Assignment 14: Magnetic Force

Due: 8:00am on Friday, February 24, 2012

Note: To understand how points are awarded, read your instructor's Grading Policy.

Exercise 27.14

The magnetic field in a certain region is 0.128 , and its direction is that of the + -axis in the Figure

.

Part A

What is the magnetic flux across the surface abcd in the figure?

ANSWER:

= 0

Correct

Part B

What is the magnetic flux across the surface befc?

ANSWER:

= −1.15×10

−2

Correct

Part C

What is the magnetic flux across the surface aefd?

ANSWER:

= 1.15×10

−2

Correct

Part D

What is the net flux through all five surfaces that enclose the shaded volume?

ANSWER:

= 0

Correct

± Force on Moving Charges in a Magnetic Field

Learning Goal: To understand the force on a charge moving in a magnetic field.

Magnets exert forces on other magnets even though they are separated by some distance. Usually the force on a

magnet (or piece of magnetized matter) is pictured as the interaction of that magnet with the magnetic field at its

location (the field being generated by other magnets or currents). More fundamentally, the force arises from the

interaction of individual moving charges within a magnet with the local magnetic field. This force is written

, where is the force, is the individual charge (which can be negative), is its velocity, and is the local

magnetic field.

This force is nonintuitive, as it involves the vector product (or cross product) of the vectors and . In the

following questions we assume that the coordinate system being used has the conventional arrangement of the

axes, such that it satisfies , where , , and are the unit vectors along the respective axes (x, y and z).

Let's go through the right-hand rule. Starting with the generic vector cross-product equation point your

forefinger of your right hand in the direction of , and point your middle finger in the direction of . Your thumb

will then be pointing in the direction of .

Part A

Consider the specific example of a positive charge moving in the +x direction with the local magnetic field in

the +y direction. In which direction is the magnetic force acting on the particle?

Express your answer using unit vectors (e.g., - ). (Recall that is written \hat i (or alternatively i_unit can be

used.))

ANSWER:

Direction of = Correct

Part B

Now consider the example of a positive charge moving in the +x direction with the local magnetic field in the +z

direction. In which direction is the magnetic force acting on the particle?

Express your answer using unit vectors.

ANSWER:

Direction of =

Correct

Part C

Now consider the example of a positive charge moving in the xy plane with velocity (i.e., with

magnitude at angle with respect to the x axis). If the local magnetic field is in the +z direction, what is the

direction of the magnetic force acting on the particle?

Hint C.1 Finding the cross product

Hint not displayed

Express the direction of the force in terms of , as a linear combination of unit vectors, , , and .

ANSWER:

Direction of =

Correct

Part D

First find the magnitude of the force on a positive charge in the case that the velocity (of magnitude ) and

the magnetic field (of magnitude ) are perpendicular.

Express your answer in terms of , , , and other quantities given in the problem statement.

ANSWER:

=

Correct

Part E

Now consider the example of a positive charge moving in the -z direction with speed with the local magnetic

field of magnitude in the +z direction. Find , the magnitude of the magnetic force acting on the particle.

Express your answer in terms of , , , and other quantities given in the problem statement.

ANSWER:

=

0

Correct

There is no magnetic force on a charge moving parallel or antiparallel to the magnetic field. Equivalently, the

magnetic force is proportional to the component of velocity perpendicular to the magnetic field.

Part F

Now consider the case in which the positive charge is moving in the yz plane with a speed at an angle with

the z axis as shown (with the magnetic field still in the +z direction with

magnitude ). Find the magnetic force on the charge.

Hint F.1 Direction of force

Hint not displayed

Hint F.2 Relevant component of velocity

Hint not displayed

Express the magnetic force in terms of given variables like , , , , and unit vectors.

ANSWER:

=

Correct

Charge Moving in a Cyclotron Orbit

Learning Goal: To understand why charged particles move in circles perpendicular to a magnetic field and why

the frequency is an invariant.

A particle of charge and mass moves in a region of space where there is a uniform magnetic field (i.e., a

magnetic field of magnitude in the +z direction). In this problem, neglect

any forces on the particle other than the magnetic force.

Part A

At a given moment the particle is moving in the +x direction (and the magnetic field is always in the +z

direction). If is positive, what is the direction of the force on the particle due to the magnetic field?

Hint A.1 The right-hand rule for magnetic force

Hint not displayed

ANSWER:

x direction

x direction

y direction

y direction

z direction

z direction

Correct

Part B

This force will cause the path of the particle to curve. Therefore, at a later time, the direction of the force will

____________.

ANSWER:

have a component along the direction of motion

remain perpendicular to the direction of motion

have a component against the direction of motion

first have a component along the direction of motion; then against it; then along it; etc.

