Department of Physics and Applied Physics95.144 Danylov Lecture 16

Lecture 16

Chapter 32

Magnetic ForceCyclotron motion

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII

Lecture Capture: http://echo360.uml.edu/danylov201415/physics2spring.html

Physics II

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Electric Field Magnetic Field

From Coulomb’s law1

4̂

Biot-Savart law

4̂

Gauss’s Law

∙

Ampere’s Law

∙

So, now we know how to find magnetic fields using Bio-Savart and Ampere’s laws.Now, the question is “how does a magnetic field interact with material (which consists of charges and current)?”

Magnetic force on a moving charge

Magnetic force on current

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Magnetic force on a moving charge

Department of Physics and Applied Physics95.144 Danylov Lecture 16

The Magnetic Force on a Moving ChargeAfter Oersted’s discovery, there were many other experiments with magnetic fields, currents, charges, etc. It was found that B exerts a force on a moving charge.

The magnetic force on a charge q as it moves through a magnetic field B with velocity v is:

where is the angle between v and B.

Department of Physics and Applied Physics95.144 Danylov Lecture 16

The Magnetic Force on a Moving Charge

A magnetic field has no effect on a charge moving parallel/antiparallel to the field

ConcepTest B field of qA) To the rightB) To the left C) Into the screenD) Out of the screenE) The magnetic force is zero

The direction of the magnetic force on the proton is

ConcepTest B field of qA) UpwardB) DownwardC) Into the screenD) Out of the screenE) The magnetic force is zero

The direction of the magnetic force on the proton is

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Cyclotron MotionMany important applications of magnetism involve the motion of

charged particles in a perpendicular magnetic field

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Cyclotron motionThe figure shows a positive charge moving in a plane that is

perpendicular to a uniform magnetic field.

B into page

Since F is always perpendicular to v, F changes the direction of the velocity, but not it is magnitude.It means q experiences only the centripetal accelerationThus, the charge undergoes uniform circular motion.

This motion is called the cyclotron motion of a charged particle in a magnetic field.

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Cyclotron radius

B

Newton’s second law for a radial direction,

The frequency of the cyclotron motion.

The radius of the cyclotron orbit:

The period of the cyclotron motion:

If B=0, then rcyc=0, which is a straight line

Note! The cyclotron frequency does not depend on v.

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Cyclotron motion

B

∥

The figure shows a more general situation in which the charged particle’s velocity is not exactly perpendicular to B.

The component of v parallel to Bis not affected by the field, so the charged particle spirals around the magnetic field lines in a helical trajectory.

The radius of the helix is determined by v, the component of v perpendicular to B.

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Aurora (Northern lights)

Charged particles(solar wind)

Department of Physics and Applied Physics95.144 Danylov Lecture 16

The Cyclotron

The first practical particle accelerator, invented in the 1930s, was the cyclotron.

Cyclotrons remain important for many applications of nuclear physics, such as the creation of radioisotopes for medicine.

The relevant equation for us is: .According to this equation, the bigger the charge, the smaller the radius.

ConcepTest Mass Spectrometer x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xA B

Two particles of the same massenter a magnetic field with the same speed and follow the paths shown. Which particle has the bigger charge?

C) both charges are equalD) impossible to tell from the picture

Follow-up: What is the sign of the charges in the picture?

If q > 0, then the force is to the left (our case)

, , ,

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Cloud/Bubble chamber to detect charged particles

(contains a supersaturated vapor of water)

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Magnetic force on Current-Carrying

Wires

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Magnetic Forces on Current-Carrying Wires We saw that a magnetic field exerts a force on a charge. A current consists of many moving charges, so a magnetic field should exert a force on a current.

The magnetic force on a segment dl:

The magnetic force of a charge:

If we apply this formula for a straight wire with a current I, then we will get:

Where l is a part of wire exposed to the magnetic field

Department of Physics and Applied Physics95.144 Danylov Lecture 16

There’s no force on a current-carrying wire parallel to a magnetic field.

Slide 32-139

Magnetic Forces on Current-Carrying Wires

∥

Department of Physics and Applied Physics95.144 Danylov Lecture 16

A current perpendicular to the field experiences a force in the direction of the right-hand rule.

Slide 32-140

Magnetic Forces on Current-Carrying Wires

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Magnetic Forces Between Parallel Current-Carrying Wires: Current in Same Direction

We’ll treat I2 as a source of the magnetic field

Let’s find a force on current I1

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Slide 32-147

Magnetic Forces Between Parallel Current-Carrying Wires: Current in Opposite Directions

Department of Physics and Applied Physics95.144 Danylov Lecture 16

SummaryElectric Field Magnetic Field

Amperian loop

The magnetic force on a charge:

The magnetic force on a current:

From Coulomb’s law1

4̂

Bio-Savart law

4̂

Gauss’s Law

∙ ∙

The electric force on a charge:

Department of Physics and Applied Physics95.144 Danylov Lecture 16

What you should readChapter 32 (Knight)

Sections 32.7 (Skip the Hall effect) 32.8 32.5 (skip)

Department of Physics and Applied Physics95.144 Danylov Lecture 16

Thank youSee you on Friday