Cyclotron and Cyclotron Frequency

Summary

The script explores the cyclotron, an early high-energy particle accelerator developed by Ernest O. Lawrence in 1930. It describes how the cyclotron uses electric and magnetic fields to accelerate protons to high velocities and energies. The device's structure, consisting of two vacuum chambers and an alternating electric field, is explained. The protons move in a circular path due to the magnetic field and gain velocity through the electric field, eventually exiting to collide with a target, such as a tumor in medical applications. The script also delves into the cyclotron frequency equation, relating it to the particle's motion and the magnetic field's influence.

Takeaways

• 🔬 The cyclotron is an early high-energy particle accelerator developed by Ernest O. Lawrence in 1930.
• 🧲 It uses electric and magnetic fields to accelerate particles, typically protons, to high velocities and energies.
• 🏥 Modern cyclotrons are still used in medicine, particularly for treating tumors with accelerated particles.
• 🛠️ The cyclotron consists of two vacuum chambers and a magnetic field that directs particles in a circular path.
• 🔋 An electric potential difference is created between the chambers to accelerate the particles as they move between them.
• ⚛️ The magnetic field creates a force that causes the charged particles to undergo centripetal acceleration, maintaining a circular motion.
• 🔄 As the particle's velocity increases, so does the radius of its circular path, allowing it to eventually exit the cyclotron.
• 🎯 The exiting particle can be directed towards a target, such as a tumor, for medical treatment.
• 🔀 An alternating electric current is used to create an alternating electric field, which is necessary for continuous particle acceleration.
• 🔁 The frequency of the alternating voltage source must match the cyclotron frequency, which is the frequency of the particle's circular motion.
• ⚖️ The cyclotron frequency can be calculated using the equation derived from the relationship between charge, magnetic field, and mass of the particle.
• 📚 Newton's second law and the right-hand rule are applied to understand the forces acting on the particles within the cyclotron.

Q & A

What is a cyclotron and who developed it?

A cyclotron is an early high-energy particle accelerator developed by Ernest O. Lawrence in 1930. It uses electric and magnetic fields to accelerate particles, typically protons, to very high velocities and energies.

What is the primary use of cyclotrons in modern times?

In modern times, cyclotrons are primarily used in medicine for treating tumors by accelerating particles to high energies to target and destroy cancer cells.

What are the two main components of a cyclotron's structure?

The two main components of a cyclotron's structure are the vacuum chambers and the semicircular vacuum chambers, which create a magnetic field for the particles to move in a circular path.

How does the magnetic field in a cyclotron affect the charged particles?

The magnetic field in a cyclotron creates a magnetic force that acts as centripetal acceleration, causing the charged particles to move in a circular pathway within the vacuum chambers.

What role does the electric field play in the acceleration of particles in a cyclotron?

The electric field, created by a voltage difference between the two semicircular vacuum chambers, accelerates the particles linearly when they travel between the chambers, increasing their velocity.

How does the radius of curvature of a particle's path change as it accelerates in a cyclotron?

As the velocity of the particle increases, the radius of curvature of its circular path also increases, allowing the particle to eventually exit the cyclotron when it reaches a sufficient energy level.

What is the purpose of using an alternating electric current in a cyclotron?

An alternating electric current is used to create an alternating electric field that changes direction with each half-cycle, ensuring that the electric field always accelerates the particle in the direction of its motion.

What is the significance of the cyclotron frequency in the operation of a cyclotron?

The cyclotron frequency is the frequency at which the voltage source oscillates, and it must match the frequency of the particle's circular motion to maintain continuous acceleration.

How is the cyclotron frequency related to the particle's motion and magnetic field?

The cyclotron frequency is determined by the particle's charge, mass, and the strength of the magnetic field. It is the frequency at which the particle completes one full cycle of its circular motion.

What is the equation that describes the magnetic force acting on a charged particle moving in a magnetic field?

