Magnetic Dipoles. Recall that an electric dipole consists of two equal but opposite charges separated by some distance, such as in

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1 MAGNETISM History of Magnetism Bar Magnets Magnetic Dipoles Magnetic Fields Magnetic Forces on Moving Charges and Wires Electric Motors Current Loops and Electromagnets Solenoids Sources of Magnetism Spin & Orbital Dipole Moments Permanent Magnets Earth s Magnetic Field Magnetic Flux Induced Emf and Current Generators Crossed Fields

2 History of Magnetism The first known magnets were naturally occurring lodestones, a type of iron ore called magnetite (Fe 3 O 4 ). People of ancient Greece and China discovered that a lodestone would always align itself in a longitudinal direction if it was allowed to rotate freely. This property of lodestones allowed for the creation of compasses two thousand years ago, which was the first known use of the magnet. In 1263 Pierre de Maricourt mapped the magnetic field of a lodestone with a compass. He discovered that a magnet had two magnetic poles North and South poles. In the 1600's William Gilbert, physician of Queen Elizabeth I, concluded that Earth itself is a giant magnet. In 1820 the Danish physicist Hans Christian Ørsted discovered an electric current flowing through a wire can cause a compass needle to deflect, showing that magnetism and electricity were related.

3 History (cont.) In 1830 Michael Faraday (British) and Joseph Henry (American) independently discovered that a changing magnetic field produced a current in a coil of wire. Faraday, who was perhaps the greatest experimentalist of all time, came up with the idea of electric and magnetic fields. He also invented the dynamo (a generator), made major contributions to chemistry, and invented one of the first electric motors In the 19th century James Clerk Maxwell, a Scottish physicist and one of the great theoreticians of all times, mathematically unified the electric and magnetic forces. He also proposed that light was electromagnetic radiation. In the late 19 th century Pierre Curie discovered that magnets loose their magnetism above a certain temperature that later became known as the Curie point. In the 1900's scientists discover superconductivity. Superconductors are materials that have a zero resistance to a current flowing through them when they are a very low temperature. They also exclude magnetic field lines (the Meissner effect) which makes magnetic levitation possible.

4 Magnetic Dipoles Recall that an electric dipole consists of two equal but opposite charges separated by some distance, such as in _ a polar molecule. Every magnet is a magnetic dipole. A bar magnet is a simple example. Note how the E field due an electric dipole is just like the magnetic field (B field) of a bar magnet. Field lines emanate from the + or N pole and reenter the - or S pole. Although they look the same, they are different kinds of fields. E fields affect any charge in the vicinity, but a B field only affects moving charges. As with charges, opposite poles attract and like poles repel. - + Electric dipole and E field S N Magnetic dipole and B field

5 Magnetic Monopole Don t Exist We have studied electric fields to due isolated + or - charges, but as far as we know, magnetic monopole do not exist, meaning it is impossible to isolate a N or S pole. The bar magnet on the left is surrounded by iron filings, which orient themselves according to the magnetic field they are in. When we try to separate the two poles by breaking the magnet, we only succeed in producing two distinct dipoles (pic on right). Bar magnet demo

6 Magnetic Fields You have seen that electric fields and be uniform, nonuniform and symmetric, or nonuniform and asymmetric. The same is true for magnetic fields. (Later we ll see how to produce uniform magnetic fields with a current flowing through a coil called a solenoid.) Regardless of symmetry or complexity, the SI unit for any E field is the N/C, since by definition an electric field is force per unit charge. Because there are no magnetic monopoles, there is no analogous definition for B. However, regardless of symmetry or complexity, there is only one SI unit for a B field. It is called a tesla and its symbol is T. The coming slides will show how to write a tesla in terms of other SI units. The magnetic field vector is always tangent to the magnetic field. Unlike E fields, all magnetic field lines that come from the N pole must land on the S pole--no field lines go to or come from infinity.

7 Force Due to Magnetic Field The force exerted on a charged particle by a magnetic field is given by the vector cross product: F = force (vector) q = charge on the particle (scalar) v = velocity of the particle relative to field (vector) B = magnetic field (vector) F = q v B Recall that the magnitude of a cross is the product of the magnitudes of the vectors times the sine of the angle between them. So, the magnitude of the magnetic force is given by F = q v B sin where is angle between q v and B vectors.

