Cyclotrons and Synchrotrons MCQ Quiz - Objective Question with Answer for Cyclotrons and Synchrotrons - Download Free PDF

Calculation:

We are given that the cyclotron frequency is proportional to certain parameters. The cyclotron frequency (f_c) is given by the formula:

f_c = \(\frac{qB}{2πm}\)

Where:

• q is the charge of the particle,
• B is the magnetic field strength,
• m is the mass of the particle.

From the formula, it is clear that the cyclotron frequency is directly proportional to the charge (q) and inversely proportional to the mass (m) of the particle. Therefore, we can write the proportionality as:

f_c ∝ \(\frac{q}{m}\)

Thus, the correct option for cyclotron frequency proportionality is:

Final Answer: The cyclotron frequency is proportional to \(\frac{q}{m}\), which corresponds to option 2.

Explanation:

Cyclotron Acceleration:

A cyclotron accelerates charged particles by using a rapidly alternating electric field between two hollow electrodes (D1 and D2), which are called dees.

The magnetic field, represented by B, is perpendicular to the plane of the dees and causes the charged particles to move in circular paths.

The particles gain energy each time they cross the gap between D1 and D2, as the electric field accelerates them.

The magnetic field inside the dees does not affect the speed of the particles because the magnetic force only causes a change in direction, not in speed.

The acceleration of the particle occurs only in the gap between D1 and D2, as it is the region where the electric field is present to accelerate the particles.

The force due to a magnetic field is: F = qvB, where q is the charge, v is the velocity, and B is the magnetic field strength.

The electric force in the gap is: F = qE, where E is the electric field in the gap.

Given that the charged particle undergoes an increase in speed, this is due to the electric field in the gap between D1 and D2, where the particle is accelerated.

The magnetic field inside the dees only causes the particles to move in circular paths without affecting their speed. Therefore, the speed of the particle increases only when it passes through the gap between D1 and D2.

∴ The correct answer is: Only in the gap between D1 and D2.

Calculation: 

\(R=\frac{m v}{B q}=\frac{\sqrt{2 m k}}{B q} \)

\(\Rightarrow K=\frac{B^2 q^2 R^2}{2 m} \).... ( kinetic energy of a particle in a cyclotron) 

⇒ K \( =\frac{\left(1.6 \times 10^{-19}\right)^2 \times 0.6^2}{2 \times 1.6 \times 10^{-27}} \mathrm{~J}=18 \mathrm{MeV}\)

∴ The kinetic energy of the accelerated protons in the cyclotron is 18 MeV.

Calculation: 

T = \(\rm \frac{2\pi m}{Bq}\)

⇒ Frequency f = \(\rm \frac{Bq}{2\pi m}\) .....( cyclotron frequency) 

⇒ f = \(\frac{10^{-4}× 1.6× 10^{-19}}{2\pi × 9 × 10^{-31}}\)

≃ 2.8 × 10⁶ Hz

∴ The frequency of revolution of the electron is approximately 2.8 × 10⁶ Hz

Concept:

Since the electron is shot into the magnetic field normal to the direction of the field, it will experience a force perpendicular to both its velocity and the magnetic field direction. This force will cause the electron to move in a circular path with a radius given by:

r = \(\frac{mv}{qB}\)

where:

r is the radius of the circular orbit in meters (m)
v is the velocity of the electron in meters per second (m/s)
Since the force on the electron is always perpendicular to its velocity, the electron will move in a circular path with a constant speed. This means that the frequency of revolution of the electron is given by:

f = \(\frac{v}{2\pi r}\)

Substituting the expression for r from above, we get:

f = \(\frac{Bq}{2\pi m}\)

where:

f is the frequency of revolution in Hertz (Hz)
B is the magnitude of the magnetic field in Tesla (T)
q is the charge of the electron in Coulombs (C)
m is the mass of the electron in kilograms (kg)

Since v cancels out from the equation, the frequency of revolution of the electron is independent of its initial velocity and only depends on the magnitude of the magnetic field, the charge and mass of the electron.

Therefore, the frequency of revolution of the electron in its circular orbit is solely dependent on the strength of the magnetic field and the properties of the electron, as given by the above equation.

