Deuterons in a cyclotron travel in a circle with radius 32.0 cm just before emerging from the de

step 1 Hey there, this is Josh. Okay, we got a problem where we want to find out what the magnetic fiel

Deuterons in a cyclotron travel in a circle with radius 32.0...

Key Concepts

Cyclotron Principle

The cyclotron is a type of particle accelerator that uses a constant magnetic field and a rapidly alternating electric field to accelerate charged particles. The electric field provides energy increments to the particles every time they cross the gap between the dees, resulting in a spiral trajectory with increasing radius as the particle gains speed.

Cyclotron Frequency

The cyclotron frequency is the constant angular frequency at which a charged particle orbits in a uniform magnetic field, given by the relation f = qB/(2?m). It is independent of the particle's speed or radius, which is central to the design and operation of a cyclotron.

Uniform Circular Motion of Charged Particles

In a uniform magnetic field, a charged particle experiences a Lorentz force that is always perpendicular to its velocity, causing it to move in a circular path. The balance between this magnetic force and the required centripetal force leads to the relation r = mv/(qB), linking the radius of the orbit to the particle’s mass, velocity, charge, and the magnetic field strength.

Kinetic Energy Calculation

The kinetic energy of a charged particle in a cyclotron is given by the classical expression KE = 1/2 mv^2. Once the speed of the particle is determined using the relationship v = 2?rf (derived from the circumference and the cyclotron frequency), this energy formula allows for calculating the amount of energy the particle has acquired during acceleration.

Transcript

00:01 Hey there, this is josh.

00:02 Okay, we got a problem where we want to find out what the magnetic field is for this system and the kinetic energy.

00:10 Okay, so starting from equation 44 .7, we can make a few notes.

00:17 First off being this, let's write down, first off, for a, we want to know what b is.

00:28 And for part b, i want to know what my maximum kinetic energy is.

00:34 Okay, let's start with part a.

00:40 So, first off, i know that omega is equal to qb over m.

00:52 And additionally, i know that omega is equal to 2 pi -f, or how many circles and radians i cover per second.

01:03 Okay.

01:04 Now, let's set these equal to each other.

01:08 2 pi f is equal to qb over m all right and let's go ahead and look at one more thing here i can solve this for b and i get that b is equal to 2 pi fm over q and i'm just going to use the absolute value of q for now okay now if we plug all this in my math is equal to 3 .344 times 10 to the minus 27 kilograms for a duteron and q is fundamental charge.

02:03 Typically it's 1 .602 in many calculations you can generally just do it as 1 .6 times 10 to the minus 1 9 coulomes.

02:14 Kuloms is charge just as kilogram as mass and i'm going to plug all my parameters in the frequency is given in the problem statement when you do this you plug it all in you get that the magnetic field is equal to 1 .18 tesla okay that is the first part let's go ahead and look at the second part part b all right so equation 44 .8 tells me that k max is equal to q squared b squared r squared divided by two m all right we have a value for q we have value for b or we have a value for let's make sure it's in meters and divided by two mass which we have mass up here so we have everything else we need we're going to plug it in we're going to get that it is equal to five point five point four seven one times 10 from the minus 1 3 joules...

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