Magnetostatics
Magnetic Fields We saw last lecture that some substances, particularly iron, possess a property we call magnetism that exerts forces on other magnetic materials We also saw that t single magnetic charges (magnetic monopoles) did not exist We saw that magnetic fields, shown up by iron filings look similar to electric dipole fields Also that magnetic fields seem to be associated with moving charges What is this "magnetic force"? How is it related to and distinguished from the "electric" force?
Magnetic Forces Consider a positive charge q moving in the field of a magnet with velocity v, experimentally we find: 1. If q moves in the +z direction and the field points in the +y direction then the force F is in the x direction. The force is proportional to the velocity and the field 2. If q moves in the +x direction the force is in the +z direction, again proportional to and v
Magnetic Forces 3. If q moves in the +y direction there is no force 4. If q is at rest there is no force 5. The force is proportional to 6. The force is proportional to the sign and magnitude of q The magnetic force on a moving charge is proportional to q, v p and,, where v p is the velocity component perpendicular to the field, while the direction of F is perpendicular to both and v and depends on the sign of F q F qv
Lorentz Force We can add the effect of an Electric Field and get the Lorentz Force The force F on a charge q moving with velocity v through a region of space with electric field E and magnetic field is given by: F qe qv x x x x x x x x x x x x v x x x x x x q F v q F v F = 0 q
Reminder: The Cross Product The cross (vector) product of two vectors is a third vector Remember the dot (scalar) product multiplied two vectors to produce a scalar A X = C The magnitude of C is given by: C = A sin C A The direction of C is perpendicular to the plane defined by A and, and in the direction defined by the right hand rule, rotating from A to. UIUC
Reminder: The Cross Product Cartesian components of the cross product: C = A X C X = A Y Z Y A Z C C Y = A Z X Z A X C Z = A X Y X A Y A Note: X A = - A X Drawing 3-dimensional vectors, conventionally a vector going into the slide a vector coming out of the slide
Right Hand Rule
Motion in a magnetic field F qv Three points are arranged in a uniform magnetic field. The field points into the screen. Consider the force on a positively charged particle in the following conditions 1) It is located at point A and is stationary. v=0 The magnetic force is zero 2) The positive charge moves from point A toward. v in x direction, in z, RH rule lesays s F in y The direction of the magnetic force on the particle is to the left 3) The positive charge moves from point A toward C. Rotate our x axis to be along the direction A C F will be perpendicular to that line and upwards x z y
Motion of a Charge in a Magnetic Field The s s represent field lines pointing into the page. A positively charged particle of mass m and charge q is shot to the right with speed v. y the right hand rule the magnetic force on it is up. Since v is to, F = F = q v. ecause F is to v,, it has no tangential component; it is entirely centripetal. Thus F causes a centripetal acceleration. As the particle turns so do v and F, and if is uniform the particle moves in a circle. This is the basic idea behind a particle accelerator like Fermilab. Since F is a centripetal force, F = F C = mv 2 / R. Let s see how C speed, mass, charge, field strength, and radius of curvature are related: R F + q, v m F = F C qv = mv 2 /R mv R= q
Question 1 Two protons each move at speed v in the x y plane (as shown in the diagram) in a region of space which contains a constant field in the -z direction. Ignore the interaction between the two protons. What isthe relation between the magnitudes of the forces on the two protons? y v 1 2 v z x (a) F 1 < F 2 (b) F 1 = F 2 (c) F 1 > F 2 The magnetic force is given by: F qv F qvsin In both cases the angle between v and is 90 Therefore F 1 = F 2.
Question 2 Two protons each move at speed v in the x y plane (as shown in the diagram) in a region of space which contains a constant field in the z direction. Ignore the interaction between the two protons. What is F 2x,the x component of the force on the second proton? y v 1 2 v z x (a) F 2x < 0 (b) F 2x = 0 (c) F 2x > 0 To determine the direction of the force, we use the right-hand rule. F qv The directions of the forces are shown in the diagram F 2x < 0
Question 3 Two protons each move at speed v in the x y plane (as shown in the diagram) in a region of space which contains a constant field in the z direction. Ignore the interaction between the two protons. Inside the field, the speed of each proton: y v 1 2 v z x (a) decreases (b) increases (c) stays the same Although the proton does experience a force (which deflects it), this is always to v. Therefore, there is no possibility to do work W F l Flcos So kinetic energy is constant and v is constant
Trajectory in a Constant Field Suppose charge q enters -field with velocity v as shown below. What will be the path q follows? x x x x x x x x x x x x x x x x x x x x x x x vx x x x x x x x x x x x q x v F F R Force is always to velocity and. Path will be circle. F will be the centripetal force needed to keep the charge in its circular orbit, radius R.
Radius of Circular Orbit Lorentz force: F qv centripetal acc: x x x x x x x x x x x x x x x x x x x x x x x vx a v R 2 Newton's 2nd Law: x x x x x x v F x x x x x x F q R F ma R qv m mv v R q consequences! 2 This is an important result, with useful experimental
Ratio of charge to mass for an electron In 1897 J.J.Thomson J used a cathode ray tube to measure e/m for an electron Used an electric and magnetic field in opposition to cancel force and thus deflection of the electron Electron accelerated through a voltage V by the electron gun 1 2 giving it kinetic energy mv ev 2 Velocity when it enters the fields 2eV v m
e/m for an electron In an electric field alone the spot is deflected Then apply a magnetic field at right angles (Force in opposite direction) until the deflection is reduced to zero Force due to electric field Force due to magnetic field When the fields cancel E v 2eV m e m 2 E 2V 2 F ˆ E eek F ev evk ˆ F F qekˆ qvkˆ v e m 1.758820 10 11 C kg E
Measurement of particle energies Many experiments using particles measure their velocity y( (energy or momentum) by measuring their curvature in a magnetic field Cloud chambers ubble chambers Magnetic spectrometers
Magnetic dipole moment A current loop behaves like a little bar magnet aligning gwith a magnetic field. The magnitude of the dipole moment is NiA The direction of the dipole moment vector is given by right hand rule i
iot Savart Law Moving charges are affected by magnetic fields; similarly moving charges (currents) create magnetic fields.
