Chapter 19 Magnetism

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) Magnetic poles cannot be isolated – if a permanent magnet 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

Magnetism 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 how 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)

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 in directions to the geographic north pole and the magnetic south 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 Fields Nikola Tesla 1856 – 1943 When moving through a magnetic field, a charged particle experiences a magnetic force 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 blue x’s indicate the magnetic field when it is directed into the page (the x represents the tail of the arrow) Blue dots would be used to represent the field directed out of the page (the • represents the head of the arrow)

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)

Chapter 19 Problem 6 A proton moves perpendicularly to a uniform magnetic field at 1.0 × 107 m/s and exhibits an acceleration of 2.0 × 1013 m/s2 in the +x-direction when its velocity is in the +z-direction. Determine the magnitude and direction of the field.

Chapter 19 Problem 40 A particle with charge q and kinetic energy KE travels in a uniform magnetic field of magnitude B. If the particle moves in a circular path of radius R, find expressions for (a) its speed and (b) its mass.

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 current-carrying 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

Chapter 19 Problem 21 Consider the system pictured in the figure. A 15-cm length of conductor of mass 15 g, free to move vertically, is placed between two thin, vertical conductors, and a uniform magnetic field acts perpendicular to the page. When a 5.0-A current is directed as shown in the figure, the horizontal wire moves upward at constant velocity in the presence of gravity. (a) What forces act on the horizontal wire, and under what condition is the wire able to move upward at constant velocity? (b) Find the magnitude and direction of the minimum magnetic field required to move the wire at constant speed. (c) What happens if the magnetic field exceeds this minimum value? (The wire slides without friction on the two vertical conductors.)

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 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

Chapter 19 Problem 30 A copper wire is 8.00 m long and has a cross-sectional area of 1.00 × 10−4 m2. The wire forms a one-turn loop in the shape of square and is then connected to a battery that applies a potential difference of 0.100 V. If the loop is placed in a uniform magnetic field of magnitude 0.400 T, what is the maximum torque that can act on it? The resistivity of copper is 1.70 × 10−8 Ω · m.

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 Fields – Long Straight Wire A current-carrying wire produces a magnetic field The compass needle points in the direction of the magnetic field produced by the current (tangential to the circle) Right Hand Rule #2: Grasp the wire in your right hand and point your thumb in the direction of the current Your fingers will curl in the direction of the field

Magnetic Fields – Long Straight Wire The magnitude of the field at a distance r from a wire carrying a current of I is µo = 4  x 10-7 T.m / A: permeability of free space

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  B|| Δℓ = µo I

Ampère’s Law for a Long Straight Wire Use a closed circular path The circumference of the circle is 2 r  B|| Δℓ = µo I B  Δℓ = B 2 r = µo I

Chapter 19 Problem 51 A wire carries a 7.00-A current along the x-axis, and another wire carries a 6.00-A current along the y-axis, as shown in the figure. What is the magnetic field at point P, located at x = 4.00 m, y = 3.00 m?

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 19 Problem 57 A wire with a weight per unit length of 0.080 N/m is suspended directly above a second wire. The top wire carries a current of 30.0 A and the bottom wire carries a current of 60.0 A. Find the distance of separation between the wires so that the top wire will be held in place by magnetic repulsion.

Magnetic Field of a Current Loop The strength of a magnetic field produced by a wire can be enhanced by forming the wire into a loop All the segments, Δx, contribute to the field, increasing its strength

Magnetic Field of a Current Loop The magnitude of the magnetic field at the center of a circular loop with a radius R and carrying current I is With N loops in the coil, this becomes

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 The same result can be obtained by applying Ampère’s Law to the solenoid

Magnetic Field of a Solenoid A cross-sectional view of a long 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 mA 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

Answers to Even Numbered Problems Chapter 19: Problem 2 toward the left; into the page; out of the page; toward top of page; into the page; out of the page the answers for part (b) are reversed from those given in part (a)

Answers to Even Numbered Problems Chapter 19: Problem 26 4.33 × 10−3 N m

Answers to Even Numbered Problems Chapter 19: Problem 44 toward the left out of the page lower left to upper right

Answers to Even Numbered Problems Chapter 19: Problem 48 40.0 μT into the page 5.00 μT out of the page 1.67 μT out of the page