Correct

Part C

The fact that the magnetic field generates a force perpendicular to the instantaneous velocity of the particle has

implications for the work that the field does on the particle. As a consequence, if only the magnetic field acts on

the particle, its kinetic energy will ____________.

ANSWER:

increase over time

decrease over time

remain constant

oscillate

Correct

Part D

If the resulting trajectory of the charged particle is a circle, what is , the angular frequency of the circular

motion?

Hint D.1 How to approach the problem

Hint not displayed

Hint D.2 Determine the magnetic force

Hint not displayed

Hint D.3 Determine the acceleration of the particle

Hint not displayed

Hint D.4 Express the angular speed in terms of the linear speed

Hint not displayed

Express in terms of , , and .

ANSWER:

=

Correct

Note that this result for the frequency does not depend on the radius of the circle. Although it appeared in the

equations of force and motion, it canceled out. This implies that the frequency (but not the linear speed) of the

particle is invariant with orbit size.

The first particle accelerator built, the cyclotron, was based on the fact that the frequency of a charged particle

orbiting in a uniform field is independent of the radius. In the cyclotron, radio frequency voltage is applied across

a gap between the two sides of the conducting vacuum chamber in which the protons circulate owing to an

external magnetic field. Particles in phase with this voltage are accelerated each time they cross the gap (because

the field reverses while they make half a circle) and reach energies of millions of electron volts after several

thousand round trips.

Motion of Electrons in a Magnetic Field

An electron of mass and charge is moving through a uniform magnetic field in vacuum.

At the origin, it has velocity , where and . A screen is

mounted perpendicular to the x axis at a distance from the origin.

Throughout, you can assume that the effect of gravity is negligible.

Part A

First, suppose . Find the y coordinate of the point at which the electron strikes the screen.

Hint A.1 Forces acting on electron

Hint not displayed

Hint A.2 Two-dimensional kinematics

Hint not displayed

Express your answer in terms of and the velocity components and .

ANSWER:

=

Correct

Part B

Now suppose , and another electron is projected in the same manner. Which of the following is the most

accurate qualitative description of the electron's motion once it enters the region of nonzero magnetic field?

ANSWER:

The electron decelerates before coming to a halt and turning around while always moving

along a straight line.

The electron's motion will be unaffected. (It will continue moving in a straight line with the

same constant velocity.)

The electron moves in a circle in the xy plane.

The electron moves along a helical path about an axis parallel to the field lines with constant

radius and constant velocity in the x direction.

Correct

Part C

The motion of the electron can be broken down into two parts: 1. constant motion in the x direction plus 2. a

periodic part, the projection of which in the yz plane is circular. Find the angular velocity of the electron

associated with the circular component of its motion.

Hint C.1 Use Newton's 2nd law to determine the force

Hint not displayed

Hint C.2 Find the magnitude of the Lorentz force

Hint not displayed

Hint C.3 Express in terms of

Hint not displayed

Hint C.4 Express the acceleration in terms of

Hint not displayed

Express the magnitude of the angular velocity in terms of , the magnitude of the electric charge , and other

known quantities.

ANSWER:

=

Correct

Charged Particles Moving in a Magnetic Field Ranking Task

Five equal-mass particles (A–E) enter a region of uniform magnetic field directed into the page. They follow the

trajectories illustrated in the figure.

Part A

Which particle (if any) is neutral?

Hint A.1 Neutral particles

Hint not displayed

ANSWER:

particle A

particle B

particle C

particle D

particle E

none

Correct

Part B

Which particle (if any) is negatively charged?

Hint B.1 Find the direction of the magnetic force

Hint not displayed

ANSWER:

particle A

particle B

particle C

particle D

particle E

none

Correct

Part C

Rank the particles on the basis of their speed.

Hint C.1 Determining velocity based on particle trajectories

Hint not displayed

Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:

Correct

Part D

Rank the particles A, B, C, and E on the basis of their speed.

Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:

Correct

Part E

Now assume that particles A, B, C, and E all have the same magnitude of electric charge. Rank the particles A, B,

C, and E on the basis of their speed.

Hint E.1 Charged particle trajectories in magnetic fields

Hint not displayed

Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:

Correct

Electromagnetic Velocity Filter

When a particle with charge moves across a magnetic field of magnitude , it experiences a force to the side. If

the proper electric field is simultaneously applied, the electric force on the charge will be in such a direction as to

cancel the magnetic force with the result that the particle will travel in a straight line. The balancing condition

provides a relationship involving the velocity of the particle. In this problem you will figure out how to arrange

the fields to create this balance and then determine this relationship.

Part A

Consider the arrangement of ion source and electric field plates shown in the figure.

The ion source sends particles with velocity along the positive x axis.

