Magnetic Field And Its Sources

CHAPTER FOUR: Magnetic Field and Its Sources 4.1. Magnetic Field 4.2. Sources of Magnetic Fields

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•Calculate Calculate the force exerted by magnetic fields

4.2

•Calculate the magnetic field due to different current and moving charge configurations

•Calculate kinematical quantities in problems involving motion in magnetic fields

•Define Ampere’s Law •Define Gauss’ Law for Magnetism 2

Outline: 1. The Magnetic Phenomenon 2. The Force of Magnetic Fields 3. The Gauss and the Tesla 4 4. Magnetic Field Lines 5. Motion in Magnetic Fields 6. Applications of the Velocity Selector `

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Ancient Greeks (around 2000 y g ) were aware that years ago) magnetite (Fe3O4) attracts pieces of iron. There h are written references f to the use of magnets for navigation during the 12th century. century In 1269, Pierre de Maricourt discovered using simple observations, the existence of magnetic “poles”. In 1600, William Gilbert discovered that the earth itself is a natural magnet. 4

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Magnetic Poles

◦ Are the basic elements of the magnetic phenomenon ◦ It comes with two varieties that exist as pairs: the “ “NORTH ( )” and (N)” d the h “SOUTH (S)” poles ◦ No matter what the shape nor the size, all magnets have two poles which can can’tt be isolated like electric charges!

Magnetic Fields emerges from the N pole and enters the S pole 5

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Magnetic Interaction: The Fundamental Law of Magnetostatics:

“Like Poles Repel, Unlike Poles Attract”

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g The magnetic effect and interaction can be better studied by analyzing the force exerted by a magnetic field on a test p specimen

The Magnetic Field only i interacts with i h the h following tests: 1 1. Moving Charges 2. Current Carrying Wires `

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The north-pole end of a bar magnet is held near a positively charged piece of plastic. Is the plastic (a) attracted, (b) repelled, ll d or (c) unaffected by the magnet?

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Ans: (c)

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When a test specimen is in a magnetic field, it experiences p a magnetic g force. The Magnetic Force on a moving test charge Th Magnetic M i F The Force on a current carrying wire 10

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Since the force is a vector, we can determine it’s magnit magnitude de and direction. direction For the magnitude:

If the g given are in NEWSUD config

•|F |FB| = qvB B sinθ i θ or |FB| = ILB sinθ i θ

If the given are in Unit Vector config If the given are in the simple 2-D config

•|FB| = qvB sinθ or |FB| = ILB sinθ 11

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There is ZERO FORCE if: v(or L) and B are parallel (θ=0) (θ 0) or are anti parallel (θ=180)

Proof: It’s simple: take the sine of the angles! 12

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For the direction:

If the given are in NEWSUD config

• Use the Right Hand Rule for 3 Dimensions

If the given are in Unit Vector Notation If the g given are in simple 2-D config

• Use the Right g Hand Rule for 2 Dimensions

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Steps: 1. Point your 4 fingers to the direction of v (or L). 2 Curl your 4 fingers 2. to the direction of B. 3 Release your 3. thumb and this will be the direction of the force! 14

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First, First assume that the charge is positive and apply the Right Hand R l to obtain Rule b i the h direction of the force, Then, the direction of g on a a force acting negative charge is opposite to that of the direction attained for the positive charge 16

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Some Conventions: y-axis

z-axis or “page” directions

x-axis x axis

X in or –z axis

out or +z axis

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FB

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FB=0

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The Gauss and the Tesla serves as unit for Magnetic Field!

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The Gauss-Tesla Relation:

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The Tesla can be represented by:

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A proton is moving with a velocity of 10Mm/s. It experiences i a magnetic ti field of 0.6G which is directed downward and northward, h d making k an angle of 70o with the o o ta horizontal. Find the magnetic force proton on the proton.

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A wire segment 3mm long carries a current of 3A in the x-direction. It lies in a magnetic field y of 0.02T that is in the xyplane and makes an angle of 30o with the x-axis, as shown h in the h figure. f What is the magnetic f t d on th i force exerted the wire segment. 21

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Magnetic Field Lines are very similar to Electric Field lines in the following aspects ◦ The direction of the field is the direction of the field lines ◦ The h magnitude d off the h field f ld is the h density d off the h lines l ◦ Field Lines never cross

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Magnetic Field Lines are very different to Electric Field lines in the following aspects

◦ Electric field lines are in the direction of the electric force while magnetic field lines are in a perpendicular direction of the magnetic g force ◦ Electric field lines have beginnings and ends, while magnetic field lines form closed loops 22

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For the motion of charges in magnetic field, field we consider two situations: The Charges are moving in a pure Magnetic Field The Charges are moving in a E-B crossed fields

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v is always perpendicular to FB. From this we have general ideas: FB only l changes h the th direction of the velocity but not the magnitude it d FB does no work on the charge FB therefore does not change the kinetic energy of the charge!

