Chapter 26 Magnetic Fields Magnets In each magnet

Chapter 26 Magnetic Fields

Magnets • In each magnet there are two poles present (the ends where objects are most strongly attracted): north and south • Like (unlike) poles repel (attract) each other (similar to electric charges), and the force between two poles varies as the inverse square of the distance between them • Magnetic poles cannot be isolated – if a permanent magnetic is cut in half, you will still have a north and a south pole (unlike electric charges) • There is some theoretical basis for monopoles, but none have been detected

Magnets • 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

Magnets • An unmagnetized piece of iron can be magnetized by stroking it with a magnet (like stroking an object to charge an object) • Magnetism can be induced – if a piece of iron, for example, is placed near a strong permanent magnet, it will become magnetized • Soft magnetic materials (such as iron) are easily magnetized and also tend to lose their magnetism easily • Hard magnetic materials (such as cobalt and nickel) are difficult to magnetize and they tend to retain their magnetism

Magnetic Fields • The region of space surrounding a moving charge includes a magnetic field (the charge will also be surrounded by an electric field) • A magnetic field surrounds a properly magnetized magnetic material • A magnetic field is a vector quantity symbolized by B • Its direction is given by the direction a north pole of a compass needle pointing in that location • Magnetic field lines can be used to show the field lines, as traced out by a compass, would look

Magnetic Field Lines • A compass can be used to show the direction of the magnetic field lines

Magnetic Field Lines • Iron filings can also be used to show the pattern of the magnetic field lines • The direction of the field is the direction a north pole would point • Unlike poles (compare to the electric field produced by an electric dipole)

Magnetic Field Lines • Iron filings can also be used to show the pattern of the magnetic field lines • The direction of the field is the direction a north pole would point • Unlike poles (compare to the electric field produced by an electric dipole) • Like poles (compare to the electric field produced by like charges)

Magnetic Fields • When moving through a magnetic field, a charged Nikola Tesla particle experiences a magnetic force 1856 – 1943 • This force has a maximum (zero) value when the charge moves perpendicularly to (along) the magnetic field lines • Magnetic field is defined in terms of the magnetic force exerted on a test charge moving in the field with velocity v • The SI unit: Tesla (T)

Magnetic Fields • Conventional laboratory magnets: ~ 2. 5 T • Superconducting magnets ~ 30 T • Earth’s magnetic field ~ 5 x 10 -5 T

Direction of Magnetic Force • Experiments show that the direction of the magnetic force is always perpendicular to both v and B • Fmax occurs when v is perpendicular to B and F = 0 when v is parallel to B • Right Hand Rule #1 (for a + charge): Place your fingers in the direction of v and curl the fingers in the direction of B – your thumb points in the direction of F • If the charge is negative, the force points in the opposite direction

Direction of Magnetic Force • The x’s indicate the magnetic field when it is directed into the page (the x represents the tail of the arrow) • The dots would be used to represent the field directed out of the page (the • represents the head of the arrow)

Differences Between Electric and Magnetic Fields • The electric force acts along the direction of the electric field, whereas the magnetic force acts perpendicular to the magnetic field • The electric force acts on a charged particle regardless of whether the particle is moving, while the magnetic force acts on a charged particle only when the particle is in motion • The electric force does work in displacing a charged particle, whereas the magnetic force associated with a steady magnetic field does no work when a particle is displaced (because the force is perpendicular to the displacement)

Force on a Charged Particle in a Magnetic Field • Consider a particle moving in an external magnetic field so that its velocity is 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

Force on a Charged Particle in a Magnetic Field • This expression is known as the cyclotron equation • r is proportional to the momentum of the particle and inversely proportional to the magnetic field • If the particle’s velocity is not perpendicular to the field, the path followed by the particle is a spiral (helix)

Particle in a Nonuniform Magnetic Field • The motion is complex

Charged Particles Moving in Electric and Magnetic Fields • In many applications, charged particles 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:

Chapter 26 Problem 23 Microwaves in a microwave oven are produced by electrons circling in a magnetic field at a frequency of 2. 4 GHz. (a) What’s the magnetic field strength? (b) The electrons’ motion takes place inside a special tube called a magnetron. If the magnetron can accommodate electron orbits with maximum diameter 2. 5 mm, what’s the maximum electron energy?

