Magnetic Field and Magnetic Forces
Lecture Notes 20
Magnetic Field and Magnetic Forces
Goals
■ To observe and visualize magnetic fields and forces. ■ To study the motion of a charged particle in a magnetic field. ■ To evaluate the magnetic force on a current-carrying conductor. ■ To determine the force and torque produced with a magnet and current-carrying loop of wire (the DC motor). ■ To study the fields generated by long, straight conductors. ■ To observe the changes in the field with the conductor in loops (forming the solenoid). ■ To calculate the magnetic field at selected points in space. ■ To understand magnetism via magnetic moments.
Historical Notes - Lodestone (magnetite mineral):
The name "magnet" comes from lodestones found in a place called Magnesia. A piece of intensely magnetic magnetite that was used as an early form of magnetic compass. Only magnetite with a particular crystalline structure, lodestone, can act as a natural magnet and attract and magnetize iron. In China, the earliest literary reference to magnetism lies in a 4th century BC book called Book of the Devil Valley Master: "The lodestone makes iron come or it attracts it." The earliest mention of the attraction of a needle appears in a work composed between 20 and 100 AD. From amusement and magic-like to application − “leading stone” (lodestone) – compasses and navigation. By the 12th century the Chinese were known to use the lodestone compass for navigation. Today’s powerful lodestones − and magnetic materials are ubiquitous.
What if no lodestones existed? ► The Chinese would certainly not have invented the magnetic compass. ► Magnetism would have been discovered much later, and one wonders how. ► Lacking the compass, the great voyages of discovery could hardly have taken place--Columbus, De Gama, Magellan and the rest. ► The history of the world might have been quite different! Magnetic Jewellery • Magnetic energy is the strongest natural force in the universe and the power of magnets is one of the most basic powers of nature. • The use of magnet therapy for health and well-being has an ancient history dating back thousands of years. • Ancient Egyptians used loadstones to prolong life and improve health. It is said that Cleopatra wore a polished lodestone on her third eye, in the belief that it helped maintain her youth and beauty. • In more recent times, Paracelsus (1493-1541) considered to be the father of modern medicine, believed that the "life force' of the body was most influenced by the force found in magnets. • In Europe, Russia, China, Japan and many other countries, convinced of the benefits, millions of people continue to use magnet therapy. • Today, we are experiencing an exciting revival of this ancient therapy. Resulting from the impact of more and more clinical studies and anecdotal evidence, 120 million people worldwide spend over $1.5 billion globally on the therapeutic benefits of magnets.
Two experiments, different scale: Although the magnet on the left is an electromagnet (huge) and the one on the right is a permanent magnet (small), the idea is the same.
Observations: ► You get a hint of magnetic field whenever you attach a note to a refrigerator door with a small magnet. ► Accidentally erase a computer disk by bringing it near a magnet. ► Credit cards, VCRs,…etc ► A familiar type of magnet: “electromagnets”. A wire coil is wound around an iron core and a current is sent through the coil. The strength of the magnetic field is determined by the size of the current. ☼ In industry, such magnets are used for sorting scrap iron among many other things. ► You are probably more familiar with “permanent magnets” (like the refrigeratordoor type). They do not need current to have a magnetic field. ► Magnets: A magnet is an object made of certain materials which create a magnetic field.
Electricity-Magnetism-Electromagnetism: From these modest origins, the sciences of electricity and magnetism developed separately for centuries…until 1820 AD. In 1820, Oersted found a connection between them: an electric current in a wire can deflect a magnetic compass needle. The new science of “electromagnetism” was developed by Faraday, Henry, and Maxwell (who, in the mid-19th century put electromagnetism on a sound theoretical basis)…
20.1 Magnetism The behavior of bar magnets ■ Notice the general behavior trends of attraction and repulsion, dipole or monopole.
The simplest magnetic structure that can exist is a magnetic dipole. Magnetic monopoles do not exist (as far as we know) ∴We cannot define the magnetic field as we do for the electric field (E=Fe/q.)
Magnetic properties of materials can be traced back to their atoms and electrons.
The clustering of the lines (of e.g. iron filings) at the ends of a magnet suggests that one end is a “source” of the lines (the field diverges from it) and the other end is a “sink” of the lines (the field converges toward it).
By convention: north pole and south pole, respectively. The magnet with its two poles is an example of a “magnetic dipole”. Note: A magnetized bar is magnetically weak in the middle Puzzle Determine which of the two bars is magnetized (without using any other material.)
A compass will align with fields
The compass will align with
whatever average field is strongest. As shown in figure, the field caused by the current in the wire is stronger than that any background field from the earth.
Absent
the current-carrying wire, the compass would align with the earth’s magnetic field. This allows a consistent direction to be determined by someone with the need for navigation.
Iron filings will align as a compass does
Each small filing lines up tangent to the field lines allowing a visual demonstration
The magnetic field lines and pattern of iron filings in the vicinity of a bar magnet and the magnetic field lines in the gap of a horseshoe magnet.
“Earth is a huge magnet”
● It can be represented by a huge bar magnet (a magnetic dipole.) (William Gilbert 1540-1603)
● The dipole axis makes an angle about 11.5o with the rotation axis. ● It intersects Earth’s surface at the “geomagnetic north pole” and the “geomagnetic south pole”. ● The “north magnetic pole” is really the south pole of Earth’s magnetic dipole.