The correct answer is option (1)

Concept:

Cyclotron:

• A cyclotron is a device used to accelerated positively charged particles (like α-particles, deuterons, etc.) to acquire enough energy to carry out nuclear disintegration, etc.
• It is based on the fact that the electric field accelerates a charged particle and the magnetic field keeps it revolving in circular orbits of constant frequency.
• The cyclotron frequency is given as,

\(⇒ f=\frac{Bq}{2\pi m}\)

Where B = magnetic field intensity, q = charge, and m = mass of the charged particle

Explanation:

• We know that a cyclotron is a device used to accelerated positively charged particles (like α-particles, deuterons, etc.).
• An electron cannot be accelerated by a cyclotron, because it has a very small mass. Due to this, its speed becomes very high and it quickly goes out of the step with the oscillating electric field.
• A neutron cannot be accelerated by a cyclotron, because cyclotron can accelerate only the charged particle and neutron is not a charged particle.

CONCEPT:

• A cyclotron is a charged particle accelerator, which was invented by Ernest O. Lawrence in 1934.
• A cyclotron accelerates ion particles (Charge particle i.e., Positive (+) or negative (-) charges) outwards from the center and it does that in a spiral path.
• The particles are accelerated by a rapidly varying electric field (A.C current) between two semi-circular hollow D.
• And the magnetic field will be responsible for keeping this charged particle in stable orbit inside the D
• The structure is as shown below

EXPLANATION:

• As explained above, in a cyclotron we use square wave, high-frequency A.C current.
• Hence by fluctuating the alternating voltage among circular dee we can accelerate charged particle.
• Thus option 4 is correct among all

The correct answer is option 1) i.e. inversely proportional to its mass

CONCEPT:

• A cyclotron is a particle accelerator that accelerates charged particles to higher energies.
  • ​The working principle of a cyclotron is the Lorentz force.
  • The charged particle is placed at the center of the arrangement between the poles of two electromagnets.
  • The magnetic field causes the particle to trace a circular path as it experiences a Lorentz force.
  • An alternating voltage is applied across the setup, which further accelerates the particle.
• Lorentz force is the force experienced by electrically charged particles moving in a magnetic field.
  • The magnitude of the magnetic force (F) on a charge (q) moving at a speed (v) in a magnetic field of strength B is given by​

⇒ F = qvB

EXPLANATION:

• When a charged particle moves perpendicular to the magnetic field (θ = 90∘), it follows a curved path and undergoes circular motion.
• Here, the magnetic force supplies the centripetal force which keeps the particle in a circular motion.
• The centripetal force, 

​\(⇒ F_C = \frac{mv^2}{r}\) ​

Where m is the mass of the particle, v is the velocity of the particle, and r is the radius of the circular path traced by the particle.

\(\therefore F = F_C⇒qvB = \frac{mv^2}{r}\)

\(⇒\frac{v}{r} = \frac{qB}{m}\)      ----(1)

• Angular velocity of the body is given as, 

​\(⇒ ω = \frac{v}{r} \) 

Where r = radius and ω = 2πf, where f is the frequency.

Substituting (1) in the equation for ω,

\(\Rightarrow \omega = \frac{qB}{m} = 2\pi f\)

\(\Rightarrow f =\frac{qB}{2\pi m}\)

\(\Rightarrow f \propto \frac{1}{m}\)

• The angular frequency of the charged particle in a cyclotron is inversely proportional to its mass.

CONCEPT:

• A cyclotron is a machine that is used to accelerate charged particles or ions to high energies.
• A simple cyclotron consists of a large circular magnet providing a constant magnetic field across the gap between the pole-faces.
• A charged particle is injected in the middle of the magnetic field and the particle is accelerated in the magnetic field in a circular shape.

EXPLANATION:

• Electrons cannot be accelerated in a cyclotron. A large increase in their energy increases their velocity to a very large extent. This throws the electrons out of step with the oscillating field. So option 2 and 3 are not correct.
• Thus, Cyclotron only accelerates protons and Ions to high energies. So option 1 is correct.
• Neutrons do not carry any charge with them so can’t be accelerated. So option 4 is not correct.

CONCEPT:

CYCLOTRON:

• A cyclotron is a device used to accelerated positively charged particles (like α -particles, deuterons etc) to acquire enough energy to carry out nuclear disintegration, etc. 
• It is based on the fact that the electric field accelerates a charged particle and the magnetic field keeps it revolving in circular orbits of constant frequency.
• The frequency of rotation of the charged particle in the magnetic field is given by

\(\Rightarrow f = \frac{{qB}}{{2\pi m}}\)

Where  q = Charge of the particle, B = Applied magnetic field and m = mass of the particle 

CALCULATION:

• Hence net magnetic force required by the magnetic field to keep the charged particle in the spiral orbit and can be expressed as

\(\Rightarrow \frac{{m{v^2}}}{R} = qvB\)

• The velocity of charge particle at any instance in a cyclotron is given as

\(\Rightarrow v = \frac{{qBR}}{m}\)

• Hence kinetic energy of the particle will be

CONCEPT:

• A cyclotron is a machine that is used to accelerate charged particles or ions to high energies.
• A simple cyclotron consists of a large circular magnet providing a constant magnetic field across the gap between the pole-faces.
• A charged particle is injected in the middle of the magnetic field and the particle is accelerated in the magnetic field in a circular shape.