8 Cross Product Review Let v 1 = x 1, y 1, z 1 and v 2 = x 2, y 2, z 2. By definition, the cross product of these vectors (pronounced v 1 cross v 2 ) is given by the following determinant. i j k v 1 v 2 = x 1 y 1 z 1 x 2 y 2 z 2 = (y 1 z 2 - y 2 z 1 ) i - (x 1 z 2 - x 2 z 1 ) j + (x 1 y 2 - x 2 y 1 ) k Note that the cross product of two vectors is a vector itself that is to each of the original vectors. i, j, and k are the unit vectors pointing, along the positive x, y, and z axes, respectively. (See the vector presentation for a review of determinants.)

9 Right Hand Rule Review A quick way to determine the direction of a cross product is to use the right hand rule. To find a b, place the knife edge of your right hand (pinky side) along a and curl your hand toward b, making a fist. Your thumb then points in the direction of a b. It can be proven that the magnitude of is given by: a b a b = a b sin where is the angle between a and b. a b a b

10 1 N = 1 C (m/ s) (T) Magnetic Field Units F = q v B sin From the formula for magnetic force we can find a relationship between the tesla and other SI units. The sine of an angle has no units, so 1 T = 1 N C (m / s) 1 N A m A magnetic field of one tesla is very powerful magnetic field. Sometimes it may be convenient to use the gauss, which is equal to 1/10,000 of a tesla. Earth s magnetic field, at the surface, varies but has the strength of about one gauss. =

11 Direction of Magnetic Field & Force Near the poles, where the field lines are close together, the field is very strong (so the field vector are drawn longer). Anywhere in the field the mag. field vector is always tangent to the mag. field line there. The + charge in the pic in moving into the page. Since q is +, the q v vector is also into the page. The - charge is moving to the right, so the q v vector is to the left. The mag. force vector is always to plane formed by the q v vector and the B vector. The force on the - charge is into the page. If a charge is motionless relative to the field, there is no magnetic force on it, but if either a magnet is moving or a charge is moving, there could a force on the charge. If a charge moves parallel to a magnetic field, there is no magnetic force on it, since sin 0 = 0. B F + B - v

12 Magnetic Field & Force Practice Find the direction of the magnetic force or velocity: 1. A + charge at P is moving out of the page. 2. A - charge at Q is moving out of the page. 3. A - charge at Q is moving to the right. 4. A + charge at Q is moving up. 5. A - charge at R is moving up and to the left. 6. A + charge at R is moving down and to the right. 7. A - charge at R feels a force into the page. 8. A + charge at P feels a force out of the page. 9. A - charge at Q feels an upward force. Q P R

13 Magnetic Force Sample Problem This magnet is similar to a parallel plate capacitor in that there is a strong uniform field between its poles with some fringing on the sides. Suppose the magnetic field strength inside is 0.07 T and a 4.3 mc charge is moving through the field at right angle to the field lines. How strong and which way is the magnetic force on the charge? Answer: S F = q v B F = q v B since sin 90 = m/s N N So, F = N directed out of the page.

14 Motion of a Charge in a Magnetic Field The s represent field lines pointing into the page. A positively charged particle of mass m and charge q is shot to the right with speed v. By the right hand rule the magnetic force on it is up. Since v is to B, F = F B = q v B. Because F is to v, it has no tangential component; it is entirely centripetal. Thus F causes a centripetal acceleration. As the particle turns so do v and F, and if B is uniform the particle moves in a circle. This is the basic idea behind a particle accelerator like Fermilab. Since F is a centripetal force, F = F C = m v 2 / R. Let s see how speed, mass, charge, field strength, and radius of curvature are related: R F B + q, m v F B = F C q v B = m v 2 / R m v R = q B

15 Continued Magnetic Force on a Current Carrying Wire A section of wire carrying current to the right is shown in a uniform magnetic field. We can imagine positive charges moving to right, each feeling a magnetic force out of the page. This will cause the wire to bow outwards. Shown on the right is the view as seen when looking at the N pole from above. The dots represent a uniform mag. field coming out of the page. The mag. force on the wire is proportional to the field strength, the current, and the length of the wire. S N I I B

16 Magnetic Force on a Wire (cont.) Current is the flow of positive charge. As a certain amount of charge, q, moves with speed v through a wire of length L, the force of this quantity of charge is: F = q v B Over the time period t required for the charge to traverse the length of the wire, we have: F = (q / t ) v t B Since q / t = I and v t = L, we can write: F = I L B I B where L is a vector of magnitude L pointing in the direction of I.