■ Our Earth itself has a magnetic field This field is not very strong but it is consistent.
● The “field declination” is the angle (left or right) between geographic north (which is toward 90o latitude) and the horizontal component of the field. The “field inclination” is the angle (up or down) between a horizontal plane and the field’s direction. ● We can use a “compass” and a “dip meter” to determine these two angles. ● The field observed at any location on the surface of Earth varies with time. In fact, Earth’s field has reversed its polarity about every million years.
Examples of Magnetic Fields Fields are created in a variety of ways and observed in a variety of places.
20.2 Magnetic Field and Magnetic Force Magnetic Forces External Forces Gravitational Force Normal Force Frictional Forces Tension Force Restoring Force of a Spring Collisional Forces Electrostatic Force Magnetic Force
Newton’s Second Law
ΣF = m a
The magnetic force, like the other forces we have encountered, may contribute to the net force that acts on an object
Review: When a charge is placed in an electric field, it experiences a force, according to
F = qE
Definition of the magnetic field The “magnetic field B” at a point is along the tangent to a field line. Its direction is that of the force on the north pole of a bar magnet, or the direction in which a compass needle points. The strength of the field is proportional to the number of lines passing through a unit area normal to the field . Therefore , B is also called the “magnetic flux density” .
There are two ways to set up a magnetic field: (1) Moving electrically charged particles. (2) Elementary particles such as electrons have an intrinsic magnetic field around them; that is these fields are a basic characteristic of the particle, just as are their mass and electric charge (or lack of charge).
The magnetic fields of the electrons in certain materials add together to give a net magnetic field around the material. This is true for the material in permanent magnets (which is good, because they can then hold notes to a refrigerator door). In other materials, the magnetic fields of all the electrons cancel out, giving no net magnetic field surrounding the material. This is true for the material in your body (which is also good, because otherwise you might be slammed up against a refrigerator door every time you passed by).
The force that a magnetic field exerts on a moving charge Experimentally, we find that when a charged particle (either alone or part of a current) moves through a magnetic field, a force due to the field can act on the particle.
The following conditions must be met for a charge to experience a magnetic force when placed in a magnetic field: 1. The charge must be moving. 2. The velocity of the charge must have a component that is perpendicular to the direction of the magnetic field.
The effect of an existing magnetic field on a charge depends on the charges direction of motion relative to the field. The magnetic force F on a particle with charge q and velocity v in a magnetic field B is perpendicular to both v and B and given by :
F = qv×B
(0 ≤ φ ≤ 180o )
Its magnitude is F = q vB sin φ , where φ is the angle between v and B .
The SI unit of the magnetic field is the tesla ( T ) • Note that 1 T ≡ N.s/(C.m), and since C/s = 1 A, 1 T ≡ 1 N/(A.m). • Since the tesla is a large unit, a (cgs) unit called the gauss ( G ) is often used where 1 T = 104 G Some approximate magnetic fields At the surface of a neutron star 108 T , At Earth’s surface 10−8 T , In interstellar space 10−10 T
Example 1 Magnetic Forces on Charged Particles A proton in a particle accelerator has a speed of 5.0x106 m/s. The proton encounters a magnetic field whose magnitude is 0.40 T and whose direction makes and angle of 30.0 degrees with respect to the proton’s velocity (see part (c) of the figure). Find (a) the magnitude and direction of the force on the proton and (b) the acceleration of the proton. (c) What would be the force and acceleration of the particle were an electron?
(a)
(
)(
)
(
F = qo vB sin θ = 1.60 ×10 −19 C 5.0 ×106 m s (0.40T )sin 30.0 = 1.6 ×10 −13 N
(b)
1.6 ×10 −13 N F 13 2 = = 9 . 6 × 10 m s a= mp 1.67 ×10 − 27 kg
(c)
Magnitude is the same, but direction is opposite.
1.6 ×10 −13 N F 17 2 = = 1 . 8 × 10 m s a= me 9.11×10 −31 kg
)
The Right Hand Rule Right Hand Rule No. 1. Extend the right hand so the fingers point along the direction of the magnetic field and the thumb points along the velocity of the charge. The palm of the hand then faces in the direction of the magnetic force that acts on a positive charge. If the moving charge is negative, the direction of the force is opposite to that predicted by RHR-1.
Using the right hand rule, one may determine the direction of the field produced by a moving positive charge.
The effect of the sign of a moving charge on the magnetic force exerted on it.
The effect of the sign of a moving charge Positive and negative charges will feel opposite effects from a magnetic field.
Motion of a charge in an electric field
Motion of a charge in a magnetic field
Concepts at a glance Magnetic Force
Electrostatic Force
Electric Field
Test Charge
Moving Test Charge
Magnetic Field
When a particle is subject to both electric and magnetic fields, the total force on it is:
F = q(E +v×B)
It is convenient to symbolize the direction of B as: × × × ×
× × × ×
× × × ×
× × × ×
× × × ×
× × × ×
× × × ×
× × × ×
B is into the page
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
B is out of the page
velocity selector Magnetic fields can alter ionic movement A velocity selector is a device for measuring the velocity of a charged particle. It consists of a tube in which an electric field E is oriented to a magnetic field, and the field magnitudes are adjusted so that the electric and magnetic forces acting on the particle balance. How should an electric field be applied so that the force it applies to the particle can balance the magnetic force?
