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**Magnetic Forces and Magnetic Fields**

Chapter 22 Magnetic Forces and Magnetic Fields

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**A Brief History of Magnetism**

13th century BC Chinese used a compass Uses a magnetic needle Probably an invention of Arab or Indian origin 800 BC Greeks Discovered magnetite attracts pieces of iron

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**A Brief History of Magnetism, 2**

1269 Pierre de Maricourt found that the direction of a needle near a spherical natural magnet formed lines that encircled the sphere The lines also passed through two points diametrically opposed to each other He called the points poles

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**A Brief History of Magnetism, 3**

1600 William Gilbert Expanded experiments with magnetism to a variety of materials Suggested the earth itself was a large permanent magnet 1750 John Michell Magnetic poles exert attractive or repulsive forces on each other These forces vary as the inverse square of the separation

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**A Brief History of Magnetism, 4**

1819 Hans Christian Oersted Pictured, 1777 – 1851 Discovered the relationship between electricity and magnetism An electric current in a wire deflected a nearby compass needle André-Marie Ampère Deduced quantitative laws of magnetic forces between current-carrying conductors Suggested electric current loops of molecular size are responsible for all magnetic phenomena

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**A Brief History of Magnetism, final**

Faraday and Henry Further connections between electricity and magnetism A changing magnetic field creates an electric field Maxwell A changing electric field produces a magnetic field

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**Electric and Magnetic Fields**

An electric field surrounds any stationary electric charge The region of space surrounding a moving charge includes a magnetic field In addition to the electric field A magnetic field also surrounds any material with permanent magnetism Both fields are vector fields

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**Magnetic Poles Every magnet, regardless of its shape, has two poles**

Called north and south poles Poles exert forces on one another Similar to the way electric charges exert forces on each other Like poles repel each other N-N or S-S Unlike poles attract each other N-S

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Magnetic Poles, cont The poles received their names due to the way a magnet behaves in the Earth’s magnetic field If a bar magnet is suspended so that it can move freely, it will rotate The magnetic north pole points toward the earth’s north geographic pole This means the earth’s north geographic pole is a magnetic south pole Similarly, the earth’s south geographic pole is a magnetic north pole

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Magnetic Poles, final The force between two poles varies as the inverse square of the distance between them A single magnetic pole has never been isolated In other words, magnetic poles are always found in pairs There is some theoretical basis for the existence of monopoles – single poles

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**Magnetic Fields A vector quantity Symbolized by**

Direction is given by the direction a north pole of a compass needle points in that location Magnetic field lines can be used to show how the field lines, as traced out by a compass, would look

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**Magnetic Field Lines, Bar Magnet Example**

The compass can be used to trace the field lines The lines outside the magnet point from the North pole to the South pole

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**Magnetic Field Lines, Bar Magnet**

Iron filings are used to show the pattern of the magnetic field lines The direction of the field is the direction a north pole would point

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**Magnetic Field Lines, Unlike Poles**

Iron filings are used to show the pattern of the magnetic field lines The direction of the field is the direction a north pole would point Compare to the electric field produced by an electric dipole

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**Magnetic Field Lines, Like Poles**

Iron filings are used to show the pattern of the magnetic field lines The direction of the field is the direction a north pole would point Compare to the electric field produced by like charges

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**Definition of Magnetic Field**

The magnetic field at some point in space can be defined in terms of the magnetic force, The magnetic force will be exerted on a charged particle moving with a velocity,

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**Characteristics of the Magnetic Force**

The magnitude of the force exerted on the particle is proportional to the charge, q, and to the speed, v, of the particle When a charged particle moves parallel to the magnetic field vector, the magnetic force acting on the particle is zero

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**Characteristics of the Magnetic Force, cont**

When the particle’s velocity vector makes any angle q ¹ 0 with the field, the magnetic force acts in a direction perpendicular to both the speed and the field The magnetic force is perpendicular to the plane formed by

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**Characteristics of the Magnetic Force, final**

The force exerted on a negative charge is directed opposite to the force on a positive charge moving in the same direction If the velocity vector makes an angle q with the magnetic field, the magnitude of the force is proportional to sin q

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More About Direction The force is perpendicular to both the field and the velocity Oppositely directed forces exerted on oppositely charged particles will cause the particles to move in opposite directions

