Magnetism - ageespringer.files.wordpress.com · Magnetism • The phenomenon of magnetism was first observed thousands of years ago in several ancient cultures. • The term “magnet”

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Magnetism

Unit 5

Where we are…

•  We have completed the segment of the course dedicated to electricity.

•  We now begin our study of magnetism.

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Magnetism

•  The phenomenon of magnetism was first observed thousands of years ago in several ancient cultures.

•  The term “magnet” is derived Magnesia, a region in Asia Minor with rocks that were found to attract each other.

•  In modern life, magnets have many uses, from electric generators to computer memory.

Magnetism

•  Any magnet, regardless of shape, has two ends where the magnetic effect is strongest.

•  These ends are called the poles of the magnet.

•  If a bar magnet is suspended and free to rotate, one end always points north.

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Magnetism

•  The end that points north is called the north pole of the magnet.

•  The other pole points south is called the south pole.

•  This is the principle on which a compass operates (more on that in a bit).

Magnetism

•  When two magnets are brought near one another, each magnet exerts a force on the other.

•  The force can either be attractive or repulsive.

•  The type of force depends on the orientation of the magnets.

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Magnetism

•  When two north poles are brought near each other, the force between them is repulsive.

•  Similarly, two south poles repel each other.

Magnetism

•  However, a north pole and a south pole attract each other.

•  So, we can conclude –  Like poles repel. –  Unlike poles attract.

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WARNING WARNING WARNING

•  At this point you have probably noticed the similarity between magnetic poles and electric charges.

•  HOWEVER, do not confuse the two phenomena.

•  We will see that electricity and magnetism are closely related, but are NOT the same.


•  One important difference is that positive electric charges can (and do) exist independently of negative charges.

•  However, no magnetic pole has ever been observed independently of the other pole.

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•  If you cut a magnet in half, you do not get separate N and S poles.

•  Instead you get two magnets, each with a N pole and a S pole.

Ferromagnetism

•  Materials like iron that exhibit strong, permanent magnetic effects are relatively rare in nature.

•  Such materials are called ferromagnetic (after the Latin ferrum).

•  Other ferromagnetic materials include cobald, nickel, and gadolinium.

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The Magnetic Field

The Magnetic Field

•  When we studied electrostatics, we introduced the electric field as the source of the electric force.

•  We can also define a magnetic field that gives rise to the magnetic force.

•  The magnetic force is the result of the interaction between the magnetic fields of two objects.

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The Magnetic Field

•  Like the electric field, the magnetic field has a magnitude (strength) and a direction at any point in space.

•  Thus, we can say it is a vector field.

The Magnetic Field

•  Like earlier, drawing vector fields is tedious.

•  We can also use field lines to represent the magnetic field.

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The Magnetic Field

•  Like with the electric field – The direction of the magnetic field is

tangent to a field line at any point. – The strength of the field is represented by

the density of the field lines.

The Magnetic Field

•  The direction of the field can be seen from the direction a compass needle points at a given point.

•  The field lines point out from the north pole and in towards the south pole.

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The Magnetic Field

•  However, unlike electric field lines – Magnetic field lines do not start on north

poles, nor do they end on south poles. –  Instead the field lines continue inside the

magnet.

The Magnetic Field

•  Based on this, we arrive at an important conclusion:

Magnetic field lines always form closed loops.

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Earth’s Magnetic Field


•  The magnetic field we are most familiar with is Earth’s magnetic field.

•  We know from experience, that a compass always points towards geographic north.

•  But we have just seen that a compass points towards the south pole of a bar magnet.

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•  The north geographic pole of the Earth is actually the south magnetic pole.

•  Also, magnetic south is not located exactly at geographic north.

The Compass

•  A compass is a simple bar magnet that is suspended through its center of mass.

•  This leaves the magnet free to rotate.

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The Compass

•  As we have seen, the compass needle points in the direction of the local magnetic field.

•  Usually this is Earth’s magnetic field.