17 Electric Motor I I F } d I I B Current along with a magnetic field can produce torque. This is the basic idea behind an electric motor. Above is a wire loop (purple) carrying a current provided by some power source like a battery. The current loop is submerged in an external field. From F = I L B, the force vectors in black are perpendicular to their wire segments. The net force on the loop is zero, but the net torque about the center is nonzero. The forces on the left and right wires produce no torque since the moment arm is zero for each (they point right at the center). However, the force F on the top wire (in the background) has a moment arm d, so it produces a torque F d. The bottom wire (in the foreground) produces the same torque. These torques work together to rotate the loop, converting electrical energy into mechanical energy. Continued

18 Electric Motor (cont.) As the loop turns it eventually reaches a vertical position (the plane of the loop parallel to the field). This is when the moment arms of the forces on the top and bottom wires are the longest, so this is where the torque is at a max. 90 later the loop will be perpendicular to the field. Here all moment arms and all torques are zero. This is the equilibrium point. The angular momentum of the loop, however, will allow it to swing right through this position. Now is when the current must change direction, otherwise the torques will attempt to bring the loop back to the equilibrium. This would amount to simple harmonic motion of the loop, which is not particularly useful. If the current changes direction every time the loop reach equilibrium, the loop will spin around in the same direction indefinitely. Although a battery only pumps current in one direction, the change in direction of current can be accomplished with help of a commutator, as can be seen with these animations: Animation 1 Animation 2

19 Electromagnets: Straight Wire Permanent magnets aren t the only things that produce magnetic fields. Moving charges themselves produce magnetic fields. We just saw that a current carrying wire feels a force when inside an external magnetic field. It also produces its own magnetic field. A long straight wire produces circular field lines centered on the wire. To find the direction of the field, we use another right hand rule: point your thumb in the direction of the current; the way your fingers of your right hand wrap is the direction of the magnetic field. B diminishes with distance from the wire. The pics at the right show cross sections of a current carrying wire. I I out of page, B counterclockwise I into page, B clockwise B

20 Straight Wire Practice Draw some magnetic field lines (loops in this case) along the wire. I Using x s and dots to represent vectors out of and into the page, show the magnetic field for the same wire. Note B diminishes with distance from the wire. B out of page B into page I

21 Current Loops and Magnetic Fields The magnetic field inside a current loop tends to be strong; outside, it tends to be weak. Here s why: Using the right hand rule we see that each length of wire contributes to a B field into the page (all lengths reinforcing one another). Outside the loop, say at P, the field is weak since the left side of the wire produces a field out of the page, but the right side produces a field into the page. Explain why the field is weak above the top wire. The situation is the same with a circular loop. The effect is magnified with multiple turns of wire. Yet another right hand rule helps with current loops: Wrap your right hand in the direction of the loop and your thumb points in the direction of B inside. This is reminiscent of angular momentum for a spinning body. I I P weak field outside I strong field inside loop, directed into page I strong field into page I weak field

22 Current Loops and Bar Magnets Notice how similar the magnetic field of a current loop is to that of a simple bar magnet. Wrap your right hand along the loop in the direction of the current and your thumb points in the direction of the north pole of your electromagnet. Note also how the field lines are very close together inside the loop, just as they are when they thread through a bar magnet. I

23 Solenoids Solenoids are one of the most common electromagnets. Solenoids consist of a tightly wrapped coil of wire, sometimes around an iron core. The multiple loops and the iron magnify the effect of the single loop electromagnet. A solenoid behaves as just like a simple bar magnet but only when current is flowing. The greater the current and the more turns per unit length, the greater the field inside. An ideal solenoid has a perfectly uniform magnetic field inside and zero field outside.

24 How Solenoids Work The cross section of a solenoid is shown. At point P inside the solenoid, the B field is a vector sum of the fields due to each section of wire. Note from the table that each section of wire produce a field vector with a component to the right, resulting in a strong field inside. In the ideal case the magnetic field would be uniform inside and zero outside. Wire Sections Direction of B 1-3 Up & Right 4 Right 5-8 Down & Right 9-11 Down & Right 12 Right Up & Right B = B P x x x x x x x x I out of the page I into the page

25 Solenoids and Bar Magnets A solenoid produces a magnetic field just like a simple bar magnet. Since it consists of many current loops, the resemblance to a bar magnet s field is much better than that of a single current loop.

26 Sources of Magnetism We have seen charges in motion (as in a current) produce magnetic fields. This is one source of magnetism. Another source is the electron itself. Electrons behave as if they were tiny magnets. Quantum mechanics is required to explain fully the magnetic properties of electrons, but it is helpful to relate these properties back to the motion of charges. Every electron in an atom behaves as a magnet in two ways, each having two magnetic dipole moments: Spin magnetic dipole moment - due to the rotation of an electron. Orbital magnetic dipole moment - due to the revolution of an electron about the nucleus. Note: Electrons are not actually little balls that rotate and revolve like planets, but imagining them this way is useful when explaining magnetism without quantum mechanics.