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**Force on a Charge Moving in a Magnetic Field, Formula**

The characteristics can be summarized in a vector equation is the magnetic force q is the charge is the velocity of the moving charge is the magnetic field

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**Units of Magnetic Field**

The SI unit of magnetic field is the Tesla (T) The cgs unit is a Gauss (G) 1 T = 104 G

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**Directions – Right Hand Rule #1**

Depends on the right-hand rule for cross products The fingers point in the direction of the velocity The palm faces the field Curl your fingers in the direction of field The thumb points in the direction of the cross product, which is the direction of force For a positive charge, opposite the direction for a negative charge

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**Direction – Right Hand Rule #2**

Alternative to Rule #1 Thumb is the direction of the velocity Fingers are in the direction of the field Palm is in the direction of force On a positive particle Force on a negative charge is opposite You can think of this as your hand pushing the particle

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**More About Magnitude of the Force**

The magnitude of the magnetic force on a charged particle is FB = |q| v B sin q q is the angle between the velocity and the field The force is zero when the velocity and the field are parallel or antiparallel q = 0 or 180o The force is a maximum when the velocity and the field are perpendicular q = 90o

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**Differences Between Electric and Magnetic Fields**

Direction of force The electric force acts parallel or antiparallel to the electric field The magnetic force acts perpendicular to the magnetic field Motion The electric force acts on a charged particle regardless of its velocity The magnetic force acts on a charged particle only when the particle is in motion and the force is proportional to the velocity

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**More Differences Between Electric and Magnetic Fields**

Work The electric force does work in displacing a charged particle The magnetic force associated with a steady magnetic field does no work when a particle is displaced This is because the force is perpendicular to the displacement

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Work in Fields, cont The kinetic energy of a charged particle moving through a constant magnetic field cannot be altered by the magnetic field alone When a charged particle moves with a velocity through a magnetic field, the field can alter the direction of the velocity, but not the speed or the kinetic energy

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**Notation Note The dots indicate the direction is out of the page**

The dots represent the tips of the arrows coming toward you The crosses indicate the direction is into the page The crosses represent the feathered tails of the arrows

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**Charged Particle in a Magnetic Field**

Consider a particle moving in an external magnetic field with its velocity perpendicular to the field The force is always directed toward the center of the circular path The magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particle

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**Force on a Charged Particle**

Using Newton’s Second Law, you can equate the magnetic and centripetal forces: Solving for r: r is proportional to the linear momentum of the particle and inversely proportional to the magnetic field and the charge

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**More About Motion of Charged Particle**

The angular speed of the particle is The angular speed, w, is also referred to as the cyclotron frequency The period of the motion is

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**Motion of a Particle, General**

If a charged particle moves in a magnetic field at some arbitrary angle with respect to the field, its path is a helix Same equations apply, with

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**Bending of an Electron Beam**

Electrons are accelerated from rest through a potential difference Conservation of Energy will give v Other parameters can be found

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**Charged Particle Moving in Electric and Magnetic Fields**

In many applications, the charged particle will move in the presence of both magnetic and electric fields In that case, the total force is the sum of the forces due to the individual fields In general: This force is called the Lorenz force It is the vector sum of the electric force and the magnetic force

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Velocity Selector Used when all the particles need to move with the same velocity A uniform electric field is perpendicular to a uniform magnetic field

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**Velocity Selector, cont**

When the force due to the electric field is equal but opposite to the force due to the magnetic field, the particle moves in a straight line This occurs for velocities of value v = E / B

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**Velocity Selector, final**

Only those particles with the given speed will pass through the two fields undeflected The magnetic force exerted on particles moving at speed greater than this is stronger than the electric field and the particles will be deflected upward Those moving more slowly will be deflected downward

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Mass Spectrometer A mass spectrometer separates ions according to their mass-to-charge ratio A beam of ions passes through a velocity selector and enters a second magnetic field

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**Mass Spectrometer, cont**

After entering the second magnetic field, the ions move in a semicircle of radius r before striking a detector at P If the ions are positively charged, they deflect upward If the ions are negatively charged, they deflect downward This version is known as the Bainbridge Mass Spectrometer

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**Mass Spectrometer, final**

Analyzing the forces on the particles in the mass spectrometer gives Typically, ions with the same charge are used and the mass is measured

