees.

\(\therefore r_{max}=R\)

Reference:

Reviews of Modern Physics. Lancaster, P.A.: Published for the American Physical Society by the American Institute of Physics, 1952. Print.

Wehr, M. Russell, and James A. Richards. Physics of the Atom. Reading, MA: Addison-Wesley Pub., 1984. Print.

Young, Hugh D., and Roger A. Freedman. University Physics. Boston, MA: Pearson Custom, 2008. Print.

Adhikari, P.B. A Textbook of Physics. 2070 ed. Vol. II. Kathmandu: Sukunda Publication, 2070. Print.