Ch28 magnetic fields

Rogers: Lectures based on Halliday, Resnick and Walker’s Fundamentals of Physics, Copyright 2005 by Wiley and Sons

Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to Copyright Protection. 1

Ch28-1/45

Ch 28 Magnetic fields

(exclude 4-crossed fields, 5-Hall effect 7-Cyclotrons)2 lectures

We are surrounded magnets and their effects.

electromagnets - next chapter discusses how current in a wire creates a magnetic field

Electrons have intrinsic magnetic fields (like charge, mass)

Permanent magnets (ch 32) occur when these intrinsic magnetic fields have a net overall magnetic field.

First we define B, the magnetic field

Ch28-2/45

Definition of magnetic field B

Electric field is defined in terms of force on an element of charge.

There are no magnetic monopoles (i.e., isolated magnetic charges) => we must define B differently.

We use magnetic force exerted on a moving chargeto define B

FB is zero in the same direction as B.

In general



Ch28-3/45

Refresher on vector (or cross) products

since sin90o = 1

If above is true, => a right-handed coordinate system

The product is perpendicular to both a and b.

Ch28-4/45

Direction of resultant vector is determined by the right hand rule.

To find direction of fingers of

right hand sweep towards

Vector is directed along thumb.

Force vector is

-in same direction as thumb for a +ve charge q

-in opposite direction for a -ve charge.

Refresher on vector (or cross) products



Ch28-5/45

No component of in direction of motion

=> cannot change particle’s speed, v

It accelerates particle by changing its direction.

Defn of B (cont)

There is no force in direction of (φ=0)

Force acting on a charged particle moving with velocity through a magnetic field is always perpendicular to both and

Ch28-6/45

Defn of B (cont)

SI unit of B is tesla (T).

Old unit is gauss (G)



Ch28-7/45

Some values of B

Some approximate magnetic field values

Surface neutron star 108 T

Near large electromagnet 1.5 - 9 T (MRI)

Near small bar magnet 10-2 T

At Earth’s surface 10-4 T = 1 G

In interstellar space 10-10 T

Best magnetically shielded room 10-14 T

Ch28-8/45

An example from particle physics

Tracks of 2 electrons and positronin a bubble chamber

Uniform magnetic field out of page.

Event is triggered by an unseen photon (no charge) from top. Triplet production.

Which way are forces? L, R, U, D

electrons

positron



Ch28-9/45

An example from particle physics

Tracks of 2 electrons and positronin a bubble chamber

Uniform magnetic field out of page.

Event is triggered by an unseen photon (no charge) from top. Triplet production.

Which way are forces? L, R, U, D

electrons

positronAns: to left in direction of motion since -ve charge

Ans: to right in direction of motion since +ve charge

Ch28-10/45

What is direction of magnetic force acting on moving particles: +-x, +-y, +-z, 0?

Is the coordinate system right handed?



Ch28-11/45

What is direction of magnetic force acting on moving particles: +-x, +-y, +-z, 0?

Ans (a): +z axisAns (b): -x axis (negative charge)Ans (c): 0 since anti-parallel.

Is the coordinate system right handed?Ans: yes

Ch28-12/45

Magnetic field lines

Represent magnetic fields by field lines as for electric fields.

1) direction of B is tangent to field line at all points

2) field strength proportional to number of lines /unit area | lines

3) they exist inside a magnet (unlike electric field lines inside a conductor)

4) they leave a magnet from North pole and enter via South pole



Ch28-13/45

Magnetic polesMagnets have two poles.

Referred to as a magnetic dipole.

Bar, horseshoe and C magnets shown.

Opposite magnetic poles attract each other and like poles repel each other

Earth’s magnetic field is produced in its core.

Compass is a bar magnet on a pivot.

Compass North end points to North pole

i.e. Earth’s “geomagnetic North Pole” is actually South pole of a magnet => field lines enter there.

Ch28-14/45

Sample problemUniform magnetic field B with magnitude 1.2 mT, is directed up throughout a lab chamber. A proton with kinetic energy 5.3 MeV enters moving horizontally from S to N. (a) What magnetic force acts on the proton as it enters? (b) What acceleration is produced by this force?

Ignore Earth’s magnetic field. mp = 1.67E-27 kg.

Key ideas: Use

First find v from

v/c about 11%, so becoming relativistic.



Ch28-15/45

Direction of force? Key ideas: v is S to N, field is out of page (shown as dots in figure, x would mean into page)

Right hand rule, +ve charge => force acts to right.

Sample problem (cont)

Very small force acting on very small mass

=> considerable acceleration (but no change in speed!)

What would happen to an electron in same situation?

Ch28-16/45

Direction of force? Key ideas: v is S to N, field is out of page (shown as dots in figure, x would mean into page)

Right hand rule, +ve charge => force acts to right.

Sample problem (cont)

Very small force acting on very small mass

=> considerable acceleration (but no change in speed!)

What would happen to an electron in same situation?

Ans: Force in opposite direction and acceleration 1836 times greater since mass much lighter (ignoring relativistic effects - which we cannot really do)



Ch28-17/45

Circulating charged particles

What does it mean if an object is moving in circle at constant velocity?

