Magnetic Fields and Forces

Getting to the heart of the matter:

Charges in motion create magnetic fields. (Stationary charges do not.) Magnetic fields can cause forces on charges in motion. (But not on stationary charges.) Since one or more charges in motion are called “current”, we can restate these more clearly as: (1) Magnetic fields are caused by currents. (2) Currents in magnetic fields can experience magnetic forces. We are intentionally using the word “can” here since, for certain alignments, magnetic fields may not cause a force on a given current.

Magnetism is caused by “electricity in motion”. As you know, magnets have “North and South poles”, that lead to attraction and repulsion. But, there are no free “magnetic charges (N, S)” associated with these poles. We’ll see that these poles are caused by electric current distributions, and that electricity and magnetism are unified interactions. This was one of the great physics discoveries of the 19th century. The connection between electricity and magnetism

Hans Christian Ørsted 1777-1851

Danish physicist and chemist, He is best known for discovering the relationship between electricity and magnetism known as electromagnetism.

While preparing for an evening lecture on April 21st 1820, Ørsted developed an experiment which provided evidence that surprised him. As he was setting up his materials, he noticed a compass needle deflected from magnetic north when the electric current from the battery he was using was switched on and off. This deflection convinced him that magnetic fields radiate from all sides of a live wire just as light and heat do, and that it confirmed a direct relationship between electricity and magnetism. Three months later he began more intensive investigations, and soon thereafter published his findings, proving that an electric current produces a magnetic field as it flows through a wire. But magnetism was known and applied from ancient times…

(From Wikipedia) Lodestone refers to either:

(1) Magnetite, a magnetic mineral form of Fe3O4, one of several iron oxides. (2) A piece of intensely magnetic magnetite that was used as an early form of magnetic compass.

Iron, steel and ordinary magnetite are attracted to a magnetic field, including the Earth's magnetic field. 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 (鬼谷子): "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 (Louen-heng): "A lodestone attracts a needle." By the 12th century the Chinese were known to use the lodestone compass for navigation. Fun with magnets: the rules of attraction and repulsion

Conclusion: Opposite poles attract, like poles repel. (Same rule as for electric charges.) Magnets can “magnetize” some materials, creating “induced magnetization”

What does this tell us about How does this happen? the induced poles? There are no free magnetic poles

If you try to separate the N and S poles of a magnet by cutting it in half, new N and S poles will appear so that each new magnet always has a NS pair!

This will make sense once we define the magnetic poles and see how they are created. How will we measure magnetic fields?

We cannot use a test charge since Q stationary charges experience no force from a magnetic field.

We can use a small “test magnet” to find the direction of the magnetic field, since the test magnet will align parallel to the field lines. The compass is an example of a “test magnet”

The action of the test magnet. Historically, mapping Earth’s Similar to an electric dipole in an magnetic field has been of great electric field. interest and utility: The result: Earth’s magnetic field is a “magnetic dipole”

Electric currents circulating in the Earth’s interior create a magnetic South pole near the geographic North pole and a magnetic North pole near the geographic South.

Notice that, outside the Earth, the shape of the magnetic field is very similar to the shape of an electric dipole field with a negative charge at “S” and a positive charge at “N”, where the field lines emerge. The shape is similar but the physics is different! The Earth’s geomagnetic field: measurement + simulation

The pattern of magnetic field lines inside the Earth is much more complicated than those inside the magnets we’ll study. This is because our “geodynamo” consists of a complex network of currents, mostly in the outer core, driven by the Earth’s rotation. Fortunately, the magnetic field pattern near and beyond the surface is an almost perfect dipole. (Out further, it is distorted by the “solar wind” of charged particles.) Now, to the basics:

The magnetic field ⎡ N ⎤ is a vector field r Units: 1 T = 1 tesla = (like E), and this is B ⎢A⋅m⎥ its symbol: ⎣ ⎦ (Nikola Tesla,1856-1943. We meet him later. )

In this chapter, we will be focusing on magnetic forces on charges and currents moving in a given magnetic field. How magnetic fields are produced will be considered in detail in the next chapter, “Magnetic sources”. But, in order know what magnetic field configurations to expect, we will look at a few pictures of basic field configurations, and the current distributions that cause them. Æ Magnetic field of a line of current, I.

Å Demo. Also magnetic field lines Magnetic field of a line of current. Right-hand rule, and 2D view of test magnets. Magnetic field of a line of current. 2D end view and 3D view.

This B field has cylindrical symmetry. Compare and contrast to E field of line charge. Magnetic dipole field due to a current loop, or a magnet.

This B field has dipole form at long distances. Magnetic field due to a solenoid (series of loops)

This B field has dipole Inside, this B field is almost uniform form at long distances. throughout the solenoid interior !

