7The Earth’s Magnetic Field The spinning iron core of the earth produces a magnetic field.
8Magnetic FieldsMagnetic field is a vector. It has direction and can be cancelled by another field.
9Magnetic Field Units The SI unit of magnetic field is the Tesla (T). Dimensional analysis:1 T = 1 N·s/(C·m) = 1 V·s /m2Sometimes we use a unit called a Gauss (G):1 T = 104 GThe earth’s magnetic field is about 0.5 G.
10Magnetic Force on Moving Charges A charged particle in a static magnetic field will experience a magnetic force only if the particle is moving.If a charge q with velocity v moves in a magnetic field B and v makes an angle q w.r.t. B, then the magnitude of the force on the charge is:
11Finding the Direction of Magnetic Force The direction of the magnetic force is always perpendicular to both andFmax occurs when is perpendicular toF = 0 when is parallel to
12Notation To represent the z-axis on an xy plane of paper. • B out of the pageB into the page
13Right Hand RuleDraw vectors v and B with their tails at the location of the charge q.Point fingers of right hand along velocity vector v.Curl fingers towards Magnetic field vector B.Thumb points in direction of magnetic force F on q, perpendicular to both v and B.
14Use RHR to find F on moving charged particle for each situation a through f
15Motion of Charges in B Fields If a charged particle is moving perpendicular to a uniform magnetic field, its trajectory will be a circle because the force F=qvB is always perpendicular to the motion, and therefore centripetal.
16If path of charge is not perpendicular to field where it enters field at some angle, charge will follow a spiral path (helix) which will spiral around the B-field.The component of velocity that is parallel to the magnetic field is unaffected. Its circlular motion will drift at a constant speed along the magnetic fieldThe perp component of the velocity to B-field causes the particle to executes uniform circular motion perpendicular to the magnetic field.
17Velocity SelectorA particle is accelerated through a potential difference where it then enters a magnetic and electric field. If the forces are balanced, the particle will move only horizontally between the plates (assuming negligible gravity)
18Mass SpectrometerThe mass spectrometer utilizes the velocity selector and is an instrument which can measure the masses and relative concentrations of atoms and molecules. The radius of turn yields the mass of the particles.
19An ion is accelerated through a voltage of 1000V with a charge of 1 An ion is accelerated through a voltage of 1000V with a charge of 1.6x10-19C after which it enters a chamber with a uniform B-field equal to 80mT. The ion follows a semi-circle path and strikes a photographic plate 1.62m from where it entered. Find its mass.
20Hall EffectIf an electric current flows through a conductor in a magnetic field, the magnetic field exerts a force on the moving charge carriers which tends to push them to one side of the conductor. A buildup of charge at the sides of the conductor will balance this magnetic influence w/ E-field, producing a measurable voltage between the two sides of the conductor. The charge carries will then just pass through in a straight line.
21Force on a WireSimilar to the force on a moving charge in a B field, we have for a conductor of length L carrying a conventional current of I in a B field. The force experienced by the conductor is:
22A thin, massless, uniform rod with length 0 A thin, massless, uniform rod with length 0.20m is attached to a frictionless hinge at point ‘P’. A horizontal spring (k = 4.8 N/m) connects the other end of the rod to a vertical wall. A uniform B-field equal to 0.34T is shown and a 6.5A current exists in the rod directed towards the hinge. How much energy is stored in the spring?BI53.1o‘P’FSFM
23Torque on a current loop A wire loop is freely pivoting in a uniform B-field (+x) as shown. Each side is length ‘a’.Ba
24Electric MotorAn electric motor converts electrical energy to mechanical energyThe mechanical energy is in the form of rotational kinetic energyWhen area vector of loop is parallel to field there is no torqueAfter the loop moves a ½ turn, the current needs to switch direction to keep it rotating. Inertia will carry it past the edge (top of rotation) and at that moment if the current is switched, the loop keeps going. If the current isn’t switched and it passes the edge, it will rotate the other way and get nowhere
25Biot Savart LawThis law is seen as the magnetic equivalent of Coulomb's Law (brute force way of doing it vs Gauss) . This finds B at a point P, a distance r from the differential element of current Ids.m0 = 4p 10-7 Tm/A, magnetic permeability of free spaceCurrent makes B-fields
26Biot-Savart Law – Set-Up The magnetic field is at some point PThe length element isThe wire is carrying a steady current of Iis the field created by the currentin the length segment dsTo find the total field, sum up thecontributions from all the currentelements IThe integral is over the entirecurrent distribution
27B-Field due to a long straight wire along y-axis
28ExamplebPIaConsider wire bent into the shape shown above. Find B at ‘P’ if current is flowing clockwise in wire.