Magnetic Force on a Current Carrying Wire • The current is a collection of many charged particles in motion • The magnetic force is exerted on each moving charge in the wire • The total force is the sum of all the magnetic forces on all the individual charges producing the current • Therefore a force is exerted on a currentcarrying wire placed in a magnetic field:

Magnetic Force on a Current Carrying Wire • The direction of the force is given by right hand rule #1, placing your fingers in the direction of I instead of v

Magnetic Force on a Current Carrying Wire of an Arbitrary Shape • For a small segment of the wire, the force exerted on this segment is • The total force is

Chapter 26 Problem 28 A wire with mass per unit length 75 g/m runs horizontally at right angles to a horizontal magnetic field. A 6. 2 -A current in the wire results in its being suspended against gravity. What’s the magnetic field strength?

Biot-Savart Law • Biot and Savart arrived at a mathematical expression that gives the magnetic field at some point in space due to a current • The magnetic field is d. B at some point P; the length element is ds; the wire is carrying a steady current of I Jean-Baptiste Biot 1774 – 1862 Félix Savart 1791 – 1841

Biot-Savart Law • Vector d. B is perpendicular to both ds and to the unit vector directed from ds toward P • The magnitude of d. B is inversely proportional to r 2, where r is the distance from ds to P • The magnitude of d. B is proportional to the current and to the magnitude ds of the length element

Biot-Savart Law • The magnitude of d. B is proportional to sinq, where q is the angle between the vectors ds and • The observations are summarized in the mathematical equation called the Biot. Savart law (magnetic field due to the current -carrying conductor): • µo = 4 x 10 -7 T. m / A: permeability of free space

Biot-Savart Law • To find the total field, sum up the contributions from all the current elements

Biot-Savart Law • The magnitude of the magnetic field varies as the inverse square of the distance from the ds element • The electric field due to a point charge also varies as the inverse square of the distance from the charge • 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 • The current element producing a magnetic field is part of an extended current distribution

A Long, Straight Conductor • The thin, straight wire is carrying a constant current

A Long, Straight Conductor • The thin, straight wire is carrying a constant current • If the conductor is an infinitely long, straight wire, θ 1 = π/2 and θ 2 = – π/2 , and the field becomes

A Long, Straight Conductor • The magnetic field lines are circles concentric with the wire • The field lines lie in planes perpendicular to the wire • The magnitude of the field is constant on any circle of radius a • Right Hand Rule #2: Grasp the wire in your right hand point your thumb in the direction of the current and your fingers will curl in the direction of the field

A Curved Wire Segment • Find the field at point O due to the wire segment (I, a are constants) • The field at the center of the full circle loop

Magnetic Field of a Current Loop

Magnetic Field of a Current Loop • The field contribution from a current element I dl = I dx • For large distances (x >> a), this reduces to

Chapter 26 Problem 30 A single-turn wire loop is 2. 0 cm in diameter and carries a 650 -m. A current. Find the magnetic field strength (a) at the loop center and (b) on the loop axis, 20 cm from the center.

Torque on a Current Loop

Torque on a Current Loop • Applies to any shape loop • Torque has a maximum value when q = 90° • Torque is zero when the field is perpendicular to the plane of the loop

Magnetic Moment • The vector is called the magnetic dipole moment of the coil • Its magnitude is given by μ = IAN • The vector always points perpendicular to the plane of the loop(s) • The equation for the magnetic torque can be written as τ = BIAN sinθ = μB sinθ • The angle is between the moment and the field

Potential Energy • The potential energy of the system of a magnetic dipole in a magnetic field depends on the orientation of the dipole in the magnetic field • Umin = – μB and occurs when the dipole moment is in the same direction as the field • Umax = + μB and occurs when the dipole moment is in the direction opposite the field

Chapter 26 Problem 35 A single-turn square wire loop 5. 0 cm on a side carries a 450 -m. A current. (a) What’s the loop’s magnetic dipole moment? (b) If the loop is in a uniform 1. 4 -T magnetic field with its dipole moment vector at 40° to the field, what’s the magnitude of the torque it experiences?