They encounter electric field plates spaced a distance apart that generate a uniform electric field of magnitude

in the +y direction. To cancel the resulting electric force with a magnetic force, a magnetic field (not shown) must

be added in which direction? Using the right-hand rule, you can see that the positive z axis is directed out of the

screen.

Hint A.1 Method for determining direction

Hint not displayed

Hint A.2 Right-hand rule

Hint not displayed

Choose the direction of .

ANSWER:

Correct

Part B

Now find the magnitude of the magnetic field that will cause the charge to travel in a straight line under the

combined action of electric and magnetic fields.

Hint B.1 Find the magnetic force

Hint not displayed

Hint B.2 Find the force due to the electric field

Hint not displayed

Express the magnetic field that will just balance the applied electric field in terms of some or all of the

variables , , and .

ANSWER:

=

Correct

Part C

It may seem strange that the selected velocity does not depend on either the mass or the charge of the particle.

(For example, would the velocity of a neutral particle be selected by passage through this device?) The

explanation of this is that the mass and the charge control the resolution of the device--particles with the wrong

velocity will be accelerated away from the straight line and will not pass through the exit slit. If the acceleration

depends strongly on the velocity, then particles with just slightly wrong velocities will feel a substantial

transverse acceleration and will not exit the selector. Because the acceleration depends on the mass and charge,

these influence the sharpness (resolution) of the transmitted particles.

Assume that you want a velocity selector that will allow particles of velocity to pass straight through without

deflection while also providing the best possible velocity resolution. You set the electric and magnetic fields to

select the velocity . To obtain the best possible velocity resolution (the narrowest distribution of velocities of

the transmitted particles) you would want to use particles with __________.

Hint C.1 Use Newton's law

Hint not displayed

Assume that the selector is short enough so that particles that move away from the axis do not have time to come

back to it.

ANSWER:

both and large

large and small

small and large

both and small

Correct

You want particles with the incorrect velocity to have the maximum possible deviation in the y direction so that

they will not go through a slit placed at the right end. The deviation will be maximum when the acceleration is

maximum. The acceleration is directly proportional to and inversely proportional to :

.

So for maximum deviation, should be large and small.

Exercise 27.28

Part A

What is the speed of a beam of electrons when the simultaneous influence of an electric field of and a

magnetic field of , with both fields normal to the beam and to each other, produces no deflection of the

electrons?

ANSWER:

=

3.38×106

Correct

Part B

When the electric field is removed, what is the radius of the electron orbit?

ANSWER:

=

4.16×10−3

Correct

Part C

What is the period of the orbit?

ANSWER:

=

7.74×10−9

Correct

Mass Spectrometer

J. J. Thomson is best known for his discoveries about the nature of cathode rays. Another important contribution of

his was the invention, together with one of his students, of the mass spectrometer. The ratio of mass to (positive)

charge of an ion may be accurately determined in a mass spectrometer. In essence, the spectrometer consists of

two regions: one that accelerates the ion through a potential and a second that measures its radius of curvature in

a perpendicular magnetic field.

The ion begins at potential and is accelerated toward zero potential. When the particle exits the region with the

electric field it will have obtained a speed .

Part A

With what speed does the ion exit the acceleration region?

Hint A.1 Suggested general method

Hint not displayed

Hint A.2 Initial energy

Hint not displayed

Hint A.3 Final energy

Hint not displayed

Find the speed in terms of , , , and any constants.

ANSWER:

=

Correct

Part B

After being accelerated, the particle enters a uniform magnetic field of strength and travels in a circle of radius

(determined by observing where it hits on a screen--as shown in the figure). The results of this experiment

allow one to find in terms of the experimentally measured quantities such as the particle radius, the magnetic

field, and the applied voltage.

What is ?

Hint B.1 Cyclotron frequency

Hint not displayed

Hint B.2 Relationship of and

Hint not displayed

Hint B.3 Putting it all together

Hint not displayed

Express in terms of , , , and any necessary constants.

ANSWER:

= Correct

By sending atoms of various elements through a mass spectrometer, Thomson's student, Francis Aston,

discovered that some elements actually contained atoms with several different masses. Atoms of the same

element with different masses can only be explained by the existence of a third subatomic particle in addition to

protons and electrons: the neutron.

Problem 27.66

A particle of charge is moving at speed in the z-direction through a region of uniform magnetic field . The

magnetic force on the particle is , where is a positive constant.

Part A

Determine the component .

Express your answer in terms of the variables , , and .

ANSWER:

=

Correct

Part B

Determine the component .

Express your answer in terms of the variables , , and .

ANSWER:

=

Correct

Part C

Can you determine the component form the given information?

ANSWER:

yes

no

Correct

Part D

If it is given in addition that the magnetic field has magnitude determine the magnitude of the last

component .

Express your answer in terms of the variables , , and .

ANSWER:

= Correct