SPECIAL CASE: When v is perpendicular to B, the charge undergoes UCM

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p In this special case,, the magnetic force provides the centripetal force necessary for the centripetal acceleration in circular i l motion. ti We use Newton’s Second Law to relate l the h quantities. ii

Cyclotron Radius

T and f are known as the cyclotron l period i d and d frequency respectively. The h cyclotron l period d and d Cyclotron Period frequency depend on the charge-to-mass ratio q/m but are independent of r and v of the particle!

Cyclotron Frequency

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There are two interesting motion paths and behaviors involved with moving charges in magnetic fields THE HELIX: Happens when charges enter B with a nonperpendicular velocity

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THE BOTTLE:

Happens when the magnetic field is not uniform (being strong at ends, d weak k at the h center

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The magnetic force can be balanced by an electric field by choosing a correct configuration, such as the figure to the right h Once there is a balance of the force, we have a region of crossed d -fields! fi ld ! To achieve the balance, the velocity must be chosen!

“Lorentz Force”

“Velocity Selector” 27

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With the velocity selector selector, the charged particle will traverse the crossed fields undeflected! d fl d!

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If it enters the field with a v greater than the selector

Deflect to FB

If it enters the field with a v lesser than the selector

Deflect to FE

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The velocity selector for crossed-fields crossed fields have very important applications that were discovered during g the late 19th and early y 20th century.

In this course, two applications will be discussed: 1 Th 1. The M Mass S Spectrometer 2. The Cyclotron `

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The mass spectrometer, first g y Francis William designed by Aston in 1919, was developed as a means of measuring the masses of isotopes. Such measurements are an important way of determining b th the both th presence off isotopes i t and their abundance in nature. For example, F l naturall magnesium has been found to have mass ratios of 24:25:26. The mass ratios are computed from the radius of curvature! Each element, by the way has a unique q/m ratio!

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58Ni

26 ion of charge +e and mass 9.62 9 62 x 10-26 kg is accelerated through a potential difference of 3kV and deflected in a magnetic field of 0.12T. (a) Find the radius of curvature of the orbit of the ion. (b) Find the difference in the radii of curvature of 58Ni ions and 60Ni ions. (Assume that the 58/60 ) mass ratio is 58/60.)

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The cyclotron was invented by E.O. Lawrence and M.S. MS Livingston in 1934 to accelerate particles such as protons or deuterons to high kinetic energies*.

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A cyclotron for accelerating protons has a magnetic field of 1.5T and a maximum radius of 0.5m 0 5m (a) What is the cyclotron frequency? (b) What is the kinetic energy of the protons when they emerge?

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1. Due to Moving Charges Ch 2. Due to Currents: Biot Savart Law

◦ 2.1 Current Loops ◦ 2.2 Current of Solenoids ◦ 2.3 2 3 Current of Straight Wires ◦ 2.4 Current of Toroids

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3. Gauss’ 3 Gauss Law for Magnetism 4. Ampere’s Law 5. Magnetism in Matter

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Permanent Magnets were the earliest known sources of magnetism. Oersted announced his discovery p needle is deflected that a compass by an electric current. Jean Baptiste Biot and Felix Savart announced d th the results lt off their th i measurements of the force on a magnet near a long currenty g wire and analyzed y results carrying in terms of the magnetic field. Andre-Marie Ampere extended th these experiments i t and d showed h d that current elements also experience a force in the presence g field and that two of a magnetic currents exert forces on each other.

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There are two possible sources of magnetic fields 1. Moving Charges 1 2. Currents in Wires Our Quest is to find the ti fi ld iin a magnetic field certain field point P!

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You will almost always encounter the permeability of free space during your computations in this chapter, chapter so might as well introduce it here:

In your calculators, press π first before multiplying it to 4 x 10-7!

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When a point charge q moves with a velocity v, it produces a magnetic f ld B at the field h field f ld point P given by:

If the given are in UV Notation

If the given are in basic geo notation

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A point charge of magnitude q = 4.5 nC is moving with speed v = 3 6 x 107 m/s 3.6 / parallel ll l to the x-axis along the line y = 3m 3m. Find the magnetic fields g produced by this charge (x = -4m, y = 3m) at (1) the origin (2) the h point i (0 (0,3m) 3 ) (3) the point (0, 6m) Ans: (1) 3.89 x 10-10 T in the paper. (2) 0 (Why?). (3) 3.89 x 10-10 T out the paper

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Since currents are basically moving charges y in wires,, we can only extend our earlier formula into a more definite law: To compute for the magnetic field f ld caused db by currents on a certain field point P, we implement BIOT-SAVART LAW! 40

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Throughout this chapter, chapter we shall encounter four basic sources of magnetic fields that produces the field by carrying currents!