The Compass

•  However, if there is a stronger magnetic field nearby, the compass will point in the direction of that field.

•  So, a compass can be used to detect magnetic fields.

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Homework

•  Read sections 20-1 and 20-2 in the book.

Uniform Magnetic Fields

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•  In many examples, we will use a uniform magnetic field to simplify the problem conceptually and mathematically.

•  Like with uniform E fields, a uniform magnetic field stays constant in both magnitude and direction from one point to another.


•  A uniform magnetic field is not easy to produce over a large area.

•  However, the field between two flat, parallel poles is uniform if the area of the faces is large compared to their separation.

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Electric Currents and Magnetic Fields

Currents and Magnetic Fields

•  Yesterday, we saw that a compass is affected by a nearby magnetic field.

•  Today, we are going to start by looking at what happens to a compass when it is brought near a wire with a current running through it.

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•  What happens to the compass when it is brought near the wire when the current is turned off?

•  What happens once the current is turned on?


•  From these results, we can conclude

•  We can also infer that the field due to a current running through a straight wire is circular.

An electric current produces a magnetic field.

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(A) Right-Hand Rule

•  In order to determine the direction of the magnetic field, physicists use the right-hand rule.

•  This is one of several right-hand rules used in physics.

(A) Right-Hand Rule

•  To find the direction of the field, – Place your right hand

along the wire so that your thumb points in the direction of the current.

– Curl your fingers around the wire.

– This is the direction of the magnetic field.

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Magnetic Field of a Loop of Current

•  Another common magnetic field results from a circular loop of current.

•  The field around each point in the wire is circular.

•  The fields in the middle combine.

Notation of the Magnetic Field

•  Just like the electric field, the magnetic field has a variable used in equations.

•  It is not M, as this would be confused with mass.

•  Instead, we use to represent the magnetic field.

€

B

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Magnetic Force on an Electric Current

Force on a Wire

•  We have already seen that the magnetic force is the result of the interaction between two magnetic fields.

•  We have also seen that a current can generate a magnetic field.

•  So, we can infer that a wire with current will experience a force in a magnetic field.

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COOL VIDEO

Questions

•  In the video, what direction does the B field point?

•  What happened when the current was turned on?

•  What direction did the force point?

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Force on a Wire

•  The force on a current-carrying wire is perpendicular to the wire, and also perpendicular to the B field.

Another Right-Hand Rule

•  To find the direction of the force, we use a (different) right hand rule.

•  Start by placing your hand so your fingers point in the direction of the current.

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Another Right-Hand Rule

•  Then, curl your fingers in the direction of the B field. If you find yourself trying to curl your fingers backwards, turn your hand upside down.

•  The direction your thumb points is the direction of the force.

Force in a Uniform B Field

•  We would like to know how to calculate the force on a segment of wire in a uniform magnetic field.

•  The derivation of this force law requires calculus and the cross-product of vectors.

•  We will be skipping the derivation.

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Force in a Uniform B Field

•  However, if the B field is uniform, and the current is constant, we can deduce a fairly simple force law.

•  The force is proportional to the amount of current, I, the length of the wire, l, and the strength of the magnetic field B.

•  The force also depends on the angle the wire makes with the field.

Force in a Uniform B Field •  The force law is

•  This is known as the Lorentz Force applied to a current in a constant magnetic field.

€

F = IBsinθMagnitude of the force on the wire

Current in the wire

Length of the wire in the field

Strength of the magnetic field

Angle between I and B

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Some Things to Notice

•  The strength of the force depends greatly on the angle.

•  If I and B are parallel, the angle is zero, and the force is also zero.

•  If I and B are perpendicular, the angle is 90, and the force is

€

Fmax = IB

Conclusion: Finding the Magnetic Force

•  So, to find the magnetic force on a wire, use the Lorentz Force law to find the magnitude.

•  Then use the right-hand rule to find the direction of the force.

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Units of B

•  The SI unit for the magnetic field, B, is the tesla (T).

