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Modern Physics

Cyclotron

A cyclotron is a compact particle accelerator that uses a constant magnetic field and a high-frequency alternating electric field to accelerate charged particles to high energies. Particles move in expanding spiral paths within two hollow, D-shaped metallic electrodes called dees.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Working principle, resonance condition, maximum kinetic energy formula, and frequency independence of the cyclotron for non-relativistic particles.

PGT extension

Relativistic mass variation effects, phase stability, high-frequency modifications (synchro-cyclotron), and particle beam focusing mechanisms.

Prerequisites

Lorentz magnetic force on a moving charge (F = q(v × B))
Centripetal force mechanics in uniform circular motion
Alternating current circuits and resonance phenomena

Construction and Basic Components

Consists of two hollow, semi-circular metal chambers called dees (D1 and D2) connected to a high-frequency alternating potential difference.
Placed inside an evacuated chamber between the pole pieces of a massive electromagnet that provides a uniform, perpendicular magnetic field.
A charged particle source is located at the exact center in the narrow gap between the two dees.

Working Principle

A charged particle injected into the gap experiences an electric field and is accelerated into one of the dees.
Inside the dee, the electric field is zero (shielding effect), and the uniform magnetic field forces the particle to travel in a semi-circular path.
The frequency of the alternating electric field is synchronized with the revolution frequency of the particle so that the electric field reverses polarity precisely when the particle crosses the gap, ensuring continuous acceleration.

Resonance Condition

For efficient acceleration, the frequency of the alternating applied voltage must exactly match the cyclotron frequency of the charged particle.
The time taken to complete a semi-circular path inside a dee is independent of the speed and radius of the particle trajectory.
This independence allows a constant oscillator frequency to be used for non-relativistic particles throughout their acceleration cycle.

Limitations of a Conventional Cyclotron

At very high velocities (relativistic speeds), the relativistic mass of the particle increases, causing a decrease in its revolution frequency.
As a result, the particle falls out of step (synchronism) with the alternating electric field.
Consequently, conventional cyclotrons are unsuitable for accelerating electrons (due to their small mass and rapid onset of relativistic effects) or protons to very high (GeV) energies without modification.

Derivations

Cyclotron Frequency and Time Period

Conditions: Non-relativistic charged particle moving in a uniform perpendicular magnetic field

Equate the magnetic Lorentz force to the required centripetal force: qv$B = \frac{mv^{2}}{r}$
Solve for the radius of the circular path: r = \frac{mv}{qB}
Express the angular frequency as omega = \frac{v}{r} = \frac{qB}{m}, and the frequency as f = \frac{omega}{2pi} = \frac{qB}{2pi m}

Result: f = \frac{qB}{2pi m} and T = \frac{2pi m}{qB}

Maximum Kinetic Energy of Accelerated Particles

Conditions: Particle exits the cyclotron at the maximum possible radius r = R (radius of the dees)

From the radius formula r = \frac{mv}{qB}, express the maximum velocity v_{max} at r = R as $v_{max} = \frac{qBR}{m}$
Substitute v_{max} into the kinetic energy formula $K_{max} = \frac{1}{2}mv_{max}^{2}$
Simplify the resulting expression: $K_{max} = \frac{1}{2}m\left(\frac{qBR}{m}\right)^{2}$

