Magnetic Forces and Fields (Chapters 29-30) Magnetism Magnetic Materials and Sources Magnetic Field, Magnetic Force Force on Moving Electric Charges Lorentz Force Force on Current Carrying Wires Applications Electromagnetic motors Torques on magnetic dipole moments μ Sources of Magnetic Field Magnetic field of a moving charge Current Carrying Wires iot-savart law Loops, Coils and Solenoids Ampère s Law Microscopic Nature of Magnetism
Magnetism Magnets and relation to electricity Magnets are objects exhibiting magnetic behavior or magnetism A magnet exhibits the strongest magnetism at extremities called magnetic poles: any magnet has two poles, conventionally dubbed north and south Like poles repel each other and unlike poles attract each other Unlike charges, magnetic poles cannot be isolated into monopoles: if a permanent magnetic is cut in half repeatedly, the parts will still have a north and a south pole How can one magnetize objects? Magnetism can be induced: either by stroking an unmagnetized piece of magnetizable material with a magnet, or by placing it near a strong permanent magnet Soft magnetic materials, such as iron, are easily magnetized - They also tend to lose their magnetism easily Hard magnetic materials, such as cobalt and nickel, are difficult to magnetize - They tend to retain their magnetism The region of space surrounding a moving charge includes a magnetic field as well as by an electric field: so, magnetism and electricity cannot be separated They are interrelated into the integrated field of electromagnetism: the first breakthrough in the great effort of developing unified theories about fundamental interactions (Maxwell, beginning of the XX-th century).
Magnetic Field Operational definition and field lines Like the sources of electric field, any magnetic material produces a magnetic field that surrounds it and extends to infinity. The symbol used to represent this vector is Let s first describe this vector using an operational definition: Def: The magnetic field in each location in the surroundings of a magnetic source is a vector with the direction given by the direction of the north pole of a compass needle placed in the respective location Similar to the electric field, a magnetic field can be patterned using field lines: the vector in a point is tangent to the line passing through that point, and the density of lines represents the strength of the field However, while electric field lines start and end on electric charges (electric monopoles), the magnetic field lines form closed loops (since there are no magnetic monopoles) Thus, the magnetic field lines should be seen as closing the loops through the body of the magnet: that is, the magnetic field inside magnets is not zero Ex: A compass can be used to trace the magnetic field lines
Magnetic Field Magnetic field lines for various magnetic sources A compass can be used to probe the magnetic field lines produced by various source, and they will always form closed loops Later we ll look at these sources more systematically: Notice the similar pattern
Magnetic Field Example: Earth s Magnetic Field The Earth s geographic north pole is closed to a (slowly migrating) magnetic south pole The Earth s magnetic field resembles that of a huge bar magnet deep in the Earth s interior slightly tilted with respect to the axis of rotation of the planet The mechanism of Earth s magnetism is not very well understood There cannot be large masses of permanently magnetized materials since the high temperatures of the core prevent materials from retaining permanent magnetization The most likely source is believed to be electric currents in the liquid part of the planetary core The direction of the Earth s magnetic field reverses every few million years The origin of the reversals is not well understood in detail, albeit there are models describing how it may happen
Magnetic Force On a moving charge Magnetic fields act on moving charges with magnetic forces. We ll study this effect in two (related) cases: 1. Moving Charged Particles 2. Current Carrying Wires 1. Magnetic Force on a Moving Point Charge Consider a test charge q moving in a field with velocity v making an angle θ with : the particle will be acted by a magnetic force F (sometimes called Lorentz force) of Magnitude: F qv F qvsin Direction: given by a right hand rule (let s call it #1): The expression for the magnetic force leads to a definition for magnetic field unit called Tesla (T) N Tesla (T) S C m s A popular alternative is the cgs unit, Gauss (G) (useful for small fields): 1T = 10 4 Gauss v v θ F q>0 F q>0 F q<0
Notation: Vectors perpendicular on page/board/slide: Outward nward Exercises: 1. Force direction: Find the direction of the force on an electron moving through the magnetic fields represented below. 2. Field direction: Find the direction of the magnetic field acting on a proton moving as represented by the adjacent velocity and force vectors. (Assume that the velocity is perpendicular on the magnetic field.) Problem: 1. Charge moving in a magnetic field: What velocity would a proton need to circle Earth 800 km above the magnetic equator, where Earth's magnetic field is directed horizontally north and has a magnitude of 4.0010-8 T?