27 Spin Magnetic Dipole Moment Just as electrons have the intrinsic properties of mass and charge, they have an intrinsic property called spin. This means that electrons, by their very nature, possess these three attributes. You re already comfortable with the notions of charge and mass. To understand spin it will be helpful to think of an electron as a rotating sphere or planet. However, this is no more than a helpful visual tool. Imagine an electron as a soccer ball smeared with negative charge rotating about an axis. By the right hand rule, the angular momentum of the ball due to its rotation points down. But since its charge is negative, the spinning ball is like a little current loop flowing in the direction opposite its rotation, and the ball becomes an electromagnet with the N pole up. For an electron we would say its spin magnetic dipole moment vector, μ s, points up. Because of its spin, an electron is like a little bar magnet μ s I N S

28 Orbital Magnetic Dipole Moment Imagine now a planet that not only rotates but also revolves around its star. If the planet had a net charge, its rotation would give it a spin magnetic dipole moment, and its revolution would give it an orbital magnetic dipole moment. Charge in motion once again produces a magnetic field. Since an electron s charge is negative, its orbit is like a current loop in the opposite direction. By the right hand rule, the angular momentum vector in the pic below would point down and the orbital magnetic dipole moment, μ orb, points up. An orbiting electron behaves like a tiny electromagnet with its N pole in the direction of μ orb. Remember, though, that in reality electrons are not like little planets. In quantum mechanics, instead of circular orbits we speak of electrons behaving like waves and we can only talk of their positions in terms of probabilities. μ orb N - I S

29 Materials and Magnetism Each electron in an atom has two magnetic dipole moments associated with it, one for spin, and one for orbit. Each is a vector. These two dipole moments combine vectorially for each electron. The resultant vectors from each electron then combine for the whole atom, often canceling each other out. For most materials the net dipole moment for each atom is about zero. For some materials each atom has a nonzero dipole moment, but because the atoms have all different orientations, the material as a whole remains nonmagnetic. Ferromagnetic materials, like iron, are comprised of atoms that each have net dipole moment. Furthermore, all the atoms have the same alignment, at least within very tiny regions called domains. The domains can have different orientations, though, leaving the iron nonmagnetic except when placed in an external field. Permanent magnets are produced when the domains in a ferromagnetic material are aligned.

30 Each atom in a ferromagnetic material like iron is like a little magnet, and these magnets are all aligned in tiny regions called domains. At high temps these domains can align in the presence of an external field (like Earth s) leaving a permanent magnet. This happens at the Mid-Atlantic Ridge beneath the Atlantic Ocean. Permanent Magnets Lets melt the iron, and bring in a magnetic field. Now, when we let the solid cool down, and take away the external magnetic field, we have formed a permanent magnet in the same direction as external field. Temp Melting point Domains Bar Magnet

31 Earth s Magnetic Field N M 11.5 N G μ orb Earth s field looks similar to what we d expect if there were a giant bar magnet imbedded inside it, but the dipole axis of this magnet is offset from the axis of rotation by Also, the south pole of this magnet is near the geographic north pole, N G. A compass points in the direction of the magnetic north pole, N M, around which the field lines reenter Earth s surface. (Magnetic north is actually the south pole of Earth s magnetic dipole.) N M, which is currently located in Greenland, drifts about over the centuries. About every million years Earth s field reverses entirely, as we know from the orientations of magnetic fields near the Mid-Atlantic Ridge. The field is likely due to the motion of charged particles in the fluid outer core, and it protects us from an otherwise deadly solar wind.

32 Magnetic Fields: Overview Although the magnetic properties of electrons must ultimately be explained with quantum mechanics, we can think of magnetism arising whenever we have charge in motion. This motion can be that of an electron (either spinning or orbiting) or it can be in the form of a current. Remember: moving charges produce magnetic fields, and external magnetic fields exert a magnetic force on moving charges (at least if the charge has a component of its velocity perpendicular to the field).

33 Magnetic Flux Magnetic flux, informally speaking, is a measure of the amount of magnetic field lines going through an area. If the field is uniform, flux is given by: Ф B = B A = B A cos The area vector in the dot product is a vector that points perpendicular to the surface and has a magnitude equal to the area of the surface. Imagine you re trying to orient a window so as to allow the maximum amount of light to pass through it. To do this you would, of course, align A with the light rays. With = 0, cos = 1, and the number of light rays passing through the window (the flux) is a max. Note: with the window oriented parallel to the rays, = 90 and Ф B = 0 (no light enters the window). The SI unit for magnetic flux is the tesla-square meter: T m 2. This is also know as a weber (Wb). A