Result: $K_{max} = \frac{q^{2}B^{2}R^{2}}{2m}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Magnetic Radius	$$r = \frac{mv}{qB}$$	m = mass, v = velocity, q = charge, B = magnetic field, r = radius m	v << c, uniform B
Cyclotron Frequency	$$f = \frac{qB}{2pi m}$$	q = charge, B = magnetic field, m = mass, f = frequency Hz	Non-relativistic speeds
Angular Frequency	$$\omega = \frac{qB}{m}$$	q = charge, B = magnetic field, m = mass, omega = angular frequency rad/s	Non-relativistic speeds
Time Period of Revolution	$$T = \frac{2pi m}{qB}$$	m = mass, q = charge, B = magnetic field, T = time period s	Independent of radius and velocity
Maximum Velocity	$$v_{max} = \frac{qBR}{m}$$	q = charge, B = magnetic field, R = outer radius of dee, m = mass m/s	Particle reaches dee periphery
Maximum Kinetic Energy	$$K_{max} = \frac{q^{2}B^{2}R^{2}}{2m}$$	q = charge, B = magnetic field, R = dee radius, m = mass J	Non-relativistic limit
Relativistic Cyclotron Frequency	$$f_{rel} = \frac{qB}{2pi \gamma m_{0}} = \frac{qB}{2pi m_{0}}\sqrt{1 - \frac{v^{2}}{c^{2}}}$$	$m_{0} = rest mass$, gamma = Lorentz factor, c = speed of light Hz	High velocity approaching c
Total Energy Gain per Revolution	$$\delta E = 2qV_{0}$$	q = particle charge, $V_{0} = peak alternating voltage across the gap$ J	Two gaps crossed per full revolution

Graphs and experiments

Spiral Trajectory of Particles in a Cyclotron

Draw: Two semicircular dees with a gap in between, showing a spiral path starting from the center and expanding outward toward the edge.

Interpret: Illustrates that as velocity increases due to gap acceleration, the radius of curvature increases proportionally, resulting in a widening spiral.

Exam use: Frequently asked in conceptual questions regarding path geometry and radius scaling.

Kinetic Energy vs. Radius Squared

Draw: A linear graph plotting K_{max} on the y-axis against R^{2} on the x-axis, passing through the origin.

Interpret: Demonstrates the quadratic dependence of maximum kinetic energy on the dee radius (K propto R^{2}).

Exam use: Used in numerical data interpretation and proportionality analysis.

Frequency vs. Energy (Relativistic Deviation)

Draw: A horizontal straight line representing constant frequency in non-relativistic theory, and a downward curving line showing frequency dropping at high energies for relativistic particles.

Interpret: Highlights why conventional cyclotrons fail for high-energy electrons or protons due to loss of resonance caused by mass increase.

Exam use: PGT-level conceptual testing on limitations and synchro-cyclotron corrections.

Solved numericals

TGT core

A cyclotron is operating with a magnetic field of 1.5 T to accelerate protons (mas$s = 1.67 \times 10^{-27} kg$, charg$e = 1.6 \times 10^{-19} C)$. Calculate the frequency of the alternating electric field required.

Identify given values: B = 1.5 T, $q = 1.6 \times 10^{-19} C$, $m = 1.67 \times 10^{-27} kg$.
Use the cyclotron frequency formula: f = \frac{qB}{2pi m}.
Substitute values: $f = \frac{(1.6 \times 10^{-19})(1.5)}{2 \times 3.1416 \times (1.67 \times 10^{-27})}$.
Calculate result: f approx 2.28 \times 10^{7} Hz = 22.8 MHz.

Answer: 22.8 MHz

PGT extension

Find the maximum kinetic energy (in MeV) of deuterons accelerated by a cyclotron with a dee radius of 0.5 m and a magnetic field of 1.2 T. (Mass of deutero$n = 3.34 \times 10^{-27} kg$, charg$e = 1.6 \times 10^{-19} C)$.

Identify given parameters: B = 1.2 T, R = 0.5 m, $q = 1.6 \times 10^{-19} C$, $m = 3.34 \times 10^{-27} kg$.
Use the maximum kinetic energy formula: $K_{max} = \frac{q^{2}B^{2}R^{2}}{2m}$.
Substitute values: $K_{max} = \frac{(1.6 \times 10^{-19})^{2}(1.2)^{2}(0.5)^{2}}{2 \times 3.34 \times 10^{-27}}$.
Compute joules: $K_{max} = \frac{(2.56 \times 10^{-38})(1.44)(0.25)}{6.68 \times 10^{-27}} = \frac{9.216 \times 10^{-39}}{6.68 \times 10^{-27}} \approx 1.38 \times 10^{-12} J$.
Convert to MeV by dividing by 1.6 \times 10^{-13}: $K_{max} = \frac{1.38 \times 10^{-12}}{1.6 \times 10^{-13}} \approx 8.62 MeV$.