• You can create electrostatic lenses that will focus or alter the path or velocity of ions or electrons. This is the foundation of modern mass spectroscopy. • Refer to the Conceptual Analysis on page 666 and Example 20.3 on page 670.
20.3 Motion of a charged particle in a magnetic field
mv qvB = Fc = r mv r= qB
2
Introduction of helical motion
Such motion can be imparted to ions given velocities both parallel and perpendicular to the applied field.
Example The drawing shows a top view of four interconnected chambers. A negative charge is fired into chamber 1. By turning on separate magnetic fields in each chamber, the charge can be made to exit from chamber 4, as shown. (a) Describe how the magnetic field in each chamber should be directed. (b) If the speed of the charge is v when it enters chamber 1, what is the speed of the charge when it exits chamber 4? Why?
20.4 Mass Spectrometers
One type of mass Spectrometers
mv mv r= = qB eB magnitude of electron charge
KE=PE 1 2
mv = eV 2
er 2 2 B m = 2V
The mass spectrum of naturally occurring neon, showing three isotopes.
Conceptual Example Particle Tracks in a Bubble Chamber The figure shows the bubble-chamber tracks from an event that begins at point A. At this point a gamma ray travels in from the left, spontaneously transforms into two charged particles. The particles move away from point A, producing two spiral tracks. A third charged particle is knocked out of a hydrogen atom and moves forward, producing the long track. The magnetic field is directed out of the paper. Determine the sign of each particle and which particle is moving most rapidly.
Example: Motion of a proton – parallel plate capacitor. Conservation of energy:
1 1 2 mv B + U B = mv A2 + U A 2 2 2e(VA − VB ) ∴ vB = m
mvB ∴r = eB
The Work done on a charged particle moving through electric and magnetic fields An electric field applies a force to a positively charged particle.
Consequently the path of the particle bends in the direction of the force. Displacement is in the direction of the electric force. The force does work on the particle. According to the work-energy theorem, the K.E. and, hence, the speed increase.
In contrast, The magnetic force always acts perpendicular to the motion of the charge. Consequently, the displacement never has a component in the direction of the magnetic force.
The magnetic force cannot do work and hence cannot change the kinetic energy of a charged particle, although the force does alter the direction of the motion.
20.5 Magnetic Force on a Current-Carrying Conductor When a conductor, often a wire, carrying current is exposed to an external magnetic field, a force is exerted on the conductor.
F= I × B
The magnetic force on the moving charges pushes the wire to the right.
F = qvB sin θ
∆q F = ( v∆t )B sin θ t L ∆ I
F = ILB sin θ
Applications of force on a conductor Novel applications have been devised to make use of the force that a magnetic field exerts on a conductor carrying current.
Magnetic rail gun
Magnetic Bird Perch
Magnetic Levitation: The world’s lightest bird is the bee hummingbird (mass=1.6 g) (less than that of a penny). Design a perch for this bird i.e. find the current I for a given ℓ and B. What happens as soon as the bird leaves the perch?
20.6 Force and Torque on a Current Loop The two forces on the loop have equal magnitude but an application of RHR-1 shows that they are opposite in direction.
The loop tends to rotate such that its normal becomes aligned with the magnetic field.
F = IaB
F ' = IbB cos φ = τ (= IaB )( b sin φ ) IAB sin φ
For a solenoid with N turns : τ = NIAB sin φ
Net torque = τ = ILB( 12 w sin φ ) + ILB( 12 w sin φ ) = IAB sin φ
magnetic moment
τ = NIA B sin φ number of turns of wire
magnetic moment of the loop µ = IA
A current-carrying solenoid in a uniform magnetic field experiences a torque
Example The Torque Exerted on a Current-Carrying Coil A coil of wire has an area of 2.0x10-4m2, consists of 100 loops or turns, and contains a current of 0.045 A. The coil is placed in a uniform magnetic field of magnitude 0.15 T. (a) Determine the magnetic moment of the coil. (b) Find the maximum torque that the magnetic field can exert on the coil. magnetic moment
(a)
NIA = (100)(0.045 A ) 2.0 ×10 − 4 m 2 = 9.0 ×10 − 4 A ⋅ m 2 magnetic moment
(
)
2 −4 −4 (b) τ = NIA B sin φ = 9.0 × 10 A ⋅ m (0.15 T ) sin 90 = 1.4 × 10 N ⋅ m
(
)
The Direct-Current Motor
The basic components of a DC motor.
If the conductor is a loop, the torque can create an electric motor.
The Direct-Current Motor
20.7 Magnetic Fields Produced by Electric Currents RHR #2 (RHR for the magnetic field): Curl the fingers of the right hand into the shape of a half-circle. Point the thumb in the direction of the conventional current I, and the tips of the fingers will point in the direction of the magnetic field B.
A Long straight conductor:
µo I B= 2π r µo is the permeability of free space µo = 4 π×10−7 T.m/A
The magnetic field that surrounds a current-carrying wire can exert a force on a moving charge.
Magnetic Field of A Current Carrying Wire: Here you can check the right-hand rule. Written by Walter Fendt.
Placed over a compass, the wire would cause the compass needle to deflect. This was the classic demonstration done by Oersted as he demonstrated the effect.
Example A Current Exerts a Magnetic Force on a Moving Charge The long straight wire carries a current of 3.0 A. A particle has a charge of +6.5x10-6 C and is moving parallel to the wire at a distance of 0.050 m. The speed of the particle is 280 m/s. Determine the magnitude and direction of the magnetic force on the particle.