The magnetic force (F) acting on a charged particle with charge (q) moving with velocity (v) in a magnetic field (B) is given by the equation F = q * v * B, where the force is perpendicular to both the velocity and the magnetic field.

How can the period of a particle's circular motion in a cyclotron be calculated?

The period (t) of a particle's circular motion can be calculated by dividing the circumference of the circle (2 * π * r) by the velocity of the particle (v), which is given by the equation v = q * B * r / m.

Outlines

00:00

🔬 Introduction to Cyclotrons and Their Medical Applications

The first paragraph introduces the cyclotron, an early high-energy particle accelerator developed by Ernest O. Lawrence in 1930. It explains how the cyclotron uses electric and magnetic fields to accelerate protons to high velocities and energies. The script describes the cyclotron's structure, including its two vacuum chambers and the magnetic field within them. It also explains the role of the electric field in accelerating particles and how the protons move in a circular path due to the magnetic force. The paragraph concludes with the application of cyclotrons in medicine, specifically for treating tumors by directing accelerated protons towards the tumor.

05:00

🔧 The Working Principle of Cyclotrons and Cyclotron Frequency

The second paragraph delves into the working principle of cyclotrons, focusing on the alternating electric field that accelerates the protons. It explains the necessity of an alternating voltage source to maintain the electric field's direction in sync with the proton's motion. The paragraph also introduces the concept of cyclotron frequency, which is the frequency at which the electric field oscillates to keep the protons on their acceleration path. The script provides the equation for the magnetic force acting on a charged particle in a magnetic field and applies Newton's second law of motion to derive the velocity of the particle. It then explains how to calculate the cyclotron frequency using the period of the particle's circular motion and the fundamental constants of the particle's charge, mass, and the magnetic field's strength.

Keywords

💡Cyclotron

A cyclotron is one of the earliest types of high-energy particle accelerators, developed by Ernest O. Lawrence in 1930. It uses electric and magnetic fields to accelerate particles, typically protons, to high velocities and energies. In the video, the cyclotron is described as a device with two vacuum chambers and a magnetic field that causes charged particles to move in a circular path, which is central to the theme of the video.

💡Particle Acceleration

Particle acceleration refers to the process of increasing the velocity of a particle, typically through the influence of electric and magnetic fields. In the context of the video, the cyclotron accelerates protons by using an electric field to increase their velocity as they move between the vacuum chambers, which is a key concept in understanding how the cyclotron operates.

💡Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. In the script, the magnetic field is crucial for the cyclotron's operation as it provides the force that causes the protons to move in a circular path within the vacuum chambers.

💡Electric Field

An electric field is a field that surrounds electrically charged particles and exerts force on other charged particles within the field. In the video, the electric field is created by an electric potential difference and is responsible for accelerating the protons as they move through the midsection between the vacuum chambers.

💡Centripetal Force

Centripetal force is the force that keeps an object moving in a circular path, directed towards the center of the circle. In the script, the magnetic force acting on the protons is described as a centripetal force that causes them to move in a circular pathway within the cyclotron's vacuum chambers.

💡Velocity Vector

A velocity vector is a vector quantity that specifies the direction and magnitude of velocity. In the context of the video, the velocity vector of the proton is perpendicular to the magnetic field, which is essential for the application of the magnetic force and the resulting circular motion.

💡Radius of Curvature

The radius of curvature refers to the radius of the circular path followed by a moving object. As the velocity of the proton increases in the cyclotron, so does the radius of its circular path, which is a key point in the explanation of how the cyclotron accelerates particles.

💡Alternating Electric Current

An alternating electric current (AC) is an electric current whose direction and magnitude change periodically with time. In the video, an alternating electric current is used to create an alternating electric field that changes direction to continue accelerating the protons in the cyclotron.

💡Cyclotron Frequency

The cyclotron frequency is the frequency at which the protons oscillate in the cyclotron's magnetic field. It is derived from the relationship between the magnetic field, the charge of the particle, and its mass. The script explains that this frequency is crucial for synchronizing the electric field changes with the proton's motion.