They are: 1. Current Loops 2. Solenoids S l id 3. Straight Wires 4. Toroids

To find the direction of the field, apply RHD

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Steps: 1. Grab the wire in such a way that your thumb is in the same direction and the current. 2. Your 4-fingers determines the direction of the magnetic ti fi field ld as it wraps around the wire 42

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Figure shows a current loop.

R

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P

To calculate the magnetic field caused by the loop at a certain point P along the central axis of the h loop, l we apply l the h formula: f l

Special Case: If P is the center of the loop, x =0, then the magnetic field at the center of the loop is

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A circular loop of radius 5.0cm has 12 turns and lies in the yz-plane, where it is centered at the origin. g It carries a current of 4A. The current is counterclockwise from the perspective of the x-axis looking to the yz-plane. Find the magnetic field at (a) center of the loop (x = 0) (b) x = 15 cm (c) x = 3 cm. 44

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g y wound A solenoid is a wire tightly into a helix off closely spaced turns as illustrated. It is used to produce a strong, strong uniform magnetic field in a region surrounded by its loops. Its role in magnetism is analogous to that of the parallel-plate capacitor, which produces a strong, uniform electric field between its plates. The magnetic field of a solenoid is essentially that of a set of N identical current loops placed side by side. We have two types of solenoids: 1. Finite Solenoids 2 IInfinite 2. fi it S Solenoids l id

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For PHYSICS 13, 13 we shall use the long solenoid approximation. n (turn density) = N/L We can calculate the magnetic field caused by solenoids at two locations: 1. At its center 2 At its ends 2.

At the center

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Find the magnetic field at the center of a solenoid of length 20 cm, radius 1.4 cm, and 600 turns that carries a current of 4A 4A.

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As with solenoids, solenoids we there are two g kinds of straight wires 1. Finite Wires 2. Infinite Wires We wish to obtain the magnetic field at a point P perpendicular f from the th line li

Field caused by finite wires. R is the distance of P from the wires Field caused by infinite wires, R is the distance of P from the wires! 48

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Find the magnetic field at the center of a q current loop p of square side L = 50cm carrying a current of 1.5A.

Picture the Problem: `

The magnetic field at the center is the sum h contributions b off the off each side! 49

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A toroid consists of loops of wire wound around a doughnutshaped h d fform. he magnetic field at a distance r from the center of the toroid are given as: If a
If rb (outer radius) 50

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We know that magnetic field lines differ from electric field lines. Magnetic field lines form closed loops. The magnetic equivalent off the h electric l charge h is called a magnetic pole. Gauss’s Gauss s Law for Magnetism is stated as:

That is, no magnetic monopoles!

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Ampere’s Law is very analogous to Gauss’s Gauss s Law for Electricity. It relates the magnetic field to the current enclosed by an imaginary loop (called Amperian Loop). Ampere’s Law works for that have a configurations g high degree of symmetry.

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Ampere s Law will only work Ampere’s if and only if the following statements hold: 1. The configuration has a very high level of symmetry 2. The current is continuous everywhere in space. Therefore, there are only three cases where Ampere’s p Law can be used: ◦ 1. Long straight lines g, tightly g y wound ◦ 2. Long, solenoids ◦ 3. Toroids

Checkpoint: In which h of the ese fourr config guration n does Am mpere’s Law ho old?

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In our discussion of the magnetism in matter, we return the atomic t tto th t i model. Electrons orbit El bi around d the h nucleus, and since we can consider it as a current, it produces a magnetic field! If we look back at the electric dipole (and the corresponding electric dipole moment), we can also make the same analogy to create a magnetic dipole (and the corresponding di magnetic i dipole moment

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We can classify matter three tt iinto t th based on their reaction to an external magnetic field!

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Diamagnets g

Magnetic Moments align to oppose the external magnetic field, thus diagmagnets are slightly repelled by magnets

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All materials have randomly oriented magnetic moments

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Paramagnets

Magnetic Moments align slightly with the external magnetic field, thus paramagnets interact weakly with magnets

Ferromagnets

Magnetic Moments align strongly with the external magnetic field, causing permanent magnetization, thus ferromagnets interact strongly with magnets

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