Electric Motor • An electric motor converts electrical energy to mechanical energy (rotational kinetic energy) • An electric motor consists of a rigid current-carrying loop that rotates when placed in a magnetic field • The torque acting on the loop will tend to rotate the loop to smaller values of θ until the torque becomes 0 at θ = 0°

Electric Motor • If the loop turns past this point and the current remains in the same direction, the torque reverses and turns the loop in the opposite direction • To provide continuous rotation in one direction, the current in the loop must periodically reverse • In ac motors, this reversal naturally occurs • In dc motors, a split-ring commutator and brushes are used

Electric Motor • Just as the loop becomes perpendicular to the magnetic field and the torque becomes 0, inertia carries the loop forward and the brushes cross the gaps in the ring, causing the current loop to reverse its direction • This provides more torque to continue the rotation • The process repeats itself • Actual motors would contain many current loops and commutators

Magnetic Force Between Two Parallel Conductors

Magnetic Force Between Two Parallel Conductors • The force (per unit length) on wire 1 due to the current in wire 1 and the magnetic field produced by wire 2: • Parallel conductors carrying currents in the same direction attract each other • Parallel conductors carrying currents in the opposite directions repel each other

Chapter 26 Problem 63 A long, straight wire carries 20 A. A 5. 0 -cm by 10 -cm rectangular wire loop carrying 500 m. A is 2. 0 cm from the wire, as shown in the figure. Find the net magnetic force on the loop.

Ampère’s Law • Ampère’s Circuital Law: a procedure for deriving the relationship between the current in an arbitrarily shaped wire and the magnetic field produced by the wire • Choose an arbitrary closed path around the current and sum all the products of B|| Δℓ around the closed path (put the thumb of your right hand in the direction of the current through the loop and your fingers curl in the direction you should integrate around the loop)

Ampère’s Law for a Long Straight Wire • Use a closed circular path • The circumference of the circle is 2 r

Ampère’s Law for a Long Straight Wire

Magnetic Field of a Solenoid

Magnetic Field of a Solenoid • If a long straight wire is bent into a coil of several closely spaced loops, the resulting device is called a solenoid • It is also known as an electromagnet since it acts like a magnet only when it carries a current • The field inside the solenoid is nearly uniform and strong – the field lines are nearly parallel, uniformly spaced, and close together • The exterior field is nonuniform, much weaker, and in the opposite direction to the field inside the solenoid

Magnetic Field of a Solenoid • The field lines of the solenoid resemble those of a bar magnet • The magnitude of the field inside a solenoid is approximately constant at all points far from its ends B = µo n I • n = N / ℓ : the number of turns per unit length • This result can be obtained by applying Ampère’s Law to the solenoid

Magnetic Field of a Solenoid • A cross-sectional view of a tightly wound solenoid • If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero • Apply Ampère’s Law to the blue dashed rectangle

Magnetic Effects of Electrons – Orbits • An individual atom should act like a magnet because of the motion of the electrons about the nucleus • Each electron circles the atom once in about every 10 -16 seconds; this would produce a current of 1. 6 m. A and a magnetic field of about 20 T at the center of the circular path • However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom • The net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials

Magnetic Effects of Electrons – Spins • Electrons also have spin (it is a quantum effect) • The classical model is to consider the electrons to spin like tops • The field due to the spinning is generally stronger than the field due to the orbital motion • Electrons usually pair up with their spins opposite each other, so their fields cancel each other, hence most materials are not naturally magnetic

