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Magnetism• The Magnetic ForceThe Magnetic Force

BvqEqF

x x x x x xB

B

Bx x x x x xx x x x x xv

v

v

x x x x x xF q

Fq

F = 0q

F F = 0

P.Ravindran, PHY041: Electricity & Magnetism 22 January 2013: Magntostatics

History of Magnetism• The first known magnets were naturally occurring lodestones, a type of iron ore called magnetite (Fe3O4). 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 i h di d h h d i lwith 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• 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 p yelectric current flowing through a wire can cause a compass needle to deflect, showing that magnetism and electricity were related.


History (cont.)• In 1830 Michael Faraday (British) and Joseph Henry (American)• 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 ll h h d f l d “f ld ” lall 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

ti f H l d th t li ht l t ti di timagnetic forces. He also proposed that light was electromagnetic radiation.

• In the late 19th century Pierre Curie discovered that magnets loose their magnetism above a certain temperature that later became known as themagnetism 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.


es (t e e ss e e ect) c a es ag et c e tat o poss b e

Magnetism

• Magnetic effects from natural magnets have been known for a long time. Recorded observations from the Greeks more than 2500 years ago.y g

• The word magnetism comes from the Greek word for a certainThe word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece.

• Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron itcould impart this property (magnetize) to a piece of iron it touched.

• Small sliver of lodestone suspended with a string will always align itself in a north‐south direction—it detects the earth’s magnetic


field.

Magnetic DipolesRecall that an electric dipole consists of two equal but opposite charges separated by some distance, such as in a polar molecule. +‐

, p

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

Electric dipole and E field

reenter the ‐ or S pole.

Although they look the same, they are diff t ki d f fi ld E fi ld ff tdifferent kinds of fields. E fields affect any charge in the vicinity, but a B field only affects moving charges. As with charges, NSopposite poles attract and like poles repel. NS


Magnetic dipole and B field

Magnetic Monopole Don’t Exist

W h di d l i fi ld d i l d h b fWe 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 th l di t th ti fi ld th i Wh t t tthemselves 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).


Magnetic FieldsMagnetic FieldsYou have seen that electric fields and be uniform, nonuniform d f d hand 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 amagnetic 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 y yper unit charge. Because there are no magnetic monopoles, there is no analogous definition for B. However, regardless of

l i h i l SI i f B fi ld Isymmetry 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 fieldhow 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.

Bar Magnet

B• Bar magnet ... two poles: N and SLike poles repel; Unlike poles attract.

• Magnetic Field lines: (defined in same way as electric field

lines, direction and density)

NS NS

• Does this remind you of a similar case in electrostatics?

Electric Field Linesof an Electric Dipole

Magnetic Field Lines ofMagnetic Field Lines of a bar magnet NS

Magnetic Fields

We know about the existence of magnetic fields by their effect on moving charges. The magnetic field exerts a force on the moving h

• What is the "magnetic force"? How is it distinguished from the "electric"

charge.

What is the magnetic force ? How is it distinguished from the electric force?

Let’s start with some experimental observations aboutthe magnetic force:

a) magnitude: to velocity of q

b) direction: to direction of q’s velocityq b) direction: to direction of q s velocity

c) direction: to direction of B

q

F

v

mag


B is the magnetic field vectorF mag

Lorentz Force• The force F on a charge q moving with velocity vthrough a region of space with electric field E and magnetic field B is given by:

BEF

B B B

BvqEqF

x x x x x x

x x x x x x

B

B

B

x x x x x x

x x x x x xv

q

v

q

v

qF q q

F = 0q

F


The magnetic field around a long straight wireg g g

• The diagram shows a wire carrying a current of about 5 amps

• If you sprinkle some iron filings on to the horizontal card and tap it gently, the iron filings will line up along the lines of flux as shown.


• You can place a small compass on the card to find the direction of the magnetic fielddirection of the magnetic field.

• With the current flowing up the wire, the compass ill i t t l k i hwill point counter‐clockwise, as shown.

• What will happen if you reverse the direction of the current?