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**Thomson’s e/m Experiment**

Electrons are accelerated from the cathode They are deflected by electric and magnetic fields The beam of electrons strikes a fluorescent screen

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**Thomson’s e/m Experiment, cont**

Thomson’s variation found e/me by measuring the deflection of the beam and the fields This experiment was crucial in the discovery of the electron

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Cyclotron A cyclotron is a device that can accelerate charged particles to very high speeds The energetic particles produced are used to bombard atomic nuclei and thereby produce reactions These reactions can be analyzed by researchers

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**Cyclotron, 2 D1 and D2 are called dees because of their shape**

A high frequency alternating potential is applied to the dees A uniform magnetic field is perpendicular to them

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Cyclotron, 3 A positive ion is released near the center and moves in a semicircular path The potential difference is adjusted so that the polarity of the dees is reversed in the same time interval as the particle travels around one dee This ensures the kinetic energy of the particle increases each trip

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Cyclotron, final The cyclotron’s operation is based on the fact that T is independent of the speed of the particles and of the radius of their path When the energy of the ions in a cyclotron exceeds about 20 MeV, relativistic effects come into play

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**First Cyclotron Invented by E. O. Lawrence and M. S. Livingston**

Invented in 1934

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**Force on a Current-Carrying Conductor**

A current carrying conductor experiences a force when placed in an external magnetic field The current represents a collection of many charged particles in motion The resultant magnetic force on the wire is due to the sum of the magnetic forces on the charged particles

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Force on a Wire In this case, there is no current, so there is no force Therefore, the wire remains vertical

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**Force on a Wire,cont The magnetic field is into the page**

The current is upward, along the page The force is to the left

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**Force on a Wire, final The field is into the page**

The current is downward along the page The force is to the right

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**Force on a Wire, equation**

The magnetic force is exerted on each moving charge in the wire The total force is the product of the force on one charge and the number of charges

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Force on a Wire, cont In terms of the current, this becomes l is a vector that points in the direction of the current Its magnitude is the length of the segment This applies only to a straight segment of wire in a uniform external magnetic field

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**Force on a Wire, Arbitrary Shape**

Consider a small segment of the wire, The force exerted on this segment is The total force is

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**Torque on a Current Loop**

The rectangular loop carries a current I in a uniform magnetic field No magnetic force acts on sides & The wires are parallel to the field and cross product is zero

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**Torque on a Current Loop, 2**

There is a force on sides & These sides are perpendicular to the field The magnitude of the magnetic force on these sides will be: F2 = F4 = I a B The direction of F2 is out of the page The direction of F4 is into the page

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**Torque on a Current Loop, 3**

The forces are equal and in opposite directions, but not along the same line of action The forces produce a torque around point O

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**Torque on a Current Loop, Equation**

The maximum torque is found by: The area enclosed by the loop is ab, so tmax = I A B This maximum value occurs only when the field is parallel to the plane of the loop

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**Torque on a Current Loop, General**

Assume the magnetic field makes an angle of q<90o with a line perpendicular to the plane of the loop The net torque about point O will be t = I A B sin q

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**Torque on a Current Loop, Summary**

The torque has a maximum value when the field is perpendicular to the normal to the plane of the loop The torque is zero when the field is parallel to the normal to the plane of the loop where A is perpendicular to the plane of the loop and has a magnitude equal to the area of the loop

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Direction of A The right-hand rule can be used to determine the direction of Curl your fingers in the direction of the current in the loop Your thumb points in the direction of

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**Magnetic Dipole Moment**

The product I is defined as the magnetic dipole moment, of the loop Often called the magnetic moment SI units: A m2 Torque in terms of magnetic moment:

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**Biot-Savart Law – Introduction**

Biot and Savart conducted experiments on the force exerted by an electric current on a nearby magnet They arrived at a mathematical expression that gives an expression for the magnetic field at some point in space due to a current

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**Biot-Savart Law – Set-Up**

The magnetic field is at some point P The length element is The wire is carrying a steady current of I

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**Biot-Savart Law – Observations**

The vector is perpendicular to both ds and to the unit vector directed from toward P The magnitude of is inversely proportional to r2, where r is the distance from to P

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**Biot-Savart Law – Observations, cont**