Magnetic Effects of Electrons – Domains • In some materials – ferromagnetic – the spins do not naturally cancel • Large groups of atoms in which the spins are aligned are called domains • When an external field is applied, it causes the material to become magnetized: the domains that are aligned with the field tend to grow at the expense of the others

Domains and Permanent Magnets • In hard magnetic materials, the domains remain aligned after the external field is removed • The result is a permanent magnet • In soft magnetic materials, once the external field is removed, thermal agitation causes the materials to quickly return to an unmagnetized state • With a core in a loop, the magnetic field is enhanced since the domains in the core material align, increasing the magnetic field

Ferromagnetism • Some substances exhibit strong magnetic effects called ferromagnetism (e. g. , 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

Paramagnetism • Paramagnetic substances have small but positive magnetism, which results from the presence of atoms that have permanent magnetic moments • These moments interact weakly with each other • When placed in an external magnetic field, atomic moments tend to line up with the field and the alignment process competes with thermal motion which randomizes the moment orientations

Diamagnetism • When an external magnetic field is applied to a diamagnetic substance, a weak magnetic moment is induced in the direction opposite the applied field • Diamagnetic substances are weakly repelled by a magnet

Earth’s Magnetic Field • The Earth’s geographic north (south) pole corresponds to a magnetic south (north) pole – a north (south) pole should be a “north- (south-) seeking” pole • The Earth’s magnetic field resembles that achieved by burying a huge bar magnet deep in the Earth’s interior • The most likely source of the Earth’s magnetic field – electric currents in the liquid part of the core

Earth’s Magnetic Field • The magnetic and geographic poles are not in the same exact location – magnetic declination is the difference between true north (geographic north pole) and magnetic north pole • The amount of declination varies by location on the earth’s surface • The direction of the Earth’s magnetic field reverses every few million years (the origin of these reversals is not understood)

Earth’s Magnetic Field • If a compass is free to rotate vertically as well as horizontally, it points to the earth’s surface • The angle between the horizontal and the direction of the magnetic field is called the dip angle • The farther north the device is moved, the farther from horizontal the compass needle would be • The compass needle would be horizontal at the equator and the dip angle would be 0° • The compass needle would point straight down at the south magnetic pole and the dip angle would be 90°

Magnetic Flux • Magnetic flux associated with a magnetic field is defined in a way similar to electric flux • SI unit of flux: Weber • Wb = T. m² Wilhelm Eduard Weber 1804 – 1891

Magnetic Flux • For a flat surface with an area A in a uniform magnetic field, the flux is (θ is the angle between B and the normal to the plane): ΦB = B A cos θ • When the field is perpendicular to the plane, θ = 0 and ΦB = ΦB, max = BA • When the field is parallel to the plane, θ = 90° and ΦB = 0 • The flux can be negative, for example if θ = 180°

Magnetic Flux • The value of the magnetic flux is proportional to the total number of magnetic field lines passing through area • When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum • When the area is parallel to the lines, no lines pass through the area and the flux is 0

Gauss’ Law in Magnetism • Magnetic fields do not begin or end at any point • The number of lines entering a surface equals the number of lines leaving the surface • Gauss’ law in magnetism says the magnetic flux through any closed surface is always zero:

Answers to Even Numbered Problems Chapter 26: Problem 16 (a) 3. 4 × 105 m/s (b) does not change

Answers to Even Numbered Problems Chapter 26: Problem 20 3. 9 mm

Answers to Even Numbered Problems Chapter 26: Problem 32 4. 0 A

Answers to Even Numbered Problems Chapter 26: Problem 36 480 m. T

Answers to Even Numbered Problems Chapter 26: Problem 38 24 A

Answers to Even Numbered Problems Chapter 26: Problem 42 (a) (− 1. 1 iˆ + 1. 5 ˆj + 1. 7 kˆ) × 10− 3 N (b) 0

Answers to Even Numbered Problems Chapter 26: Problem 58 10 m