=> constant force acting towards centre of circle to keep particle accelerating inwards

Figure: electrons circulating in a chamber with low pressure gas (to visualize). Magnetic field out of page. Electron gun G provides an initial speed v.

Force directed toward centre since | v and B.

There is no effect on speed.

G

Ch28-18/45

Circulating charged particles(cont)

Determine characteristics of motion of these electrons or any charged particle in a uniform magnetic field (out of page).

From Ch 6, uniform circular motion implies a centripetal force

and since B | v

=>

Solving for r:

Radius of curvature is bigger for heavy particles, bigger for higher energy particles, smaller for higher charges and stronger fields

Which e- has higher energy?

G



Ch28-19/45


We can also specify periods and frequencies

period (s)

frequency (1/s)

angular frequency(1/s)

Independent of speed of particles! (as long as non-relativistic).

Faster particles go in bigger circles.

For a given |q|/m: T etc are all same, independent of v r is same for same values of v

G

Ch28-20/45


Consider looking in direction of B (i.e. opposite of example shown) => for particles in a circular orbit, direction of +ve particles is counterclockwise, direction of -ve particles is clockwise.

So far we considered particles with velocityvector | magnetic field.

What happens when they are not | ?

G



Ch28-21/45

Helical pathsIf velocity of a charged particle has a component parallel to a (uniform) magnetic field => particle will move in a helical path about

direction of magnetic field.

Consider the two components of velocity in figure.

The | component is used to determine radius

and the || component determines pitch p (distance between adjacent turns)

Ch28-22/45

Helical paths (cont)

In magnetic bottle, charged particles follow field lines in a helical path. If field strong enough at ends, they can be reflected back (not shown infigure, but stronger field at ends has a component of force backwards).

Earth’s magnetic field lines cause stray charged particles to be trapped in Van Allen radiation belts Bounce back and forth between N and S poles (few seconds/trip).

Solar flares=> radiation disrupts reflection => electrons come down into atmosphere => aurora/northern lights (100 km high green from oxygen, pink from nitrogen -appears white when weak)



Ch28-23/45

Mass spectrometer

Mass spectrometers are used to determine atomic mass of a substance.Element 1 is a source which uses an

electric potential difference to accelerate a particle of charge q (all in vacuum).

Element 2 is a uniform magnetic field B (out of page) whichbends charged particle to a detector which measures radius of curvature

Element 3 is solving physics equations that determine mass given V, B and x

Ch28-24/45

Mass spectrometer (cont)

As particle leaves source, it has roughly no kinetic energy. +ve charge accelerates going from high to low potential. At end of applied potential

Using relationship (28-15) for r in a uniform magnetic field

Result gives m/q in terms of what we measure.



Ch28-25/45

Magnetic force on a current-carrying wireA flexible wire will be bent in presence of a magnetic field. Direction of force depends on current direction (right hand rule).

Inside wire consider e-with drift velocity down, or +ve charge with drift velocity up,

both => force to right

F = vdeB

Ch28-26/45

Magnetic force on a current-carrying wire (cont)

Consider a length of wire, L. All conduction electrons in this segment will drift past plane xx in time t = L/vd=> in that time a charge

will pass thru plane.

Recall

Drift velocity has dropped out of equation!

Eqn applies if field B | wire. It gives the magnetic force acting on a length of wire L with current i .

More generally:



Ch28-27/45

If a wire is not straight, or field not uniform, force acting on wire can be found by considering wire as series of elemental components

Perform an integration over actual wire to get

Magnetic force on a current-carrying wire (cont)

L is a length vector, magnitude L and directed along wire in same direction as conventional current.

FB is in direction of cross product because i is, by convention, in direction of +ve charge carriers.

Above eqns equivalent toi.e., either pair can be used to define B.

Eqns in terms of iL are usually used since measuring force on a wire is easier.

Ch28-28/45

Question

Current i in a wire passing thru a uniform magnetic field B. Resulting magnetic force is shown. Field is oriented so force is a maximum. What is direction of magnetic field?

(a) +ve y axis

(b) -ve z axis

(c) -ve y axis



Ch28-29/45

Question

Current i in a wire passing thru a uniform magnetic field B. Resulting magnetic force is shown. Field is oriented so force is a maximum. What is direction of magnetic field?

(a) +ve y axis

(b) -ve z axis

(c) -ve y axis Ans: => B -ve y-axis by rhr

Ch28-30/45

B must be horizontal to right so force is upwards since i is out of page (and +ve).

Since this must balance gravity where m is mass of wire. φ must be 90o to maximize force

where m/L is the linear density. Result independent of L.

This is 160 times Earth’s magnetic field.

Sample problem

A straight copper wire has a current i=28 A out of page. What are direction and magnitude of smallest magnetic field B needed to suspend the wire against gravity (linear density is 46.6 g/m).

Key ideas: To be minimal force => acts directly up and B must be | L.



Ch28-31/45

Torque on a current loop (electric motors)

A current loop in a uniform magnetic field forms a simple motor.

As shown, at ends i is parallel or anti-parallel to B => no force acting.

On sides forces are in opposite directions => rotate the current loop.