Magnetic Force Charge in an electromagnetic field Any moving charge not only that is acted by a magnetic field but it also produces a magnetic field that surrounds it and extends to infinity A test charge q moving in an electric field E and a magnetic field, with velocity making an angle θ with will be acted by a net electromagnetic force (sometimes called Lorentz force): F F F q E v electric magnetic parallel to the direction of E perpendicular on the direction of + Ex: One type of velocity selector Consider an electric field perpendicular on a magnetic field Then only the particles entering the fields with velocity perpendicular of both will be allowed to pass, which corresponds to the following condition that the particles are supposed to obey: E qe qv 0 v +
Magnetic Force Trajectory of a point charge in a magnetic field Let s look at two particular trajectories that a charged particle may have in a magnetic field 1. Consider a particle moving into an external magnetic field so that its velocity is perpendicular to the field n this case, the particle will move in a circle, with the magnetic force always directed toward the center of the circular path Equating the magnetic and centripetal forces, we can find the radius of the circle: 2 v F qv m r r mv q 2. f the particle s velocity is not perpendicular to the field, the path followed by the particle is a spiral called a helix The helix spirals along the direction of the field with a velocity given by the component of the velocity parallel with + : called cyclotron equation v + F + F v F + v v
Magnetic Force Currents in magnetic field A current is a collection of many drifting charged particles, such that a magnetic force is expected to act on a current-carrying wire placed in a magnetic field This magnetic force is the resultant of the forces acted on the individual microscopic electric carriers, but it makes more sense to integrate its effects into a unique magnetic force acted on the macroscopic current = 0 F = 0 F F Ex: Experimental observations: A current carrying vertical wire placed in a magnetic field pointing perpendicular into the slide, will be acted by a magnetic force perpendicular on the current and magnetic field: either to the left, or to the right, depending on the direction of the current
Magnetic Force On a current carrying wire 2. Magnetic Force on Current Carrying Wire Consider a straight current carrying wire of length l immersed in field, making an angle θ with : the portion dl of wire will be acted by a magnetic force df Magnitude: df d df d sin Direction: Given by right hand rule #1, but instead of aligning the fingers with the velocity, one aligns the fingers with the direction of the current Since the current flows in the direction of the positive carriers, the thumbs always indicates the direction of the force f the wire is straight, and the force is the same for each cross-section, the force on a length L of wire is F Lsin θ F F d
Problems: 2. asics of a rail gun: A rail gun looks (very)schematically as in the figure. Evaluate the speed that the projectile of mass m would achieve after traveling from rest a distance d on the rails spaced by l with a driving current with a magnetic field. Rail m Projectile l 3. Force on a semicircular current: A semicircular thin conductor of radius R carries a time dependent current y i i t e 0, where 0 and τ are positive constant. The wire is allowed to move vertically through a uniform magnetic field, as in the figure. Find the acceleration of the conductor as a time dependent function. R R θ x
Applications Torque on a Current Loop The magnetic force can be used to make electromagnetic motors by using it to rotate a current carrying loops n order to see the principles of such an arrangement, consider a loop carrying a current in an external magnetic field The two sides perpendicular on will be acted by forces opposite in direction creating a torque that will rotate the loop: 1 F a 1 1 2 sin 1 2 F2 2 asin We can immediately find an expression for the net torque τ = τ 1 + τ 2 : 1 1 F 2asin F 2asin Fa sin F b angle between the magnetic field and the perpendicular to the surface of the loop ba sin Asin We see that the torque is maximum when the magnetic field is parallel with the plane of area A (θ = 90 ), and zero when is perpendicular on A (θ = 0 ) F F
Magnetic Moment, μ The net torque exerted by a magnetic field on N current carrying loops is NA sin This magnetic torque exerted on the loop of current can be written in terms of a vector quantity called magnetic moment: sin Any loop of electric current can be associated with a magnetic moment pointing perpendicular on the plane of the loop So, we see that a magnetic dipole in a magnetic field will have the tendency to rotate either in a position with μ parallel with stable equilibrium or anti-parallel with unstable equilibrium The current doesn t have to be carried by a wire: any closed loop of moving charges will have a moment: as we shall see later, these moments explain magnetism at a microscopic scale Ex: Electrons in an atom have an orbital moment due to the their orbital motion about the nucleus A +