Next chapter ! Drawing conventions for 2D pictures of B fields

Now we return to the central topic of this chapter, magnetic forces. Magnetic force on a charged particle of velocity v in field B:

First, consider the force magnitude: F = q v⊥ B = q vBsin(φ)

1. The force is proportional to the charge. 2. The force is proportional to the speed. 3. The force is proportional to the magnetic field. 4. The force depends on the angle between v and B.

The full story is contained in the cross product: r r r F = q(v × B) r F So the force vector is also v perpendicular to both v and B. v r (We will look at evaluation of the φ B cross product in more detail.) Lorenz force equation: combining the electric and magnetic forces acting on a charged particle. r If a charged particle is traveling in a F region that has both electric and r FB r magnetic fields, the forces due to both FE fields may be added vectorially r (superposed): E r r r vv F = FE + FB φ r The resulting equation is known as the B Lorenz force equation: r r r F = q(E + vr × B)

(Ludvig Valentin Lorenz, 1829 - 1891: Danish mathematician and physicist.)

We’ll begin by considering problems where only a magnetic field is present. But later, we’ll look at the more general case. Magnetic force on a charged particle of velocity v in field B:

Again, the magnetic force equation: r r r F = q(v × B) magnitude F = q v⊥ B = q vBsin(φ)

The sin(φ) factor tells us that the particle will experience the maximum force when the vectors r are lined up as shown, with the F velocity perpendicular to the field. r B vv But the other thing to notice is that the magnetic force is always perpendicular to φ the velocity. This means that the force never acts along the particle’s direction of motion. So, F does no work on the particle! F can change the particle’s direction, but not its speed (kinetic energy). In a region with uniform magnetic field B:

As we said, alignment matters. Negative charge flips the force. We can use “right hand rules” to figure out directions associated with the cross product.

Å Cross product demonstrator Charged particle motion in a uniform B field

From the discussion above we see that, in 2D, this must be circular motion at constant speed. Why??

The magnetic force provides the radial acceleration needed to maintain circular motion: v2 F = qv B = qvB = ma = m ⊥ r r One power of v cancels, giving: mv p mv p qB = = → r = = r r qB qB

And the orbital frequency, f, is: v qB ω qB ω = = → f = π = r m 2 2π m

f is also known as “cyclotron frequency” Discuss the equations for r and f. Magnets at the LHC proton collider, CERN, Geneva, Switzerland

~8000 magnets of various types. Magnetic fields at full strength: 8.4 T. Proton energies: 7 TeV per particle. Radius of main ring: 4.3 km. Discovery of the positron (anti-electron)

These particles are in a uniform magnetic field. A high energy photon (gamma ray) has collided with an electron in a hydrogen atom. The electron proceeds forward. There was also enough energy in the collision to produce another pair of particles (electron and its anti- matter partner, the positron) via E = mc2. For each particle, from the direction in which it turns we can find the charge sign, and from the radius of curvature right after the collision we can calculate the momentum: p = qBr (The particles spiral in to smaller radii as they lose energy colliding with atoms.) Charged particle motion in a uniform B field in 3D

If vparallel = 0, then the motion is circular, parallel to the y-z plane. Otherwise, the particle coasts at constant velocity in the +x or –x direction, and the resulting trajectory is a helix. How would we calculate its pitch angle? Charged particle motion in a “magnetic bottle”

Since the magnetic field is strongest at these locations, the cyclotron frequency is highest here. The Van Allen Belts: nature’s magnetic bottle

The dipole field of the Earth is strongest near the poles, creating a magnetic bottle effect between the poles. As the particles stream into the poles (and back out again), their cyclotron frequency rises, then falls. They also radiate photons, observable as: (1) Aurorae at the poles. (2) Radio “whistlers”.

D Magnetic force on a conductor carrying a current I in a uniform magnetic field, B

We’ve considered magnetic forces on individual charges. Now we find the total force on a segment of conductor of length L carrying a current I in a uniform field B. Because the

charge carriers have drift velocity vd , each carrier feels a magnetic force. But, the electric fields in the conductor constrain them to motion in the direction of the current. The force acting on all charges in this volume is:

F = Qtotvd B = (qnAL)vd B = (nqvd )ALB = (JA)LB = ILB

Giving us the simple result: F = ILB

This can be used for any problems where the current is perpendicular to the magnetic field, and hence to the drift velocity. But we need to generalize this

formula to account for the cross product relationship between vd and B. Generalizing the force on a conductor to any orientation

For a single particle in a uniform B field: r r Cross product: F = qvr × B

Magnitude with tilt: F = qvB sin(φ)

Magnitude at 90o: F = qvB

Our derivation showed that we should replace qv by IL: Magnitude at 90o: F = ILB Magnitude with tilt: F = ILB sin(φ) r r r Cross product: F = IL × B

This last equation is general, and it makes sense, since the direction of v is now the same as the direction of the conductor, and I contains q. Evaluating the cross product