•  From the Lorentz force law, we can see

•  An older name for the tesla is the “weber per meter squared.” The two are equivalent.

€

1T =1 NA⋅ m

€

1Wbm2 =1T

Units of B

•  Magnetic fields are also sometimes given in units of gauss (G).

•  Fields given in gauss should always be converted to tesla before being used in calculations.

€

1G =10−4T

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Example: Magnetic Force

A wire is carrying 30 A of current and has a length of 12 cm. The wire passes between the poles of a magnet with a uniform field of 0.9 T. If the wire makes an angle of 60° with the field, what is the magnitude of the magnetic force on the wire?

Homework

•  Read section 20-3.

•  Do problems 1-3 on page 577.

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Magnetic Force on a Moving Charge

Magnetic Force

•  We saw on Wednesday that a magnetic field exerts a force on a wire carrying a current I.

•  What is a current?

•  Since a current is is just a flow of electrons, we might expect that a moving free charged particle would experience a force in a B field also.

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Magnetic Force

•  Let’s recall some things we’ve learned about the magnetic force:

– The force depends on the strength of the B field.

– The force depends on the strength of the current.

•  The current is determined by the strength of the charged particles and their velocity.

– So the force on a single particle depends on its charge and its velocity.

Magnetic Force

•  Let’s recall some things we’ve learned about the magnetic force:

– The force also depends on the angle between the field and the current (or velocity vector).

– Lastly, the force is perpendicular to both the field and the current (or velocity).

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Lorentz Force Law

•  Like the current example, this formula only tells us the magnitude of the force.

•  To find the direction, we use the right-hand rule.

€

F = qvBsinθMagnitude of the force on the wire

Charge of the particle

Magnitude of the particle’s velocity

Strength of the magnetic field

Angle between I and B

Yet Another Right-Hand Rule

•  Start by placing your right hand so that your fingers point in the direction of the particle’s velocity vector.

•  Then, curl your fingers in the direction of B.

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Yet Another Right-Hand Rule

•  Your thumb points in the direction of the force.

•  WARNING: This is only true if the particle is positively charged. If the particle is negatively charged, you must flip the direction of the force.

Some Things to Notice

•  Just as with currents, the strength of the force depends greatly on the angle.

•  If v and B are parallel, the angle is zero, and the force is also zero.

•  If v and B are perpendicular, the angle is 90, and the force is

€

Fmax = qvB

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Example: Magnetic Force on a Proton

A proton (m = 1.67 x 10-27 kg) has an initial velocity of 5 x 10-6 m/s when it enters a region of uniform magnetic field. The proton experiences a force of 8 x 10-14 N. If the velocity and force on the proton are as shown below,

a) What is the magnitude of the B field?

b) What is the direction of the B field?

Drawing the B Field

•  In many of the following problems, we will define B as pointing into or out of the page so that we can observe the 2D motion of the particle.

•  We notate this in the following way:

⊗ ¤ Vector pointing into the page

Vector pointing out of the page

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Path of a Charged Particle

•  Consider the following example of an electron in a uniform B field.

•  The electron experiences a force as shown.


•  But Newton’s 2nd says this force produces an acceleration on the electron.

•  This causes the direction of the velocity to change.

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•  This change in the direction of V also changes the direction of F.

•  As a result, the electron follows a circular path.


•  In fact, any charged particle moves in a circular path in a uniform B field.

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•  Also, since the trajectory is circular, we know how to calculate the acceleration:

€

ac =v 2

r

Example: Electron in a B Field Suppose an electron is moving at 2 x 107 m/s in a uniform 0.01 T magnetic field as shown.

a) What is the magnitude of the force on the electron?

b) What is the radius of the electron’s trajectory?

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Homework

•  Read section 20-4 through the section about the Aurora Borealis.

•  Do problems 9, 11, and 14 on page 578.

•  The answer to 14 will be on the blog.