💡Right-Hand Rule

The right-hand rule is a common mnemonic for understanding the direction of vectors in various physical situations, such as the direction of the magnetic force on a moving charge. In the script, the right-hand rule is applied to determine the direction of the magnetic force acting on the protons in the cyclotron.

💡Newton's Second Law of Motion

Newton's second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In the video, this law is applied to explain how the magnetic force causes the protons to accelerate in a circular path within the cyclotron.

Highlights

The cyclotron was one of the earliest high-energy particle accelerators, developed by Ernest O. Lawrence in 1930.

Cyclotrons use electric and magnetic fields to accelerate particles, usually protons, to high velocities and energies.

Though modern particle accelerators are more complex, cyclotrons are still used in medicine to treat tumors.

The cyclotron consists of two vacuum chambers and a magnetic field that causes charged particles to move in a circular path.

An electric potential difference is created between the two semicircular vacuum chambers to accelerate particles.

The magnetic field provides the centripetal force needed for particles to move in a circular motion within the vacuum chambers.

The electric field in the midsection between the chambers accelerates the particle's velocity as it moves between them.

As the particle's velocity increases, so does the radius of its circular path, allowing it to eventually exit the cyclotron.

The exiting particle can be directed towards a target, such as a tumor, for medical treatment.

The electric field must alternate direction to continue accelerating the particle in the opposite direction after each half cycle.

An alternating voltage source is used to create the alternating electric field needed for continuous particle acceleration.

The frequency of the voltage source must match the cyclotron frequency to synchronize with the particle's circular motion.

The cyclotron frequency is derived from the relationship between the particle's charge, velocity, magnetic field, and mass.

The cyclotron frequency equation is given by f = qb / (2πm), where q is the charge, b is the magnetic field strength, and m is the mass of the particle.

The period of the particle's motion is the time it takes to complete one full cycle in the cyclotron.

The period can be calculated using the circumference of the circular path divided by the particle's velocity.

The cyclotron's practical application in medicine demonstrates the significance of its historical development and ongoing use.