Applications Electromagnetic Motors An electric motor converts electrical energy into mechanical energy in the form of rotational kinetic energy As described on the previous slides, the simplest 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 f 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, such that dc-motors must use split-ring commutators and brushes Actual motors would contain many current loops and commutators
Sources of Magnetic Field Moving Charge We ve seen that magnetic fields act on moving charges (point-like and currents), so it is just natural to expect that moving charges also produce magnetic fields: a fact first discovered serendipitously by Hans Oersted in 1819 Consider a point charge moving with constant velocity v: then, the magnetic field at a position r from the particle making an angle θ with v is where µ 0 = 4 10-7 Tm/A is the magnetic permeability of free space Magnitude: ˆ 0 qv r 2 4 r q 0 4 vsin Direction: perpendicular on the plane determined by r and v. Use the following right hand rule (#2): grab the velocity in your right hand with the thumb in its direction. Then the curl of the fingers will show the direction of the field around v: clockwise for positive charges and anticlockwise for negative charges r 2 Weaker field behind + r θ v Larger field in the plane + Weaker fields ahead v
Consider a current carried along a wire. Then, the magnetic field produced by a segment dl of the current at a position r from the segment making an angle φ with dl is given by Magnitude: Sources of Magnetic Field Element of current iot-savart Law: d d 0 d 4 r 0 d 4 0 4 rˆ 2 sin 2 r Direction: perpendicular on the plane determined by r and v. Use the same right hand rule as for moving positively charged particles, but curl your right hand fingers around the current. Hence, for a certain finite length of wire d r rˆ 2 d d d r φ d d d
Problem: 4. Moving charges interacting electrically and magnetically: Two protons move with uniform speed v along parallel paths at distance r from each other. a) Find a symbolical expression for the magnetic force exerted by one proton on the other one: is it attractive or repulsive? s this always the case? b) Calculate the electric force between the charges and compare to the magnetic force. + +
Problems: 5. Straight Current: A straight wire of length 2a centered in y = 0, carries a current in positive y-direction. Calculate the magnetic field at distance r along x-axis. Useful integral: a dy y 2a 2 2 2 2 2 2 a x y xx y xx a 3 2 1 2 1 2 a a 6. Circular Current: A circular wire loop of radius a lays in the yz-plane and is traveled by a counterclockwise current. a) Calculate the magnetic field produced at a distance x along the axis of the loop. b) Using the result, find the field in the center of the ring.
Sources of Magnetic Fields Long straight wire Consider a long straight wire carrying a current, the magnetic field at a distance r perpendicular on the wire is given by: Magnitude: using the result for Problem 5: 0 1 2 2 2 a r r a 1 Direction: Given by the right hand rule #2 Comments: The magnetic field has cylindrical symmetry around the wire t gets weaker and weaker as the circles are larger and larger 0 2 r Quiz 1: A long wire carries a current as in the figure. Compared to the magnetic field at point A, the magnetic field at point is a) Half as strong, same direction. b) Half as strong, opposite direction. c) One-quarter as strong, same direction. d) One-quarter as strong, opposite direction.
Magnetic Force etween Two Parallel Conductors f two long current carrying wires are placed parallel with each other, they will interact via magnetic forces The force on wire 1 is due to magnetic field produced by wire 2 onto the current in 1, so the force per unit length L is: Comments: 2 r F 0 L L 0 2 r Parallel currents attract each other whereas anti-parallel conductors repel each other F L The force between parallel conductors can be used to redefine the Ampere (A) Def: f two long, parallel wires 1 m apart carry the same current, and the magnitude of the magnetic force per unit length is 2 x 10-7 N/m, then the current is defined to be 1 A Then the Coulomb (C) can be also defined in terms of the Ampere Def: f a conductor carries a steady current of 1 A, then the quantity of charge that flows through any cross section in 1 second is 1 C
Problems: 7. Force on a moving particle by a current carrying wire: A proton moves with speed v = 0.25 m/s parallel with a long wire carrying a current = 2.0 A, at distance r = 1.0 mm. Calculate the magnetic force on the proton. r + v e + 8. Superposition of aligned magnetic fields: Two long parallel wires carry currents 1 and 2 in opposite directions. The figure is an end view of the conductors. Calculate the magnitudes of the magnetic field in points A, and C located at given equal distances a from the closest wires. 1 2 a b c a a a a