Whiteboarding Groups

Group Members Problem 1 Robert, Piper, Anthony 4 2 Aidan, Bailey, Jacob 5 3 Sarah, Connor, John 7 4 Angi, Armen, Krystiana 8 5 Kaleb, Brie, Abbey 9 6 Rachel, Jeremiah 14 7 Miggy, Ellen 16

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Mass Spectrometer

Mass Spectrometer

•  The mass spectrometer is a device used in atomic physics to determine the mass of unknown atoms or particles.

•  The device has two components, both of which operate on principles we are familiar with.

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Mass Spectrometer

Part 1: The Velocity Selector

•  The charged particles are initially passed through a region of perpendicular E and B fields.

•  When a particle enters this region, it experiences opposing electric and magnetic forces.

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•  If one force is greater than the other, the particle will be deflected either up or down.

•  The strength of the E and B fields are adjusted until the particles travel through the region without being deflected.


•  Since the trajectory of the particles was not changed, we can conclude that no net force was exerted on the particles.

•  This means that

€

F = FE + FM = 0∑

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•  Since the net force is zero, the electric and magnetic forces must be equal.

•  We can solve this equation to find the velocity of the particle.

€

qE = qvB

€

v =EB

Part 2: Uniform B Field

•  Once the velocity of the particles has been determined, the particles pass into a region filled with a (different) uniform B field.

•  The B field causes the particles to move in a circle.

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•  The radius of the circle depends on the strength of the magnetic force, which depends on the strength of the B field.

•  The B field is adjusted until the particles move in a circle that allows them to strike the detector.


•  Once the radius of the circle and the strength of the B field have been measured, we can use Newton’s second law to find the mass.

€

F∑ = mac

qvB = m v 2

r

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•  Solving for m:

•  Plugging in the result for v from part 1:

€

m =qr " B

v

€

m =qrB " B

E

Charge of the particle

Radius of the circle

Strength of the B field from part 1 Strength of the B field from part 2 Strength of the E field from part 1

Mass Spectrometer

•  Historically, the masses of many atoms were measured this way.

•  Samples of some elements were found to separate into two or more radii.

•  This indicated that some atoms of the same element had different masses.

•  These are called isotopes, and the different mass is due to different numbers or neutrons.

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A Word About Units

•  Since atoms have very little mass, the mass of an atom is often expressed in terms of atomic mass units (u).

•  You will need to convert atomic mass units to kg before performing calculations.

€

1u =1.66 ×10−27kg

Example: Mass Spectrometry A sample of an unknown element is shot through a mass spectrometer. When the fields are set to E = 8800 V/m, B = 0.035 T, and B’ = 0.2 T, the particles move in a circle of radius 20.87 cm. If the particles carry the same charge as a proton, determine the mass of the element.

Express your answers in both kg and atomic mass units.

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Example: Mass Spectrometry Carbon atoms (mass 12 u) are mixed with atoms of another unknown element. In a mass spectrometer (E, B, and B’ held constant), the carbon atoms move in a path of radius 22.4 cm. The unknown atoms move in a circle with radius 26.2 cm. What is the mass of the unknown element?

Homework

•  Do problems 60 and 62 on page 581 of the book.

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B Field of a Wire

B Field of a Wire

•  We saw earlier that a wire carrying a current, I, generates a magnetic field.

•  We know from experiments that the field of the wire is circular about the wire itself.

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B Field of a Wire

•  Experiments have shown that the field is directly proportional to the amount of current in the wire and inversely proportional to the distance from the wire.

€

B = constant⋅Ir

B Field of a Wire

•  The constant of proportionality is written as

•  Where µ0 is called the permeability of free space.

€

constant =µ02π

€

µ0 = 4π ×10−7 T ⋅mAµ0 =1.26 ×10−6 T ⋅mA

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B Field of a Wire

•  Therefore, the magnetic field near a long, straight wire is given by

€

B =µ02π

Ir

Example: B Field Near a Wire A wire is carrying a DC current of 25 A vertically upward as shown. What is the magnitude and direction of the magnetic field at point P?