Answer: 8.62 MeV

TGT core

If the magnetic field in a cyclotron is doubled, what happens to the cyclotron frequency of a proton?

Recall the formula for cyclotron frequency: f = \frac{qB}{2pi m}.
Note that frequency is directly proportional to the magnetic field B (f propto B), assuming mass and charge remain constant.
When B is doubled, the frequency f must also double.

Answer: It gets doubled

Assumptions and limitations

Assumption: The magnetic field is perfectly uniform over the entire region of particle motion, which is difficult to maintain at the edges.
Limitation: Relativistic mass increase at high velocities causes the particle to lag behind the electric field, restricting conventional cyclotron energy limits.
Limitation: Cyclotrons cannot accelerate neutral particles (neutrons) or uncharged bodies because magnetic and electric forces act only on moving charges.

Common misconceptions

Common error: The electric field inside the dees accelerates the charged particles.

Correct: The electric field inside the dees is zero; acceleration occurs exclusively in the gap between the dees.

A hollow conductor shields its interior from external electrostatic fields.

Common error: The time period of revolution depends on the speed and radius of the particle.

Correct: The time period of a non-relativistic particle in a cyclotron is independent of its speed and radius.

As speed increases, the radius increases proportionally, keeping the time taken for a semi-circular orbit constant (T = \frac{pi m}{qB}).

Common error: Cyclotrons can accelerate electrons to very high speeds easily.

Correct: Cyclotrons are inefficient for accelerating electrons to high energies.

Electrons have a very small rest mass, causing them to reach relativistic speeds rapidly and lose synchronization with the oscillator frequency.

Quick revision

Cyclotron frequency depends only on the specific charge (q/m) and magnetic field B, independent of radius and speed for non-relativistic particles.
Magnetic Radius: r = \frac{mv}{qB}
Cyclotron Frequency: f = \frac{qB}{2pi m}

Must remember

Cyclotron is a device used to accelerate charged particles like protons, deuterons, and alpha particles to high energies.
The electric field inside the dees is zero due to electrostatic shielding; acceleration occurs only in the narrow gap between the dees.
The time period and frequency of revolution of a non-relativistic charged particle in a cyclotron are independent of its speed and radius.
The resonance condition requires that the frequency of the alternating applied voltage must exactly match the cyclotron frequency of the particle ($f = \frac{qB}{2\pi m}$).
Maximum kinetic energy acquired by a particle in a cyclotron is given by $K_{max} = \frac{q^{2}B^{2}R^{2}}{2m}$.
Conventional cyclotrons are unsuitable for accelerating electrons to high energies because electrons easily attain relativistic speeds, altering their mass and causing loss of synchronism.

Glossary

DeesTwo hollow, semi-circular metal chambers placed inside an evacuated chamber where charged particles move in semi-circular paths.

Cyclotron FrequencyThe frequency at which a charged particle revolves in a magnetic field, given by $f = \frac{qB}{2\pi m}$.

Resonance ConditionThe state where the oscillator frequency of the alternating electric field matches the revolution frequency of the charged particle.

Lorentz FactorFactor $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ used to account for relativistic mass increase at high velocities.

Synchro-cyclotronA modified cyclotron designed to compensate for relativistic mass increase by modulating the oscillator frequency.

Non-Relativistic vs. Relativistic Cyclotron

Parameter	Conventional Cyclotron (Non-Relativistic)	Relativistic Cyclotron / Synchro-cyclotron
Particle Speed	$v \ll c$	Approaching speed of light ($c$)
Mass Variation	Constant mass ($m = m_0$)	Variable mass ($m = \gamma m_0$)
Oscillator Frequency	Constant frequency	Modulated or varying frequency
Suitability for Electrons	Unsuitable	More suitable with modifications

Practice Quiz

Choose an option to reveal the explanation.

1. What is the primary function of the electric field in a cyclotron?

Explanation: The magnetic field bends the particle's path, while the alternating electric field in the gap accelerates the particle every time it crosses.

2. Why is the electric field zero inside the dees of a cyclotron?

Explanation: The dees are hollow metal chambers, creating an electrostatic shielding effect where the electric field inside is zero.