34 Changing Magnetic Flux A changing magnetic flux in a wire loop induces an electric current. The induced current is always in a direction that opposes the change in flux. These facts were discovered by Michael Faraday and represent a key connection between electricity and magnetism. One simple example of this is a magnet moving in and out of a wire loop. As a bar magnet approaches a wire loop along a line perpendicular to the loop, more and more field lines poke through the loop and the flux increases. To oppose this change in flux a current is induced in the direction shown. Note that the induced current produces its own magnetic field pointing to the right. Also note that there is no battery in the loop! This current N S will only exist when the flux inside the loop changes. When the magnet is withdrawn the flux v decreases and current is induced in the other direction. There is no Java script current when the magnet is still. I

35 Induced emf s and Currents The current induced in a loop come not from a battery but from a changing magnetic flux. We can think of the loop containing an imaginary battery that gets turned on whenever flux in the loop changes. The strength of this battery is called the emf (electromotive force); it s symbol is a script E:, and it s measured in volts. The induced current is given by: I = / R where R is the internal resistance in the loop. itself depends on the rate at which the flux inside the loop is changing. If the flux is changing at a constant rate, = - Ф B / t This Faraday s law. The negative sign here indicates the emf opposes the change in flux. The greater the change in flux the greater, the greater the induced emf, and greater the induced current.

36 Electromagnetic Induction: Practice For each scenario determine the direction of the induced emf and current. wire loop B increasing B decreasing B very large but constant B increasing B decreasing B increasing

37 Induction: Nonuniform, Static Fields y x B is static (constant in time). It is uniform in space in the y and z directions but not in the x direction. B decreases as x increases. As the rectangular loop is moved in the following directions, determine the direction of the induced emf and current as well as the direction of the net force on the loop by the field. Loop motion: 1. Left 2. Right 3. Up 4. Down 5. In 6. Out B is uniform here but only in the region shown. Beyond this region B is approximately zero. As the loop is pulled out of the field determine the direction of the induced emf and current as well as the direction of the net force on the loop. Do the same as the loop is pushed into the field.

38 Electric Generators In a motor we have seen that a current loop in an external magnetic field produces a torque on the loop. In a generator we ll see that a torque on a current loop inside a magnetic field produces a current. In summary: Motor: Current + Magnetic field Torque Generator: Torque + Magnetic field Current Turbines in a power plant are usually rotated either by a waterfall or by steam created heat produced from nuclear power or the burning of coal. As the turbines rotate, current loops turn through a magnetic field to generate electricity. This process converts mechanical energy into electrical energy. The simplest form of an electric generator is called an alternating current (AC) generator. The current produced by an AC generator switches directions every time the wire inside of it is rotated through a half turn. In the United States, generator generally have a frequency of 60 Hz, which means the current switches direction 120 times every second! A graph of the current output from an AC generator produces a sinusoidal curve due to the periodic nature of the generator s rotation. Continued

39 Electric Generator (cont.) Animation I induced B As a turbine turns (due to some power source like coal) a current loop (purple) is rotated inside a magnetic field. The field is static but as the loop turns as the number of field lines poking through it changes. Thus we have a changing flux and a corresponding induced emf and current. The pic shows a loop just after it was horizontal (perpendicular to the field). The flux is decreasing since the loop is becoming more vertical. Since fewer field lines are entering the loop, the induced current is in a direction to produce more field lines downward. Just prior to this, as the loop was approaching horizontal, the number of field lines inside it was increasing, so the current was in the other direction to oppose this change. The current changes direction twice with each turn--whenever the loop is horizontal. The result here is AC, but (direct current) DC motors exist as well in which current only flows in one direction.

40 Electric & Magnetic Fields Picture tubes in standard televisions are basically cathode ray tubes (CRT s). In a CRT electrons are shot from a hot filament into a region of crossed fields in which a magnetic field is perpendicular to an electric field. On the other side of the crossed fields is a fluorescent screen (not shown) where electrons produce spots of light when they make contact with it. J. J. Thompson used a CRT to discover the electron in When the charge enters the fields, F E is up and F B is down. By adjusting B and measuring the deflection of the electrons, Thompson determined that they were negatively charged and calculated their mass to charge ratio. Let s find a relationship between q, B, and E if there is no deflection at all: F net = 0 F B = F E q v B = q E v = - q, m v E B B E

41 How speakers work Bestiary of magnetic minerals History of magnets Magnetite Slide show Best ever site for pictures, simple explanations, etc. Another good site for how magnets work Equations and such See also: Credits How a metal detector works How a compass is oriented magnetically How Faraday did his current experiment How a hard drive works How magnet lines is working How two magnets repel and attract Nuclear spin up/down Pulsed magnets