The iot Savart law Magnetic field intensity at position P by a conductor with current I d 4 R 0 I d R 3 7 4 10 N / A 0 / : permeability of free space integrating 2 0 I d R c 3 4 R I d Q(x,y,z ) I R r r ' d P(x,y,z) In 3 D, 0 J v v 3 4 R R dv O
Ampere s Law Ampere s law is to magnetism what Gauss s law is to electrostatics.. dl 0 I enc This method works in cases with high symmetry where the properties of the field can be inferred, figured out, whatever.
Ampere s circuit law I Amper s law The line integral of around a closed path is the same as the net current I enc enclosed by the path. C d c d 0I enc ( r ) dl 0 J v ( r ) ds c s Integral form of Ampere s Law
Ampere s Law: differential form Ampere s Law d I 0 c ( r ) dl ( r ) ds c s 0 I 0 J v ( r ) ds s C S ( r ) 0 J v ( r ) ( 0 Ampere s Law: differential form ) 3 rd Maxwell equation
Magnetic flux magnetic flux ds s Unit: Wb Gauss s Law for magnetostatic field Integral form s ds 0 Differntial form 0 4 th Maxwell eq. Thers is no magnetic charge-monopole..
Current enclosed Current density J is the current per unit area through a wire. J d I J. enc enc. da
Faraday s law and Lenz s law Faraday s law describes how magnetic fields can create voltages i.e. we are now connecting magnetic and electric phenomena In words a time varying magnetic flux through a circuit loop creates a voltage difference between the ends of the loop. Lenz s law indicates the polarity of the voltage
Faraday s law V ind is the induced voltage between the ends of the loop, M is the magnetic flux through the loop dm Vind dt Flux through 1 loop is given by M 1loop Area Flux through N loops is N M. da
Faraday s law: bigpicture The fundamental point of Faraday s law is that a time-varying magnetic flux,, M, leads to an induced voltage and thus an E-field. In the briefest of terms A changing magnetic field produces an electric field.
Lenz s law The voltage induced is always such as to keep the flux through the circuit constant. The direction of the voltage is oriented to create a current in the loop such that the flux remains the same.
Magnetic Force on a Current Carrying Wire A section of wire carrying current to the right is shown in a uniform magnetic field. We can imagine positive charges moving to right, each feeling a magnetic force out of the page. This will cause the wire to bow outwards. Shown on the right is the view as seen when looking at the N pole from above. The dots represent a uniform S mag. field coming out of the page. The mag. force on the wire is proportional to the field strength, the current, and the length of the wire. N I............ I............ Continued
Magnetic Force on a Wire (cont.) Current is the flow of positive charge. As a certain amount of charge, g,q, moves with speed v through a wire of length L,, the force of this quantity of charge is: F =qv Over the time period t required for the charge to traverse the length of the wire, we have: F = (q/t)vt Since q/t = I and vt= L, we can write: F = IL where L is a vector of magnitude L pointing in the direction of I............. I............
Electric Motor I I F } d I I Current along with a magnetic field can produce torque. This is the basic idea behind an electric motor. Above is a wire loop (purple) carrying a current provided by some power source like a battery. The current loop is submerged in an external field. From F = I L, the force vectors in black are perpendicular to their wire segments. The net force on the loop is zero, but the net torque about the center is nonzero. The forces on the left and right wires produce no torque since the moment arm is zero for each (they (h point right ihat the center). However, the force F on the top wire (in the background) has a moment arm d, so it produces a torque F d. The bottom wire (in the foreground) produces the same torque. These torques work together th to rotate tt the loop, converting electrical lenergy into mechanical energy. Continued
Electric Motor (cont.) As the loop turns it eventually reaches a vertical position (the plane of the loop parallel to the field). This is when the moment arms of the forces on the top and bottom wires are the longest, so this is where the torque is at a max. 90 later the loop will be perpendicular to the field. Here all moment arms and all torques are zero. This is the equilibrium point. The angular momentum of the loop, however, will allow it to swing right through this position. Now is when the current must change direction, otherwise the torques will attempt to bring the loop back to the equilibrium. This would amount to simpleharmonic motion ofthe loop, which isnot particularly useful. If the current changes direction every time the loop reach equilibrium, the loop will spin around in the same direction indefinitely. Although a battery only pumps current in one direction, the change in direction of current can be accomplished with hl help of a commutator. tt
Electromagnets: Straight Wire Permanent magnets aren t the only things that produce magnetic fields. Moving charges themselves produce magnetic fields. We just saw that a current carrying wire feels a force when inside an external magnetic field. It also produces its own magnetic field. A long straight wire produces circular field lines centered on the wire. To find the direction of the field, we use another right hand rule: point your thumb in the direction of the current; the way your fingers of your right hand wrap is the direction of the magnetic field. diminishes with distance from the wire. The pics at the right show cross sections of a current carrying wire. I I out of page, counterclockwise I into page, clockwise