Two equations we’re using have cross products: r r r r r F = qvr × B F = IL × B The cross product is calculated by (1) filling a 3x3 matrix with x-y-z unit vectors in the first row, and putting the two vectors in order into the second and third rows, then (2) finding the determinant of this matrix. In the first equation above, we would have: ⎡iˆ ˆj kˆ ⎤ r r ⎢ ⎥ v × B = det ⎢vx vy vz ⎥ ⎢ ⎥ ⎣Bx By Bz ⎦ r r ˆ ˆ ˆ v × B = i (vy Bz − vz By ) − j(vx Bz − vz Bx ) + k(vx By − vy Bx )

Then, since F = q(v x B), if q is negative, it will flip the direction of F.

Å Example with uniform B field in z direction Define the “magnetic moment”: useful in torque and energy equations

The “magnetic moment”, μ , also called the “magnetic dipole moment”, tells us the strength of a magnetic dipole field, in the S same way that p, the electric dipole moment tells us the strength of an electric N dipole field. We’ll define it here, then use it in upcoming derivations: μ = IA

The top picture shows us the direction of the magnetic moment for a current loop; and the bottom picture, for a hydrogen atom—caused by the current of the electron traveling in its orbit. Magnetic moment for a coil of many turns

Å(Ignore the external B field for now.) Find the magnetic moment if the coil above has (1) a single turn, (2) 500 turns. The general expression? Also, note the units. Torque on a current loop in a uniform magnetic field, B

Maximum torque

F = ILB

The long sides of the loop, each of length L, experience forces F = ILB, perpendicular to the wire and to the magnetic field. There is a Zero torque torque on the loop about the y axis, similar to the torque felt by an electric dipole in a uniform electric field. Note the direction of μ when the torque is minimum and maximum. D Torque on a current loop in a uniform magnetic field, B

Discuss forces on ends, of length b. Then…

Total force on each side: F = IaB Torqueτ due to each side: F = ILB b IabBsin(φ) = F sin(φ) = side 2 2

Total torque on loop:

F = IaB τ = 2τ side = IabBsin(φ) = IABsin(φ) = μBsin(φ)

But τ, μ, and B are all vectors. We can see from the picture, and from the sin(φ) factor that they are related by the cross product: τ r r = μr × B Application of loops in a magnetic field: the DC electric motor Materials that can be magnetized

These are called “ferromagnetic materials”, and they include iron (Fe), nickel (Ni), cobalt (Co), and gadolinium (Gd).

You can see that if this magnetization is induced by an external magnetic field, the situation resembles the induced polarization of a dielectric by an electric field. A closer look at magnetization

Note: Some materials, such as “soft iron” are easy to magnetize, and when their magnetic domains align, the resulting field can be much larger than the external field causing the magnetization. In some materials, the magnetization persists

“Soft iron” and “soft steel” are examples of alloys that magnetize easily, with internal magnetic fields up to 2T. But when the external field is removed, their magnetic domains readily return to random orientation, and the magnetization disappears. These materials are useful for transformers and inductors (more later). “Hard steel” and other special ferromagnetic alloys, are harder to magnetize. But after the external field is removed, their magnetic domains retain alignment. Permanent magnets can be made from these materials, and from specialized ceramic materials.

D Magnetic flux

Recall electric flux: General expression v r Φ = E ⋅ dA = E cos(φ)dA E ∫∫ ∫∫ AA r r Uniform field, flat surface Φ E = E ⋅ A = EA⊥ = E(Acosφ)

r r Æ Magnetic flux is: General expression Φ = B ⋅ dA = Bcos(φ)dA B ∫∫ ∫∫ AA r r Uniform field, flat surface Φ B = B ⋅ A = BA⊥ = B(Acosφ)

General Uniform field, flat surface Gauss’s Law for magnetic fields

What can we calculate with magnetic flux? Not much yet, until we have seen Faraday’s Law of induced emf. But for now, we can jump directly to Gauss’s Law for magnetic fields. Start with Gauss’s Law for electric fields, the one we have seen already: v r Q Φ = E ⋅ dA = enclosed E ∫∫ A ε 0

As we have seen, magnetic forces come from electric charges in motion. There are no free magnetic charges. Magnetic field lines diverge from N poles and converge into S poles, but they do not begin or end at either pole.

Then Qmagnetic = 0, so that there cannot be enclosed charge. Gauss’s Law for magnetism is then: r r Φ = B ⋅ dA = 0 B ∫∫ A

Å Sketch a bar magnet and look at magnetic Gaussian surfaces “containing” the poles, and containing the whole magnet. Contrast to case of the electric dipole. Crossed E and B fields: a velocity selector

Å Find v in terms of E and B D The mass spectrometer: a velocity selector followed by a uniform B field to give momentum information. Why do different beam spots correspond to different masses? E

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