µo I sin θ F = qvB sin θ = qv 2π r µo I B= 2π r
20.8 Force between parallel conductors This was the classic demonstration done by Ampere as he demonstrated the effect
Current carrying wires can exert forces on each other
B -F I1 Wire 1
I2 F
Wire 2
Repulsion
µo I1 B= 2π r
B F I1 Wire 1
-F I2 Wire 2
Attraction F = I2 L B sinθ
Conceptual Example The Net Force That a Current-Carrying Wire Exerts on a Current Carrying Coil. Is the coil attracted to, or repelled by the wire?
20.9 Current Loops and solenoids A Loop of Wire
B=
µo I 2R
center of circular loop
µo I B=N 2R Center of a circular loop with N turns
Example Finding the Net Magnetic Field A long straight wire carries a current of 8.0 A and a circular loop of wire carries a current of 2.0 A and has a radius of 0.030 m. Find the magnitude and direction of the magnetic field at the center of the loop.
µ o I1 µ o I 2 µ o I1 I 2 B= − = − 2π r 2 R 2 πr R
( 4π ×10 B=
)
T ⋅ m A 8.0 A 2.0 A = 1.1×10 −5 T − 2 π (0.030 m ) 0.030 m
−7
The field lines around the bar magnet resemble those around the loop
RHR #2 shows also the north and south poles of the loop.
Magnetic Field of a Solenoid A solenoid is a long coil of wire in the shape of a helix. The field inside the solenoid and away from its ends is nearly constant in magnitude and directed parallel to the axis.
Centre of a long solenoid:
B = µ o nI
number of turns per unit length
(Optional) Toroidal Solenoid
µo NI B= 2π r
20.10 Magnetic Field Calculations
Magnetic field ∆B caused by a segment ∆ of a circular conducting loop.
µo I µo I µo I = ( ∆1 + ∆ 2= + ...) (2= B π R) 2 2 4π R 4π R 2R
Ampere’s Law Ampere’s Law for Static Magnetic Fields For any current geometry that produces a magnetic field that does not change in time
For any current geometry that produces a magnetic field that does not change in time,
∑ B// ∆ℓ = µo Iencl
where ∆ is a small segment of length along a closed path of arbitrary shape around the current, B is the component of the magnetic field parallel to ∆, I is the net current passing through the surface bounded by the path, and µo is the permeability of free space. The symbol Σ indicates that the sum of all B// ∆ℓ terms must be taken around the closed path.
Ampere’s law for an arbitrary closed curve of straight segments around a pair of conductors.
Example An Infinitely Long, Straight, Current-Carrying Wire Use Ampere’s law to obtain the magnetic field.
∑ B ∆ = µ I ||
o
B (∑ ∆ ) = µ o I B 2π r = µ o I
µo I B= 2π r
20.11 Magnetic Materials The intrinsic “spin” and orbital motion of electrons gives rise to the magnetic properties of materials. In ferromagnetic materials groups of neighboring atoms, forming magnetic domains, the spins of electrons are naturally aligned with each other.
Review Examples The triangular loop of wire shown in the drawing carries a current of I = 4.70 A. A uniform magnetic field is directed parallel to side AB of the triangle and has a magnitude of 1.80 T. (a) Find the magnitude and direction of the magnetic force exerted on each side of the triangle. (b) Determine the magnitude of the net force exerted on the triangle.
Solution The magnitude of the magnetic force exerted on a long straight wire is given by F = ILB sin θ. The direction of the magnetic force is predicted by RHR#1. The net force on the triangular loop is the vector sum of the forces on the three sides. a. The direction of the current in side AB is opposite to the direction of the magnetic field, so the angle θ between them is θ = 180°.
The magnitude of the magnetic force is F AB = ILB sin θ = ILB sin 180 ° =
0N
C
For the side BC, the angle is θ = 55.0°, and the length of the side is 2 .00 m L= = 3.49 m cos 55.0 ° The magnetic force is
b
gb
gb
g
FBC = ILB sin θ = 4 .70 A 3.49 m 1.80 T sin 55.0° = 24 .2 N
I B
An application of Right-Hand No. 1 shows that the magnetic force on side BC is directed perpendicularly out of the paper , toward the reader. For the side AC, the angle is θ = 90.0°. We see that the length of the side is L = (2.00 m) tan 55.0° = 2.86 m The magnetic force is
b
gb
gb
55.0o A
g
F AC = ILB sin θ = 4 .70 A 2 .86 m 1.80 T sin 90.0° = 24 .2 N
B 2.00 m
An application of Right-Hand No. 1 shows that the magnetic force on side AC is directed perpendicularly into the paper , away from the reader.
b. The net force is the vector sum of the forces on the three sides. Taking the positive direction as being out of the paper, the net force is
b
g
∑ F = 0 N + 24.2 N + −24.2 N = 0 N
Example The radius of a coil of wire with N turns is r = 0.22 m. A current Icoil = 2.0 A flows clockwise in the coil, as shown. A long, straight wire carrying a current Iwire = 31 A toward the left is located 0.05 m from the edge of the coil. The magnetic field at the center of the coil is zero Tesla. Determine N, the number of turns.