29Magnetic Field from a Wire, RHR #3 The magnetic field lines from a current form circles around a straight wire with the direction given by another right hand rule.The magnitude of the magnetic field a distance r from a straight wire is given by (just proven with rigorous proof)
30Force between 2 current-carrying wires What happens when current as shown in 2 parallel wires?I1aI2
33Loop of CurrentConsider a coil of radius R with current flowing CW. Find B at center of coil.X
34A loop of Current is RHR #4 Fingers are current direction and thumb is magnetic north pole
35What is the direction of the net force on the loop? A current I flows in the positive y direction in an infinite wire; a current I also flows in the loop as shown in the diagram.IdWhat is the direction of the net force on the loop?(a) F = -x(b) F = 0(c) F = +x(d) F = +y(e) F = -y
36B-field due to a current loop a distance ‘x’ away dsfrontal view of loop of current
37Ampere’s LawAmpere’s Law is to magnetic fields as Gauss’ Law was to electric fields. Both are used for high symmetry problems.Similar to drawing a Gaussian surface
38NOW apply Ampere’s Law to find B surrounding long straight current carrying wire (already did this using Biot Savart)Consider long wire with current I into page.r
39Magnetic field inside a wire Find B inside the wire with uniform current a distance r from the center where r < R
40Solenoid is a series of loops of current Solenoid is a series of loops of current. One end acts like a N & the other like a S-pole. Ideal solenoid is approached when turns are closely spaced and length is much greater than radius of turns.B-field of a solenoidCurrent would flow from left to right across top of solenoid
41Derivation of B inside solenoid Cross sectional view of solenoid showing current into page on top, out of page on bottom
44a) Bx(a) < 0 b) Bx(a) = 0 c) Bx(a) > 0 (a) Bx(b) < 0 A current I flows in an infinite straight wire in the +z direction as shown. A concentric infinite cylinder of radius R carries current 2I in the -z direction.x2IIabya) What is the magnetic field Bx at point ‘a’, just outside the cylinder as shown? Take CW as positive for B.a) Bx(a) < 0b) Bx(a) = 0c) Bx(a) > 0b) What is the magnetic field Bx at point ‘b’, just inside the cylinder as shown?(a) Bx(b) < 0b) Bx(b) = 0c) Bx(b) > 0
45I I a) BL(6a)< BR(6a) b) BL(6a)= BR(6a) c) BL(6a)> BR(6a) Two cylindrical conductors each carry current I into the page as shown. The conductor on the left is solid and has radius R=3a. The conductor on the right has a hole in the middle and carries current only between R=a and R=3a.What is the relation between the magnetic field at R = 6a for the two cases (L=left, R=right)?3aa3aIIa) BL(6a)< BR(6a)b) BL(6a)= BR(6a)c) BL(6a)> BR(6a)
46I I a) BL(2a)< BR(2a) b) BL(2a)= BR(2a) c) BL(2a)> BR(2a) Two cylindrical conductors each carry current I into the page as shown. The conductor on the left is solid and has radius R=3a. The conductor on the right has a hole in the middle and carries current only between R=a and R=3a.What is the relation between the magnetic field at R = 2a for the two cases (L=left, R=right)?3aa3aIIa) BL(2a)< BR(2a)b) BL(2a)= BR(2a)c) BL(2a)> BR(2a)
47Consider a thin, infinitely large sheet of current that carries a linear current density, λ. The current is out of the page. Find B near the sheet.Recall a similar problem with infinitely charged sheet of charge density, σ.