Trick is to reverse current flow when loop is vertical (commutator does so).

Ch28-32/45

Torque on a current loop (electric motors) (cont)

Define n, normal vector | plane of loop. Direction of n defined by another right-hand rule with fingers curling along direction of current.

Left hand figure is not like previous one since field is into page.If loop in (a) is in plane of page => net force on loop is 0 since everything is equal and opposite. But once at an angle wrt a horizontal axis there is a net force/torque.



Ch28-33/45


Looking from side 2 end of loop along it (side 4 is behind side 2) .

θ is the arbitrary angle between n and B.

Side 2: L is in direction of i, magnitude b. Angle with B is 90 - θ.

Using

rhr tells us F2 is out of page.

Side 4:L is in opposite direction => force is equal and opposite direction => no net force in or out of page.

θ90 - θ

θ

B

n

90 - θ

L

Ch28-34/45


Sides 1 & 3: L is | to B so F1 and F3both have magnitude = iaB. In opposite directions => no net movement up or down.

Forces are not on same line of action => net torque which tends to rotate loop so that n is aligned with B.

Projection of force F1 | wire is F1cos(90-θ) =F1sinθzero when θ is 0 (aligned).

Torque has a moment arm

so magnitude of torque from F1 & F3 is

θ90 - θ

θ

B

n

b/2 sinθ



Ch28-35/45


Suppose single loop is replaced by a coil with N loops or turns.

A=ab, area of coil. (NiA) are all characteristics of coil.

This eqn holds for all flat coils, no matter the shape, as long as B uniform.

In an electric motor the current must be reversed when θ is 0 (via commutator).

Ch28-36/45

Magnetic dipole moment

A current carrying coil is said to be a magnetic dipole just like a bar magnet since a magnetic field generates a torque on it.

We assign a magnetic dipole moment µ to the coil. µ has same direction as normal to coil (same rhr as before for that direction) and magnitude

N is # turns in coil, i the current in wire and A area enclosed. Rewrite eqn for τ as

where θ is angle between µ and B.

Generalize to

which is very much like



Ch28-37/45

Magnetic dipole moment (cont)

A magnetic dipole in a magnetic field => magnetic potential energy that depends on orientation. in field. For electric dipoles we showed

In analogy, for magnetic case we have

U=+µB U=-µB

Aligned µ and B => lowest energy (-µ B) => state torque wants to achieve. Higher energy when anti-parallel (+µ B)

Must work against torque to get there.

Ch28-38/45

Magnetic dipole moment (cont)

Work done on dipole by magnetic field as rotated is:

If an external agent acts on dipole during change in orientation, work done on dipole by applied torque is the -ve of work done by magnetic field on dipole, i.e.

The first eqn suggests another unit for µ, viz [J/T]

recall



Ch28-39/45

Some magnetic dipole moments

Small bar magnet 5 J/T

Earth 8.0x1022 J/T

proton 1.4x10-26 J/T

electron 9.3x10-24 J/T

There are many magnetic dipoles aside from the current carrying loop.

Some examples magnetic dipole moments

Note: electron has a much larger magnetic dipole moment than proton.

Ch28-40/45

Consider above 4 orientations of µ, a magnetic dipole moment in magnetic field B. Rank them, largest first, according to

(a) magnitude of torque on dipole

(b) potential energy of dipole.

Question



Ch28-41/45

Consider above 4 orientations of µ, a magnetic dipole moment in magnetic field B. Rank them, largest first, according to

(a) magnitude of torque on dipole

(b) potential energy of dipole.

Question

Ans: => 1&4 same& highest since cos(180-θ) is same and negative. Or because they required moving against torque which wants them to align with B.

Ans:

=> all the same

Ch28-42/45

What is the direction of the current in the coil?

How much work would an external agent have to do via a torque to rotate to 90o?

Circular coil has 250 turns, A=2.52E-4 m2, i=100µA. Coil at rest in B with magnitude B=0.85 T and µ aligned with B.

Sample problem

Ans: rhr fingers follow current & thumb points to µ=> current down screen on closest side.

Key idea: applied work = change in potential due to change in orientation.



Ch28-43/45

Following not used in class

Ch28-44/45

Sample problem: galvanometerA galvanometer measures current flow by measuring torque exerted by a uniform magnetic field on a current-carrying coil.

Example: Coil is 1.2 x 1.2cm2, 250 turns, can rotate about an axis into page. Uniform magnetic field of 0.23 T as shown so field | normal vector of coil (along pointer) and parallel plane of coil.

A spring counterbalances torque so for a given i, pointer is steady with an angulardeflection dφ. Greater current => greater deflection.

If 100 µA causes 28o deflection, what is torsional spring constant κ ( )?



Ch28-45/45

Sample problem: galvanometerWhich way does the current flow?

What is torsional spring constant κ( )?

Key idea: spring’s torque is balanced by magnetic field torque when there is a constant current. where A=area of coil, φ is the angular deflection (28deg) and θ = 90 since field always | to normal to coil.

Ans: up left, down right

Digital ammeters and voltmeters do not use a moving coil.

Ch28 magnetic fields

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