Iwire
r Solution
µ 0 I wire µ 0 I coil =N 2π r 2 Rcoil
I wire Rcoil (31)(0.22) = =4 N= π (0.05 + 0.22) I coil π (0.27)(2.0)
Icoil
Magnetic Field and Magnetic Forces
Goals
■ To observe and visualize magnetic fields and forces. ■ To study the motion of a charged particle in a magnetic field. ■ To evaluate the magnetic force on a current-carrying conductor. ■ To determine the force and torque produced with a magnet and current-carrying loop of wire (the DC motor). ■ To study the fields generated by long, straight conductors. ■ To observe the changes in the field with the conductor in loops (forming the solenoid). ■ To calculate the magnetic field at selected points in space. ■ To understand magnetism via magnetic moments.
Historical Notes - Lodestone (magnetite mineral):
The name "magnet" comes from lodestones found in a place called Magnesia. A piece of intensely magnetic magnetite that was used as an early form of magnetic compass. Only magnetite with a particular crystalline structure, lodestone, can act as a natural magnet and attract and magnetize iron. In China, the earliest literary reference to magnetism lies in a 4th century BC book called Book of the Devil Valley Master: "The lodestone makes iron come or it attracts it." The earliest mention of the attraction of a needle appears in a work composed between 20 and 100 AD. From amusement and magic-like to application − “leading stone” (lodestone) – compasses and navigation. By the 12th century the Chinese were known to use the lodestone compass for navigation. Today’s powerful lodestones − and magnetic materials are ubiquitous.
What if no lodestones existed? ► The Chinese would certainly not have invented the magnetic compass. ► Magnetism would have been discovered much later, and one wonders how. ► Lacking the compass, the great voyages of discovery could hardly have taken place--Columbus, De Gama, Magellan and the rest. ► The history of the world might have been quite different! Magnetic Jewellery • Magnetic energy is the strongest natural force in the universe and the power of magnets is one of the most basic powers of nature. • The use of magnet therapy for health and well-being has an ancient history dating back thousands of years. • Ancient Egyptians used loadstones to prolong life and improve health. It is said that Cleopatra wore a polished lodestone on her third eye, in the belief that it helped maintain her youth and beauty. • In more recent times, Paracelsus (1493-1541) considered to be the father of modern medicine, believed that the "life force' of the body was most influenced by the force found in magnets. • In Europe, Russia, China, Japan and many other countries, convinced of the benefits, millions of people continue to use magnet therapy. • Today, we are experiencing an exciting revival of this ancient therapy. Resulting from the impact of more and more clinical studies and anecdotal evidence, 120 million people worldwide spend over $1.5 billion globally on the therapeutic benefits of magnets.
Two experiments, different scale: Although the magnet on the left is an electromagnet (huge) and the one on the right is a permanent magnet (small), the idea is the same.
Observations: ► You get a hint of magnetic field whenever you attach a note to a refrigerator door with a small magnet. ► Accidentally erase a computer disk by bringing it near a magnet. ► Credit cards, VCRs,…etc ► A familiar type of magnet: “electromagnets”. A wire coil is wound around an iron core and a current is sent through the coil. The strength of the magnetic field is determined by the size of the current. ☼ In industry, such magnets are used for sorting scrap iron among many other things. ► You are probably more familiar with “permanent magnets” (like the refrigeratordoor type). They do not need current to have a magnetic field. ► Magnets: A magnet is an object made of certain materials which create a magnetic field.
Electricity-Magnetism-Electromagnetism: From these modest origins, the sciences of electricity and magnetism developed separately for centuries…until 1820 AD. In 1820, Oersted found a connection between them: an electric current in a wire can deflect a magnetic compass needle. The new science of “electromagnetism” was developed by Faraday, Henry, and Maxwell (who, in the mid-19th century put electromagnetism on a sound theoretical basis)…
20.1 Magnetism The behavior of bar magnets ■ Notice the general behavior trends of attraction and repulsion, dipole or monopole.
The simplest magnetic structure that can exist is a magnetic dipole. Magnetic monopoles do not exist (as far as we know) ∴We cannot define the magnetic field as we do for the electric field (E=Fe/q.)
Magnetic properties of materials can be traced back to their atoms and electrons.
The clustering of the lines (of e.g. iron filings) at the ends of a magnet suggests that one end is a “source” of the lines (the field diverges from it) and the other end is a “sink” of the lines (the field converges toward it).
By convention: north pole and south pole, respectively. The magnet with its two poles is an example of a “magnetic dipole”. Note: A magnetized bar is magnetically weak in the middle Puzzle Determine which of the two bars is magnetized (without using any other material.)
A compass will align with fields
The compass will align with
whatever average field is strongest. As shown in figure, the field caused by the current in the wire is stronger than that any background field from the earth.
Absent
the current-carrying wire, the compass would align with the earth’s magnetic field. This allows a consistent direction to be determined by someone with the need for navigation.
Iron filings will align as a compass does
Each small filing lines up tangent to the field lines allowing a visual demonstration
The magnetic field lines and pattern of iron filings in the vicinity of a bar magnet and the magnetic field lines in the gap of a horseshoe magnet.
“Earth is a huge magnet”
● It can be represented by a huge bar magnet (a magnetic dipole.) (William Gilbert 1540-1603)
● The dipole axis makes an angle about 11.5o with the rotation axis. ● It intersects Earth’s surface at the “geomagnetic north pole” and the “geomagnetic south pole”. ● The “north magnetic pole” is really the south pole of Earth’s magnetic dipole.
■ Our Earth itself has a magnetic field This field is not very strong but it is consistent.