• The radius of the circular path (R) is given by:

\( R=\frac{m~v}{q~B}\)

• Angular velocity (ω) is given by:

\(ω = \frac{{B\;q}}{m}\)

Where B = magnetic field in the cyclotron, q = charge of particle, m = mass of the particle, and v = velocity of particle.

EXPLANATION:

• The radius of the circular path (R) is given by:

\(\Rightarrow R=\frac{m~v}{q~B}\)

i.e., v = qBR/m

CONCEPT:

Cyclotron:

• A cyclotron is a device used to accelerated positively charged particles (like α-particles, deuterons, etc.) to acquire enough energy to carry out nuclear disintegration, etc.
• It is based on the fact that the electric field accelerates a charged particle and the magnetic field keeps it revolving in circular orbits of constant frequency.

EXPLANATION:

• In a cyclotron, the kinetic energy of the charged particle increases so the speed of the charged particle also increases.
• In a cyclotron, the charged particle moves in a circular path so its direction of motion continuously changes therefore the momentum also changes because momentum is a vector quantity.
• The time period of the charged particle in a cyclotron is given as,

\(⇒ T=\frac{2\pi m}{Bq}\)

Where B = magnetic field intensity, q = charge, and m = mass of the charged particle

• For a particular charged particle the value of m, B, and q does not change during the motion of the charged particle in a cyclotron, so the time-period does not change in the cyclotron.
• Hence, option 3 is correct.

CONCEPT:

• A cyclotron is a machine that is used to accelerate charged particles or ions to high energies.
• A simple cyclotron consists of a large circular magnet providing a constant magnetic field across the gap between the pole-faces.
• A charged particle is injected in the middle of the magnetic field and the particle is accelerated in the magnetic field on a circular path.

The frequency (f) of charge particle inside the cyclotron is given by:

\(f=\frac{q~B}{2~\pi ~m}\)

Where q is charge, B is magnetic field and m is mass of the charge particle.

EXPLANATION:

• From the above formula of frequency of moving charge inside a cyclotron, frequency is directly proportional to the magnetic field inside the cyclotron.
• As the magnetic field is doubled the frequency of charge will also become two times. So option 1 is correct.

EXPLANATION:

→Cyclotron is a device used to accelerate positively charged particles (like α particles, deuterons, etc.)

It is based on the fact that the electric field accelerates a charged particle and a perpendicular magnetic field keeps it revolving in circular orbits of constant frequency.

→Thus a small potential difference would impart enormously large velocities if the particle is made to traverse the potential difference a number of times.

∴ In a cyclotron, the charged particles experience a large force between the dees while circulating inside the dees. 

Hence, in the cyclotron, the particle is always made to accelerate. 

So, the correct answer is option (1).

Concept –

Motion of the charged particle in a uniform magnetic field –

When a charge particle q enters a magnetic field \(\vec B\) with a velocity \(\vec v\), it experiences a force

\(\vec F = q\left( {\vec v \times \vec B} \right)\)

The direction of this force is perpendicular to both \(\vec v\) and \(\vec B\). The magnitude of this force is

F = qvB sin θ

Explanation –

The direction of this force is perpendicular to both \(\vec v\) and \(\vec B\) and the magnitude of this force is given as

F = qvB sin θ

If a charge particle enters a magnetic field with velocity v such that the direction between the velocity of the charge particle q and magnetic field B is 90°, then it experiences a maximum force.

Here θ = 90°, so

F = qvB sin 90° = qvB = a maximum force

As the magnetic force acts on a particle perpendicular to its velocity, it does not do any work on the particle. It does not change the kinetic energy or the speed of particle.

The magnetic field B is perpendicular to the paper and going into it (show by small cross). A charge +q is projected with speed v in the plane of the paper. The velocity is perpendicular to the magnetic field. A force F = qvB acts on the particle perpendicular to both \(\vec v\) and \(\vec B\). This force continuously deflects the particle sideways without changing its speed and particle will move along a circle perpendicular to the field.