• Can you see that the further from the wire the circles are, the more widely separated they become? What does this tell you?

• The flux density is greatest close to the wire.• As you move away from the wire the magnetic field• As you move away from the wire the magnetic field becomes weaker.


• The right‐hand grip rule gives a simple way to remember the direction of the field: i i i i th i• imagine gripping the wire, so that your right thumb points in the direction of the current.

• your fingers then curl in th di ti f th li fthe direction of the lines of the field:


The magnetic field of a flat coilThe magnetic field of a flat coil

• The diagram shows a flat coil carrying electric current:• Again, we can investigate the shape and direction of the magnetic field using iron filings and a compass.


• Close to the wire, the lines of flux are circles.• Can you see that the lines of flux run counter‐clockwise around the left side of the coil and clockwise around the right side?clockwise around the right side?

• What happens at the center of the coil?• The fields due to the sides of the coil are in the same• The fields due to the sides of the coil are in the same direction and they combine to give a strong magnetic field.

• How would you expect the field to change, if the direction of the current flow around the coil was

d?reversed?


Magnetic Monopoles

• P h th i t ti h j t lik l t i h S h• Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or ‐magnetic charge).

• How can you isolate this magnetic charge?

Try cutting a bar magnet in half:Try cutting a bar magnet in half:

NS N NS SEven an individual electron has a magnetic “dipole”!

• Many searches for magnetic monopoles—the existence of which would explain (within framework of QM) the quantization of electric

magnetic dipole !

would explain (within framework of QM) the quantization of electric charge (argument of Dirac).

• No monopoles have ever been found: 0B dS

˜


No monopoles have ever been found:

Force Due to Magnetic FieldThe force exerted on a charged particle by a magnetic field is given by the vector cross product:

F = force (vector)F = qvB

F = force (vector)

q = charge on the particle (scalar)

v = velocity of the particle relative to field (vector)

B = magnetic field (vector)B = magnetic field (vector)

Recall that the magnitude of a cross is the product of the magnitudes of the vectors times the sine of the angle between

F = qvBsin

magnitudes of the vectors times the sine of the angle between them. So, the magnitude of the magnetic force is given by


F = qvBsinwhere is angle between qv and B vectors.

Cross Product ReviewLet v1 = x1, y1, z1 and v2 = x2, y2, z2 .By definition the cross product of these vectors (pronounced “vBy definition, the cross product of these vectors (pronounced v1cross v2”) is given by the following determinant.

v1 v2 = x1 y1 z1

i j kv1 v2 x1 y1 z1

x2 y2 z2

= (y1 z2 - y2 z1) i - (x1 z2 - x2 z1) j + (x1 y2 - x2 y1) 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


positive x, y, and z axes, respectively. (See the vector presentation for a review of determinants.)

Right Hand Rule ReviewgA 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

ab

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 ab.

a b

b, making a fist. Your thumb then points in the direction of

a b

a bIt can be proven that the magnitude of

i i ba b

b

is given by:

absin| a b | =

babsin| a b | where is the angle between

aa and b.

Magnetic Field UnitsMagnetic Field Units

F = q v B sin

1 N = 1C(m/s)(T)

q s

1 N 1C(m/s)(T)From the formula for magnetic force we can find a relationship between the tesla and other SI units The sine of an angle has nobetween the tesla and other SI units. The sine of an angle has no units, so

1 T = 1 N 1 N =

f ld f l f l f ld b

1 T C(m/s) Am

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.


Direction of Magnetic Field & ForceNear 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 is moving into the screen. Since q is +, the q v vector is also into the screen. 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. p y q

BFThe force on the ‐ charge is into the

+ ‐ v

screen. If a charge is motionless relative to the field, there is no magnetic force on it, but if either a

Bvmagnet 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, sinceforce on it, since sin 0° = 0.

Magnetic Force Sample ProblemThis 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:

F = qv B F = q v B

Sq q

since sin 90° = 1.

+5 m/s

NN So, F = 0.0015 N directed out of the pagedirected out of the page.

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