● The “field declination” is the angle (left or right) between geographic north (which is toward 90o latitude) and the horizontal component of the field. The “field inclination” is the angle (up or down) between a horizontal plane and the field’s direction. ● We can use a “compass” and a “dip meter” to determine these two angles. ● The field observed at any location on the surface of Earth varies with time. In fact, Earth’s field has reversed its polarity about every million years.
Examples of Magnetic Fields Fields are created in a variety of ways and observed in a variety of places.
20.2 Magnetic Field and Magnetic Force Magnetic Forces External Forces Gravitational Force Normal Force Frictional Forces Tension Force Restoring Force of a Spring Collisional Forces Electrostatic Force Magnetic Force
Newton’s Second Law
ΣF = m a
The magnetic force, like the other forces we have encountered, may contribute to the net force that acts on an object
Review: When a charge is placed in an electric field, it experiences a force, according to
F = qE
Definition of the magnetic field The “magnetic field B” at a point is along the tangent to a field line. Its direction is that of the force on the north pole of a bar magnet, or the direction in which a compass needle points. The strength of the field is proportional to the number of lines passing through a unit area normal to the field . Therefore , B is also called the “magnetic flux density” .
There are two ways to set up a magnetic field: (1) Moving electrically charged particles. (2) Elementary particles such as electrons have an intrinsic magnetic field around them; that is these fields are a basic characteristic of the particle, just as are their mass and electric charge (or lack of charge).
The magnetic fields of the electrons in certain materials add together to give a net magnetic field around the material. This is true for the material in permanent magnets (which is good, because they can then hold notes to a refrigerator door). In other materials, the magnetic fields of all the electrons cancel out, giving no net magnetic field surrounding the material. This is true for the material in your body (which is also good, because otherwise you might be slammed up against a refrigerator door every time you passed by).
The force that a magnetic field exerts on a moving charge Experimentally, we find that when a charged particle (either alone or part of a current) moves through a magnetic field, a force due to the field can act on the particle.
The following conditions must be met for a charge to experience a magnetic force when placed in a magnetic field: 1. The charge must be moving. 2. The velocity of the charge must have a component that is perpendicular to the direction of the magnetic field.
The effect of an existing magnetic field on a charge depends on the charges direction of motion relative to the field. The magnetic force F on a particle with charge q and velocity v in a magnetic field B is perpendicular to both v and B and given by :
F = qv×B
(0 ≤ φ ≤ 180o )
Its magnitude is F = q vB sin φ , where φ is the angle between v and B .
The SI unit of the magnetic field is the tesla ( T ) • Note that 1 T ≡ N.s/(C.m), and since C/s = 1 A, 1 T ≡ 1 N/(A.m). • Since the tesla is a large unit, a (cgs) unit called the gauss ( G ) is often used where 1 T = 104 G Some approximate magnetic fields At the surface of a neutron star 108 T , At Earth’s surface 10−8 T , In interstellar space 10−10 T
Example 1 Magnetic Forces on Charged Particles A proton in a particle accelerator has a speed of 5.0x106 m/s. The proton encounters a magnetic field whose magnitude is 0.40 T and whose direction makes and angle of 30.0 degrees with respect to the proton’s velocity (see part (c) of the figure). Find (a) the magnitude and direction of the force on the proton and (b) the acceleration of the proton. (c) What would be the force and acceleration of the particle were an electron?
(a)
(
)(
)
(
F = qo vB sin θ = 1.60 ×10 −19 C 5.0 ×106 m s (0.40T )sin 30.0 = 1.6 ×10 −13 N
(b)
1.6 ×10 −13 N F 13 2 = = 9 . 6 × 10 m s a= mp 1.67 ×10 − 27 kg
(c)
Magnitude is the same, but direction is opposite.
1.6 ×10 −13 N F 17 2 = = 1 . 8 × 10 m s a= me 9.11×10 −31 kg
)
The Right Hand Rule Right Hand Rule No. 1. Extend the right hand so the fingers point along the direction of the magnetic field and the thumb points along the velocity of the charge. The palm of the hand then faces in the direction of the magnetic force that acts on a positive charge. If the moving charge is negative, the direction of the force is opposite to that predicted by RHR-1.
Using the right hand rule, one may determine the direction of the field produced by a moving positive charge.
The effect of the sign of a moving charge on the magnetic force exerted on it.
The effect of the sign of a moving charge Positive and negative charges will feel opposite effects from a magnetic field.
Motion of a charge in an electric field
Motion of a charge in a magnetic field
Concepts at a glance Magnetic Force
Electrostatic Force
Electric Field
Test Charge
Moving Test Charge
Magnetic Field
When a particle is subject to both electric and magnetic fields, the total force on it is:
F = q(E +v×B)
It is convenient to symbolize the direction of B as: × × × ×
× × × ×
× × × ×
× × × ×
× × × ×
× × × ×
× × × ×
× × × ×
B is into the page
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
B is out of the page
velocity selector Magnetic fields can alter ionic movement A velocity selector is a device for measuring the velocity of a charged particle. It consists of a tube in which an electric field E is oriented to a magnetic field, and the field magnitudes are adjusted so that the electric and magnetic forces acting on the particle balance. How should an electric field be applied so that the force it applies to the particle can balance the magnetic force?
• You can create electrostatic lenses that will focus or alter the path or velocity of ions or electrons. This is the foundation of modern mass spectroscopy. • Refer to the Conceptual Analysis on page 666 and Example 20.3 on page 670.