The magnitude of is proportional to the current and to the magnitude ds of the length element ds The magnitude of is proportional to sin q, where q is the angle between the vectors and

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**Biot-Savart Law, Equation**

The observations are summarized in the mathematical equation called Biot-Savart Law: The Biot-Savart law gives the magnetic field only for a small length of the conductor

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**Permeability of Free Space**

The constant mo is called the permeability of free space mo = 4 p x 10-7 T. m / A The Biot-Savart Law can be written as

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Total Magnetic Field To find the total field, you need to sum up the contributions from all the current elements You need to evaluate the field by integrating over the entire current distribution The magnitude of the field will be

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**B Compared to E Distance**

The magnitude of the magnetic field varies as the inverse square of the distance from the source The electric field due to a point charge also varies as the inverse square of the distance from the charge

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**B Compared to E, 2 Direction**

The electric field created by a point charge is radial in direction The magnetic field created by a current element is perpendicular to both the length element and the unit vector

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B Compared to E, 3 Source An electric field is established by an isolated electric charge The current element that produces a magnetic field must be part of an extended current distribution Therefore you must integrate over the entire current distribution

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**B for a Long, Straight Conductor, Direction**

The magnetic field lines are circles concentric with the wire The field lines lie in planes perpendicular to to wire The magnitude of the field is constant on any circle of radius a The right hand rule for determining the direction of the field is shown

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**B for a Circular Current Loop**

The loop has a radius of R and carries a steady current of I Find at point P

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**Field at the Center of a Loop**

Consider the field at the center of the current loop At this special point, x = 0 Then,

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**Magnetic Field Lines for a Loop**

Figure a shows the magnetic field lines surrounding a current loop Figure b shows the field lines in the iron filings Figure c compares the field lines to that of a bar magnet

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**Magnetic Force Between Two Parallel Conductors**

Two parallel wires each carry a steady current The field due to the current in wire 2 exerts a force on wire 1 of F1 = I1l B2

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**Magnetic Force Between Two Parallel Conductors, cont**

Substituting the equation for B2 gives Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying current in opposite directions repel each other

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**Magnetic Force Between Two Parallel Conductors, final**

The result is often expressed as the magnetic force between the two wires, FB This can also be given as the force per unit length, FB/l

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**Definition of the Ampere**

The force between two parallel wires can be used to define the ampere When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

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**Definition of the Coulomb**

The SI unit of charge, the coulomb, is defined in terms of the ampere When a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C

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**Magnetic Field of a Wire**

A compass can be used to detect the magnetic field When there is no current in the wire, there is no field due to the current The compass needles all point toward the earth’s north pole Due to the earth’s magnetic field

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**Magnetic Field of a Wire, 2**

The wire carries a strong current The compass needles deflect in a direction tangent to the circle This shows the direction of the magnetic field produced by the wire

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**Magnetic Field of a Wire, 3**

The circular magnetic field around the wire is shown by the iron filings

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André-Marie Ampère 1775 –1836 Credited with the discovery of electromagnetism The relationship between electric currents and magnetic fields Died of pneumonia

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Ampere’s Law The product of can be evaluated for small length elements on the circular path defined by the compass needles for the long straight wire Ampere’s Law states that the line integral of around any closed path equals moI where I is the total steady current passing through any surface bounded by the closed path

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Ampere’s Law, cont Ampere’s Law describes the creation of magnetic fields by all continuous current configurations Most useful for this course if the current configuration has a high degree of symmetry Put the thumb of your right hand in the direction of the current through the amperian loop and your figures curl in the direction you should integrate around the loop

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Amperian Loops Each portion of the path satisfies one or more of the following conditions: The value of the magnetic field can be argued by symmetry to be constant over the portion of the path The dot product can be expressed as a simple algebraic product B ds The vectors are parallel

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**Amperian Loops, cont Conditions: The dot product is zero**

The vectors are perpendicular The magnetic field can be argued to be zero at all points on the portion of the path

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**Field Due to a Long Straight Wire – From Ampere’s Law**

Want to calculate the magnetic field at a distance r from the center of a wire carrying a steady current I The current is uniformly distributed through the cross section of the wire

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**Field Due to a Long Straight Wire – Results From Ampere’s Law**

Outside of the wire, r > R Inside the wire, we need I’, the current inside the amperian circle