Transcripts

one of the earliest high Energy particle

accelerators was the cyclotron the

cyclotron was developed by Ernest o

Lawrence in 1930 and it basically uses

electric and magnetic Fields to

accelerate particles usually protons to

very high velocities and very high

energies now although modern day high

Energy particles are much more

complicated than the cyclotron the cycle

is still Used in Medicine to treat

tumors so Let's take a look at the

following diagram that basically

describes the structure of our cyclotron

and how it actually works so basically

We have some type of device that creates

a magnetic field so We have two vacuum

chambers vacuum Chamber number one and

semicircular vacuum Chamber number two

now within these vacuum chambers We have

a magnetic field that exists and in this

case the magnetic field points into the

board and That is shown by these Blue

aes with the Circle around them now

between our two semicircular vacuum

chambers We have a Space and within the

Space between these two points We create

An electric potential difference of

voltage difference and this voltage

difference basically creates An electric

field as shown by the these Green arrows

so For This particular Moment or This

particular Moment In Time the electric

field points in this Direction from left

to Right Now That means that this part

has a positive Charge and this part has

a negative Charge now Let's suppose We

take our particle Let's Say A proton and

we place the proton in this location

into Chamber One so We give it some

velocity and When A charged particle in

this case the proton travels inside a

magnetic field What will happen is the

magnetic field will basically create a

magnetic force that will act on our

Charge particle and because the velocity

vector is perpendicular to the magnetic

field the magnetic force will be

perpendicular to both the velocity and

our magnetic field and that means that

magnetic force will act to create

centripetal

acceleration so that magnetic force will

cause the proton To Move in a circular

pathway When that proton is within these

two vacuum chambers so When the proton

is in either one of the two vacuum

chambers the magnitude of velocity does

not actually change However When the

proton travels within this Space Where

We have our electric field there the

velocity will increase because the

electric force as a result of that

electric field will cause that velocity

to increase so When the particle travels

between this midsection its velocity

will increase and when the velocity of a

particle Moving in a Circle increases

the radius of curvature the radius of

that Circle will also inas and so

initially the radius is some small

radius When accelerates the radius

increases and this basically continues

until The radius is large enough And The

velocity is great enough and the Energy

is great enough that the particle

actually exits our cyclotron and when

the particle exits we can play some type

of target object here for example a

tumor and that particle will collide

with that target object so once again a

particle such as a proton begins from

rest It begins To Move in the vacuum

chambers as shown two chambers are

separated by distance as shown in this

diagram and in this Space in this region

here we have an electric potential

difference that creates An electric

field That is used to accelerate that

proton linearly now the magnetic field

inside the two chambers ke the proton

Moving in a circular pathway It doesn't

actually increase the magnitude of the

velocity now as the Speed of the proton

increases its radius of curvature also

increases and eventually as We see in

the diagram the proton leaves the

Chamber and collides with our target so

If We're using this to treat tumor a

tumor We can place a tumor in this

section now notice an important point so

When the proton begins To Move initially

as it exits and begins to travel Through

The Space the electric field has to

Point this way to accelerate the proton

in this Direction in the same Direction

as Motion But when the proton finishes

The Second semicircular loop when it

exits and begins to travel this way we

don't want that electric field to Point

in this Direction because that will

actually stop it instead We want the

electric field to Point In The opposite

Direction and so to create that type of

alternating electric field We use An

alternating electric current so the

voltage source is an alternating voltage

source and it turns out that to keep our

electric field changing every half a

cycle We want the frequency of the

voltage source to be the same as the

frequency of our particle so once again

notice that the voltage source Is

alternating This Is because the electric

field Lines must change directions for

the proton to continue on its

acceleration pathway this implies that

the frequency of oscillation of the

voltage equals to the frequency of the

proton along its circular pathway and

this frequency is known as the cyclotron

frequency So what exactly is the

equation that gives us the cyclotron

frequency well Let's Begin by recalling

What happens inside our two chambers so

whenever We have a positive Charge That

Is Moving with the velocity V inside a

magnetic field B and the magnetic field

is perpendicular to our velocity that

will create a magnetic force on that

object the Charge the magnetic force

will be perpendicular And The magnitude

is given by this equation the magnetic

force acting in the Charge Uh is equal

to the Charge of that particle Q

multiplied by its velocity multiplied by

its magnetic field now to find The

Direction We can apply Right Hand rule

So basically this is our particle our

velocity is going this way the magnetic

field points into the board so that

means We Take The Right Hand We point

this way In The Direction Of The

velocity We curl our fingers into the

board to curl the fingers into our

magnetic field extend the thumb the

thumb points In The Direction of our

magnetic force as shown in this diagram

now Let's Begin by applying Uh newton's

second Law of Motion so the net force

acting on the particle is equal to the

Mass times its acceleration now because

our object isel

a circular fash that means a is equ v s

div r Where v s div r is the centripetal

acceleration the force is given by the

magnetic force

qvb so Let's Take This equation and

solve for the velocity the velocity is

equal to qrb div m notice one of the V

will cancel out now how exactly Can We

find the frequency well rec call That

frequency is related to the period so If

We find the period We can find the

frequency

what So what exactly is the period well

the period is Simply the amount of time

it takes our particle to travel One full

cycle so that means to calculate the

period t We take the distance it travels

in one Circle divided by its velocity

the distance is 2 pi r Where r is the

radius of the Circle So this is our

circumference and we Divide that by the

velocity obtained in this in part one so

We have 2 pi r div qrb div by M The m

Will Go On top the RS cancel and see the

period is equal to 2 pi M div Q m by B

now recall that the frequency is equal

to 1 div by the period so frequencies

equ to 1 div t so We plug in this

equation for t We get this so We see

that the cyclotron frequency is equal to

qb div by 2 m m m by Pi