20.3 Motion of a charged particle in a magnetic field
mv qvB = Fc = r mv r= qB
2
Introduction of helical motion
Such motion can be imparted to ions given velocities both parallel and perpendicular to the applied field.
Example The drawing shows a top view of four interconnected chambers. A negative charge is fired into chamber 1. By turning on separate magnetic fields in each chamber, the charge can be made to exit from chamber 4, as shown. (a) Describe how the magnetic field in each chamber should be directed. (b) If the speed of the charge is v when it enters chamber 1, what is the speed of the charge when it exits chamber 4? Why?
20.4 Mass Spectrometers
One type of mass Spectrometers
mv mv r= = qB eB magnitude of electron charge
KE=PE 1 2
mv = eV 2
er 2 2 B m = 2V
The mass spectrum of naturally occurring neon, showing three isotopes.
Conceptual Example Particle Tracks in a Bubble Chamber The figure shows the bubble-chamber tracks from an event that begins at point A. At this point a gamma ray travels in from the left, spontaneously transforms into two charged particles. The particles move away from point A, producing two spiral tracks. A third charged particle is knocked out of a hydrogen atom and moves forward, producing the long track. The magnetic field is directed out of the paper. Determine the sign of each particle and which particle is moving most rapidly.
Example: Motion of a proton – parallel plate capacitor. Conservation of energy:
1 1 2 mv B + U B = mv A2 + U A 2 2 2e(VA − VB ) ∴ vB = m
mvB ∴r = eB
The Work done on a charged particle moving through electric and magnetic fields An electric field applies a force to a positively charged particle.
Consequently the path of the particle bends in the direction of the force. Displacement is in the direction of the electric force. The force does work on the particle. According to the work-energy theorem, the K.E. and, hence, the speed increase.
In contrast, The magnetic force always acts perpendicular to the motion of the charge. Consequently, the displacement never has a component in the direction of the magnetic force.
The magnetic force cannot do work and hence cannot change the kinetic energy of a charged particle, although the force does alter the direction of the motion.
20.5 Magnetic Force on a Current-Carrying Conductor When a conductor, often a wire, carrying current is exposed to an external magnetic field, a force is exerted on the conductor.
F= I × B
The magnetic force on the moving charges pushes the wire to the right.
F = qvB sin θ
∆q F = ( v∆t )B sin θ t L ∆ I
F = ILB sin θ
Applications of force on a conductor Novel applications have been devised to make use of the force that a magnetic field exerts on a conductor carrying current.
Magnetic rail gun
Magnetic Bird Perch
Magnetic Levitation: The world’s lightest bird is the bee hummingbird (mass=1.6 g) (less than that of a penny). Design a perch for this bird i.e. find the current I for a given ℓ and B. What happens as soon as the bird leaves the perch?
20.6 Force and Torque on a Current Loop The two forces on the loop have equal magnitude but an application of RHR-1 shows that they are opposite in direction.
The loop tends to rotate such that its normal becomes aligned with the magnetic field.
F = IaB
F ' = IbB cos φ = τ (= IaB )( b sin φ ) IAB sin φ
For a solenoid with N turns : τ = NIAB sin φ
Net torque = τ = ILB( 12 w sin φ ) + ILB( 12 w sin φ ) = IAB sin φ
magnetic moment
τ = NIA B sin φ number of turns of wire
magnetic moment of the loop µ = IA
A current-carrying solenoid in a uniform magnetic field experiences a torque
Example The Torque Exerted on a Current-Carrying Coil A coil of wire has an area of 2.0x10-4m2, consists of 100 loops or turns, and contains a current of 0.045 A. The coil is placed in a uniform magnetic field of magnitude 0.15 T. (a) Determine the magnetic moment of the coil. (b) Find the maximum torque that the magnetic field can exert on the coil. magnetic moment
(a)
NIA = (100)(0.045 A ) 2.0 ×10 − 4 m 2 = 9.0 ×10 − 4 A ⋅ m 2 magnetic moment
(
)
2 −4 −4 (b) τ = NIA B sin φ = 9.0 × 10 A ⋅ m (0.15 T ) sin 90 = 1.4 × 10 N ⋅ m
(
)
The Direct-Current Motor
The basic components of a DC motor.
If the conductor is a loop, the torque can create an electric motor.
The Direct-Current Motor
20.7 Magnetic Fields Produced by Electric Currents RHR #2 (RHR for the magnetic field): Curl the fingers of the right hand into the shape of a half-circle. Point the thumb in the direction of the conventional current I, and the tips of the fingers will point in the direction of the magnetic field B.
A Long straight conductor:
µo I B= 2π r µo is the permeability of free space µo = 4 π×10−7 T.m/A
The magnetic field that surrounds a current-carrying wire can exert a force on a moving charge.
Magnetic Field of A Current Carrying Wire: Here you can check the right-hand rule. Written by Walter Fendt.
Placed over a compass, the wire would cause the compass needle to deflect. This was the classic demonstration done by Oersted as he demonstrated the effect.
Example A Current Exerts a Magnetic Force on a Moving Charge The long straight wire carries a current of 3.0 A. A particle has a charge of +6.5x10-6 C and is moving parallel to the wire at a distance of 0.050 m. The speed of the particle is 280 m/s. Determine the magnitude and direction of the magnetic force on the particle.