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**Field Due to a Long Straight Wire – Results Summary**

The field is proportional to r inside the wire The field varies as 1/r outside the wire Both equations are equal at r = R

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**Magnetic Field of a Toroid**

Find the field at a point at distance r from the center of the toroid The toroid has N turns of wire

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**Magnetic Field of a Solenoid**

A solenoid is a long wire wound in the form of a helix A reasonably uniform magnetic field can be produced in the space surrounded by the turns of the wire Each of the turns can be modeled as a circular loop The net magnetic field is the vector sum of all the fields due to all the turns

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**Magnetic Field of a Solenoid, Description**

The field lines in the interior are Approximately parallel to each other Uniformly distributed Close together This indicates the field is strong and almost uniform

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**Magnetic Field of a Tightly Wound Solenoid**

The field distribution is similar to that of a bar magnet As the length of the solenoid increases The interior field becomes more uniform The exterior field becomes weaker

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**Ideal Solenoid – Characteristics**

An ideal solenoid is approached when The turns are closely spaced The length is much greater than the radius of the turns For an ideal solenoid, the field outside of solenoid is negligible The field inside is uniform

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**Ampere’s Law Applied to a Solenoid**

Ampere’s Law can be used to find the interior magnetic field of the solenoid Consider a rectangle with side l parallel to the interior field and side w perpendicular to the field The side of length l inside the solenoid contributes to the field This is path 1 in the diagram

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**Ampere’s Law Applied to a Solenoid, cont**

Applying Ampere’s Law gives The total current through the rectangular path equals the current through each turn multiplied by the number of turns

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**Magnetic Field of a Solenoid, final**

Solving Ampere’s Law for the magnetic field is n = N / l is the number of turns per unit length This is valid only at points near the center of a very long solenoid

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**Magnetic Moment – Bohr Atom**

The electrons move in circular orbits The orbiting electron constitutes a tiny current loop The magnetic moment of the electron is associated with this orbital motion The angular momentum and magnetic moment are in opposite directions due to the electron’s negative charge

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**Magnetic Moments of Multiple Electrons**

In most substances, the magnetic moment of one electron is canceled by that of another electron orbiting in the opposite direction The net result is that the magnetic effect produced by the orbital motion of the electrons is either zero or very small

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Electron Spin Electrons (and other particles) have an intrinsic property called spin that also contributes to its magnetic moment The electron is not physically spinning It has an intrinsic angular momentum as if it were spinning Spin angular momentum is actually a relativistic effect

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**Electron Magnetic Moment, final**

In atoms with multiple electrons, many electrons are paired up with their spins in opposite directions The spin magnetic moments cancel Those with an “odd” electron will have a net moment Some moments are given in the table

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**Ferromagnetic Materials**

Some examples of ferromagnetic materials are Iron Cobalt Nickel Gadolinium Dysprosium They contain permanent atomic magnetic moments that tend to align parallel to each other even in a weak external magnetic field

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Domains All ferromagnetic materials are made up of microscopic regions called domains The domain is an area within which all magnetic moments are aligned The boundaries between various domains having different orientations are called domain walls

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**Domains, Unmagnetized Material**

The magnetic moments in the domains are randomly aligned The net magnetic moment is zero

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**Domains, External Field Applied**

A sample is placed in an external magnetic field The size of the domains with magnetic moments aligned with the field grows The sample is magnetized

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**Domains, External Field Applied, cont**

The material is placed in a stronger field The domains not aligned with the field become very small When the external field is removed, the material may retain most of its magnetism

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Magnetic Levitation The Electromagnetic System (EMS) is one design model for magnetic levitation The magnets supporting the vehicle are located below the track because the attractive force between these magnets and those in the track lift the vehicle

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EMS, cont The proximity detector uses magnetic induction to measure the magnet-rail separation The power supply is adjusted to maintain proper separation

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**EMS, final Disadvantages Advantage**

Instability caused by the variation of magnetic force with distance Compensated for by the proximity detector Relatively small separation between the magnets and the tracks Usually about 10 mm Track needs high maintenance Advantage Independent of speed, so wheels are not needed Wheels are in place for “emergency landing” system

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**German Transrapid – EMS Example**

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22.7 Source of magnetic field due to current

22.7 Source of magnetic field due to current

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