3. The time period of revolution of a charged particle in a cyclotron is given by which expression?

Explanation: The time period of revolution is $T = \frac{2\pi m}{qB}$, which is independent of velocity and radius.

4. How does the maximum kinetic energy of a particle depend on the outer radius $R$ of the dees?

Explanation: From the formula $K_{max} = \frac{q^{2}B^{2}R^{2}}{2m}$, the maximum kinetic energy is directly proportional to the square of the radius ($R^2$).

5. Which of the following particles cannot be accelerated to very high energies in a conventional cyclotron?

Explanation: Electrons have a very small rest mass, causing them to reach relativistic speeds rapidly, leading to a loss of synchronism with the oscillator frequency.

6. What is the resonance condition for a cyclotron operating at non-relativistic speeds?

Explanation: Resonance occurs when the oscillator frequency matches the cyclotron frequency $f = \frac{qB}{2\pi m}$.

7. If the magnetic field $B$ in a cyclotron is doubled, how does the cyclotron frequency change?

Explanation: Since cyclotron frequency $f = \frac{qB}{2\pi m}$ is directly proportional to $B$, doubling $B$ doubles the frequency.

8. What happens to the relativistic mass of a charged particle as its velocity approaches the speed of light?

Explanation: At relativistic speeds, mass increases with velocity as $m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$.

9. What is the total energy gain by a charged particle per full revolution in a cyclotron?

Explanation: The particle crosses the gap twice per full revolution, gaining $qV_0$ energy each time, resulting in a total gain of $\Delta E = 2qV_0$.

10. Which of the following describes the path of a charged particle inside a dee?

Explanation: Inside the dee, the electric field is zero and the uniform magnetic field forces the particle to travel in a semi-circular path.

11. In a conventional cyclotron, why does the time taken for a semi-circular path remain constant even as the particle's speed increases?

Explanation: Since $r = \frac{mv}{qB}$, as velocity $v$ increases, the radius $r$ increases proportionally, keeping the time period $T = \frac{\pi m}{qB}$ constant.

12. What is the angular frequency $\omega$ of a charged particle in a magnetic field $B$?

Explanation: Angular frequency is given by $\omega = 2\pi f = \frac{qB}{m}$.

13. Why are conventional cyclotrons unable to accelerate protons to very high (GeV) energies?

Explanation: At high energies, even protons attain speeds where relativistic mass increase becomes significant, causing them to fall out of step with the alternating electric field.

14. What modifications are made in a synchro-cyclotron to overcome relativistic limitations?

Explanation: In a synchro-cyclotron, the frequency of the alternating electric field is modulated (decreased) to maintain synchronism with the slower, relativistically massive particles.

15. What is the expression for maximum velocity $v_{max}$ of a particle of mass $m$ and charge $q$ in a cyclotron of outer radius $R$?

Explanation: Equating magnetic force to centripetal force at radius $R$: $\frac{mv^2}{R} = qvB$, which gives $v_{max} = \frac{qBR}{m}$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The magnetic field and the angle between the magnetic field and velocity vector of a charged particle inside the dee of a cyclotron are respectively:

Explanation: The magnetic field inside the cyclotron is uniform and perpendicular to the plane of the dees, meaning the angle between the magnetic field vector and the velocity vector of the particle in the circular plane is $90^{\circ}$.

2. If the kinetic energy of a proton is doubled, the radius of its circular path in a uniform magnetic field of a cyclotron will change by a factor of:

Explanation: Radius $r = \frac{mv}{qB} = \frac{\sqrt{2mK}}{qB}$. Since $K$ is doubled, the radius becomes $\sqrt{2}$ times the original radius.

3. In a cyclotron, a charged particle accelerates:

Explanation: Inside the dees, the electric field is zero due to shielding. Acceleration occurs exclusively in the narrow gap between the dees due to the alternating electric field.

4. The time period of a charged particle revolving in a magnetic field does not depend upon:

Explanation: The time period formula $T = \frac{2\pi m}{qB}$ shows that $T$ depends on mass, charge, and magnetic field, but is independent of velocity and radius for non-relativistic particles.