µo I sin θ F = qvB sin θ = qv 2π r µo I B= 2π r
20.8 Force between parallel conductors This was the classic demonstration done by Ampere as he demonstrated the effect
Current carrying wires can exert forces on each other
B -F I1 Wire 1
I2 F
Wire 2
Repulsion
µo I1 B= 2π r
B F I1 Wire 1
-F I2 Wire 2
Attraction F = I2 L B sinθ
Conceptual Example The Net Force That a Current-Carrying Wire Exerts on a Current Carrying Coil. Is the coil attracted to, or repelled by the wire?
20.9 Current Loops and solenoids A Loop of Wire
B=
µo I 2R
center of circular loop
µo I B=N 2R Center of a circular loop with N turns
Example Finding the Net Magnetic Field A long straight wire carries a current of 8.0 A and a circular loop of wire carries a current of 2.0 A and has a radius of 0.030 m. Find the magnitude and direction of the magnetic field at the center of the loop.
µ o I1 µ o I 2 µ o I1 I 2 B= − = − 2π r 2 R 2 πr R
( 4π ×10 B=
)
T ⋅ m A 8.0 A 2.0 A = 1.1×10 −5 T − 2 π (0.030 m ) 0.030 m
−7
The field lines around the bar magnet resemble those around the loop
RHR #2 shows also the north and south poles of the loop.
Magnetic Field of a Solenoid A solenoid is a long coil of wire in the shape of a helix. The field inside the solenoid and away from its ends is nearly constant in magnitude and directed parallel to the axis.
Centre of a long solenoid:
B = µ o nI
number of turns per unit length
(Optional) Toroidal Solenoid
µo NI B= 2π r
20.10 Magnetic Field Calculations
Magnetic field ∆B caused by a segment ∆ of a circular conducting loop.
µo I µo I µo I = ( ∆1 + ∆ 2= + ...) (2= B π R) 2 2 4π R 4π R 2R
Ampere’s Law Ampere’s Law for Static Magnetic Fields For any current geometry that produces a magnetic field that does not change in time
For any current geometry that produces a magnetic field that does not change in time,
∑ B// ∆ℓ = µo Iencl
where ∆ is a small segment of length along a closed path of arbitrary shape around the current, B is the component of the magnetic field parallel to ∆, I is the net current passing through the surface bounded by the path, and µo is the permeability of free space. The symbol Σ indicates that the sum of all B// ∆ℓ terms must be taken around the closed path.
Ampere’s law for an arbitrary closed curve of straight segments around a pair of conductors.
Example An Infinitely Long, Straight, Current-Carrying Wire Use Ampere’s law to obtain the magnetic field.
∑ B ∆ = µ I ||
o
B (∑ ∆ ) = µ o I B 2π r = µ o I
µo I B= 2π r
20.11 Magnetic Materials The intrinsic “spin” and orbital motion of electrons gives rise to the magnetic properties of materials. In ferromagnetic materials groups of neighboring atoms, forming magnetic domains, the spins of electrons are naturally aligned with each other.
Review Examples The triangular loop of wire shown in the drawing carries a current of I = 4.70 A. A uniform magnetic field is directed parallel to side AB of the triangle and has a magnitude of 1.80 T. (a) Find the magnitude and direction of the magnetic force exerted on each side of the triangle. (b) Determine the magnitude of the net force exerted on the triangle.
Solution The magnitude of the magnetic force exerted on a long straight wire is given by F = ILB sin θ. The direction of the magnetic force is predicted by RHR#1. The net force on the triangular loop is the vector sum of the forces on the three sides. a. The direction of the current in side AB is opposite to the direction of the magnetic field, so the angle θ between them is θ = 180°.
The magnitude of the magnetic force is F AB = ILB sin θ = ILB sin 180 ° =
0N
C
For the side BC, the angle is θ = 55.0°, and the length of the side is 2 .00 m L= = 3.49 m cos 55.0 ° The magnetic force is
b
gb
gb
g
FBC = ILB sin θ = 4 .70 A 3.49 m 1.80 T sin 55.0° = 24 .2 N
I B
An application of Right-Hand No. 1 shows that the magnetic force on side BC is directed perpendicularly out of the paper , toward the reader. For the side AC, the angle is θ = 90.0°. We see that the length of the side is L = (2.00 m) tan 55.0° = 2.86 m The magnetic force is
b
gb
gb
55.0o A
g
F AC = ILB sin θ = 4 .70 A 2 .86 m 1.80 T sin 90.0° = 24 .2 N
B 2.00 m
An application of Right-Hand No. 1 shows that the magnetic force on side AC is directed perpendicularly into the paper , away from the reader.
b. The net force is the vector sum of the forces on the three sides. Taking the positive direction as being out of the paper, the net force is
b
g
∑ F = 0 N + 24.2 N + −24.2 N = 0 N
Example The radius of a coil of wire with N turns is r = 0.22 m. A current Icoil = 2.0 A flows clockwise in the coil, as shown. A long, straight wire carrying a current Iwire = 31 A toward the left is located 0.05 m from the edge of the coil. The magnetic field at the center of the coil is zero Tesla. Determine N, the number of turns.
Iwire
r Solution
µ 0 I wire µ 0 I coil =N 2π r 2 Rcoil
I wire Rcoil (31)(0.22) = =4 N= π (0.05 + 0.22) I coil π (0.27)(2.0)
Icoil