Sources of Magnetic Fields Circular loop of current. Coils f we allow x 0 in the result of Problem 6, the magnetic field in the center of a circular loop is N 2 0 a 0 32 2 2 x0 2 x a N loops form a coil with the maximum field in the middle of the coil: 2a Notice that a current loop can be seen as a magnet with magnetic field lines that remind of the equipotential lines of an electric dipole: so the loop behaves like a magnetic dipole S The field produced by a loop or a coil is related to the respective magnetic moment μ Ex: Magnetic field and moment: the magnetic field in the center of a circular loop is related to its magnetic moment as given by a 0 max N a 2 2 0 3 2 a The magnetic field inside has the same direction as the magnetic moment μ
Magnetism in Materials Magnetic moments of electrons Now we are prepared to see that the magnetism of materials is microscopically mainly determined by the alignment of elementary electronic magnetic dipoles Notice first that atoms should act like magnets because of the orbital motion of the electrons about the nucleus Since each electron circles the atom once in about every 10-16 seconds, it produces a current of 1.6 ma and a magnetic field of about 20 T at the center of the orbit However, the magnetic field produced by one electron in an atom is often canceled by an oppositely revolving electron in the same atom, so the net result is that the magnetic effect produced by electrons orbiting the nucleus is either zero or very small for most materials The classical model is to consider the electrons to spin like tops but it is actually a quantum effect. The magnetic moment of an electron is given by ohr magneton: 24 9.27 10 J T Most materials are not naturally magnetic since electrons usually pair up with their spins opposite each other + spin
Paramagnets Magnetism in Materials Types of magnetism So, since the alignment of elementary magnetic dipoles associated with the electronic spins depends on the microscopic structure, various materials are classified depending on how they behave in external magnetic fields: Ex: aluminum, uranium Moments point in random directions in zero external field but in an external field ext they rotate so the net field increases The increment in field is small and paramagnetism competes with thermal motion Ferromagnets Ex: iron, nickel n some materials, large groups of atoms in which the spins are aligned form ferromagnetic domains When an external field ext is applied, the domains that are aligned with the field tend to grow at the expense of the others This causes the material to become magnetized by amounts larger than in the paramagnetic case The magnetization is slowly disappearing after removing the external field Diamagnets Ex: mercury, superconductors, animal bodies n these materials an external magnetic field induces opposite magnetic n these cases the internal magnetic field is less than the external one A diamagnet placed in an external magnetic field will have the tendency to float ext ext
Ampère s Law André-Marie Ampère found a procedure for deriving the relationship between the current in an arbitrarily shaped wire and the magnetic field produced by the wire: Ampère s Circuital Law: f a net current encl is enclosed by an arbitrary closed path, the integral of all products dl (where is the component of the magnetic field along each elementary step dl of the path) is proportional to encl : 3 d Amperian loop d 0 encl Line integral around a closed path called an Amperian loop Net current inside the path Ex: Ampere s law can be used to demonstrate the result that we obtained previously for a closed circular path around a long straight current : since the circumference of the path is 2 r, and by symmetry the field around the Amperian is expected to be everywhere constant and tangent to the path, we get: 0 d d 2r 0encl 2 r 4 1 2 encl = 2 + 3 + 4 r
Problem: 8. Magnetic field inside and outside of a current carrying conductor: A cylindrical conductor with radius R carries a current uniformly distributed over the cross-sectional area of the conductor. Confirm the relationships given below for the magnetic field in the interior and the exterior of the conductor.
f a long straight wire is wound into a coil of closely spaced loops, the resulting device is called a solenoid Direction The field lines of the solenoid resemble those of a bar magnet and the field direction is given by the right hand rule applied to the current through any of the turns Magnitude Sources of Magnetic Field Solenoids t is also known as an electromagnet since it acts like a magnet only when it carries a current The magnitude of the field inside a solenoid is constant at all points far from its ends n where n is the number of turns per unit length This expression can be obtained by applying Ampère s Law to the solenoid 0 n N S N solenoid Quiz 2: What is the direction of the field in the center of the solenoid below? S N bar magnet
Sources of Magnetic Field Solenoid field using Ampère s Law Consider a cross-sectional view of a tightly wound solenoid of turn density n, carrying a current f the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero Then we can apply Ampère s Law to a rectangular Amperian a b c d a: Amperian loop with N turns inside Comment: n reality the field is not perfectly uniform along the axis of a cylinder. N L 0N 0 n0 L Ampère s law