5. A cyclotron oscillator frequency is $10\text{ MHz}$. What should be the operating magnetic field for accelerating protons? (Mass of proton $\approx 1.67 \times 10^{-27}\text{ kg}$, charge $\approx 1.6 \times 10^{-19}\text{ C}$)

Explanation: Using $B = \frac{2\pi m f}{q} = \frac{2 \times 3.1416 \times 1.67 \times 10^{-27} \times 10^7}{1.6 \times 10^{-19}} \approx 0.655\text{ T} \approx 0.66\text{ T}$.

6. Why are electrons difficult to accelerate in a standard cyclotron?

Explanation: Due to their extremely small rest mass, electrons reach velocities close to the speed of light at relatively low kinetic energies, causing rapid loss of synchronism.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the main purpose of the massive electromagnet in a cyclotron setup?

Explanation: The electromagnet provides a uniform magnetic field perpendicular to the dees, which forces the charged particles into circular or semi-circular trajectories.

2. If the radius of the dees of a cyclotron is doubled while keeping the magnetic field and particle type constant, the maximum kinetic energy increases by a factor of:

Explanation: Since $K_{max} \propto R^2$, doubling the radius $R$ increases the maximum kinetic energy by a factor of $2^2 = 4$.

3. Which physical quantity remains constant for a non-relativistic charged particle throughout its acceleration inside a conventional cyclotron?

Explanation: For non-relativistic particles, the time period $T = \frac{2\pi m}{qB}$ is constant and independent of speed and radius.

4. In a relativistic cyclotron (such as a microtron or synchro-cyclotron), as the particle's energy increases, its cyclotron frequency:

Explanation: Due to the relativistic mass increase ($m = \gamma m_0$), the cyclotron frequency $f = \frac{qB}{2\pi \gamma m_0}$ decreases as energy and velocity increase.

5. Calculate the cyclotron frequency for $\alpha$ particles in a magnetic field of $1.5\text{ T}$. (Mass of $\alpha$ particle $\approx 6.64 \times 10^{-27}\text{ kg}$, charge $\approx 3.2 \times 10^{-19}\text{ C}$)

Explanation: $f = \frac{qB}{2\pi m} = \frac{(3.2 \times 10^{-19}) \times 1.5}{2 \times 3.1416 \times (6.64 \times 10^{-27})} \approx 1.146 \times 10^7\text{ Hz} = 11.46\text{ MHz}$.

6. What happens if the frequency of the alternating voltage source in a cyclotron drifts significantly away from the particle's revolution frequency?

Explanation: If resonance is lost, the electric field does not reverse polarity at the correct instant, so the particle is no longer continuously accelerated.

7. The maximum kinetic energy of deuterons accelerated in a cyclotron is $10\text{ MeV}$. If the magnetic field is $1\text{ T}$ and dee radius is $0.5\text{ m}$, what is the approximate mass of a deuteron?

Explanation: Using $K_{max} = \frac{q^{2}B^{2}R^{2}}{2m} \implies m = \frac{q^2 B^2 R^2}{2 K_{max}}$. Substituting $q = 1.6 \times 10^{-19}\text{ C}$, $B = 1\text{ T}$, $R = 0.5\text{ m}$, $K = 10\text{ MeV} = 1.6 \times 10^{-12}\text{ J}$ gives $m \approx 3.34 \times 10^{-27}\text{ kg}$.

8. Which of the following statements about the magnetic force acting on a charged particle inside a dee is correct?

Explanation: The magnetic force is always perpendicular to the velocity vector of the particle inside the dee, so it does no work and only provides the centripetal force for circular motion.

9. In the context of high-energy particle acceleration, why is a synchro-cyclotron preferred over a conventional cyclotron for heavy ions at high velocities?

Explanation: A synchro-cyclotron adjusts (modulates) the frequency of the applied alternating voltage to maintain resonance as the particles become relativistic and their revolution frequency drops.

10. What is the primary limitation of a cyclotron regarding neutral particles like neutrons?

Explanation: Cyclotrons rely on electric and magnetic forces ($F = q(E + v \times B)$) acting on electric charges. Neutral particles possess zero charge ($q = 0$) and cannot be accelerated by these fields.