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Example: B Field Between Two Currents

Two parallel straight wires are 10 cm apart and carry currents in opposite directions. The current in wire 1 is 5 A and the current in wire 2 is 7 A. What is the magnitude and direction the the magnetic field halfway between the two wires?

Homework


•  Do problems 26, 27, 32, and 33 on pages 578-579.

•  For 33, remember that the two magnetic fields are vectors. You will have to use vector addition to solve the problem.

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Magnetic Force Between Two Wires

Force Between Two Wires

•  We saw last week that a wire carrying current generates a magnetic field.

•  We also know that a wire experiences a force when placed in a magnetic field.

•  Therefore, two wires carrying currents must exert forces on each other.

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•  Consider two parallel wires, 1 and 2, carrying currents I1 and I2. The wires are a distance d apart.

•  Wire 1 generates a magnetic field given by

€

B1 =µ02π

I1d


•  Wire 2 experiences a force due to the field generated by wire 1.

•  Plugging in for B1.

€

F2 = I2 2B1 sin 90( )F2 = I2 2B1

€

F2 =µ02π

I1I2d 2

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•  Notice that the force on wire 2 is caused by the field due to wire 1 only.

•  Wire 2 also generates a B field, but does not exert a force on itself.

•  Wire 2 does exert a force on wire 1 though.

Direction of the Force

•  The direction of the force is found using the right-hand rule.

•  Here, B points into the board, and I points up.

•  Therefore, F points to the left.

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•  What if the current in wire 2 pointed in the opposite direction?

•  In that case, the force would point to the right.

•  Notice in both cases, the force on wire 1 is the opposite of the force on wire 2. This is consistent with Newton’s 3rd Law.


•  So, we can conclude:

Currents running in the same direction attract.

Currents running in opposite directions repel.

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Example: Force Between Two Wires

The two wires of a 2 m long appliance cord are 3 mm apart. If each wire carries 8 A of current, calculate the magnitude of the force each wire exerts on the other.

Example: Suspending a Wire

A horizontal wire carries 80 A of current. A second wire is located parallel to the first wire 20 cm below. How much current must the second wire carry so that the magnetic force on wire 2 balances out the force of gravity. Each wire is 10 cm long, and wire 2 has a density of .12 g/m.

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Homework



•  For problem 42, you will have to look up the density of copper.

Review Problems


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Solenoids and Electromagnets

This material will be on Friday’s test.

Solenoids

•  We have looked at the magnetic field of a long, straight wire.

•  We have also looked at the magnetic field of a loop of current.

•  Now, we will examine the field that results when many loops are connected together.

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Solenoids

•  A long coil of wire consisting of many loops is called a solenoid.

•  The field is essentially the sum of the fields due to each individual loop.

Solenoids

•  Notice that the magnetic field of the solenoid looks much like the field of a bar magnet.

•  As a result, we can define a N pole and a S pole for the solenoid.

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Solenoids

•  The magnetic field inside the solenoid is constant and can be calculated using the formula:

€

B =µ0NI

Example: B Field of a Solenoid A 15 cm long solenoid contains 520 coils of wire. How much current is running through the wire if the B field is measured to be 0.42 T at the center of the solenoid?

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Electromagnets

•  If the center of the solenoid is filled with an iron core, the iron becomes magnetized.

•  As a result, the magnetic field of the solenoid is increased.

Electromagnets

•  The magnetic field is often hundreds of times stronger than the field due to the current alone.

•  Such a construction is called an electromagnet.

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Electromagnets

•  Electromagnets and solenoids have a wide range of uses, including motors, circuit breakers, household circuits, and research.

Homework


•  Do problems 48-50 and 69 on pages 580-581.

Documents

Magnetism - ageespringer.files.wordpress.com · Magnetism • The phenomenon of magnetism was first observed thousands of years ago in several ancient cultures. • The term “magnet”