Physics 1202: Lecture 12 Today’s Agenda Announcements: –Lectures posted on: –HW assignments, solutions etc. Homework #4:Homework #4: –Not this week ! (time to prepare midterm) Midterm 1: –Friday Oct. 2 –Chaps. 15, 16 & 17.

Magnetic Force on a Current or

Current loop & Magnetic Dipole Moment We can define the magnetic dipole moment of a current loop as follows: direction: right-hand rule Torque on loop can then be rewritten as: Note: if loop consists of N turns,  = N A I magnitude:  A I  A I B  sin  B x. F F  w    If plane of loop is not  to field, there will be a non-zero torque on the loop! No net force

Calculation of Magnetic Field Two ways to calculate the Magnetic Field: Biot-Savart Law: Ampere's Law These are the analogous equations for the Magnetic Field! "Brute force"  I "High symmetry"  0 = 4  X T m /A: permeability (vacuum)

Magnetic Field of  Straight Wire  Direction of B: right-hand rule

Lecture 12, ACT 1 I have two wires, labeled 1 and 2, carrying equal current, into the page. We know that wire 1 produces a magnetic field, and that wire 2 has moving charges. What is the force on wire 2 from wire 1 ? (a) Force to the right (b) Force to the left (c) Force = 0 Wire 1 I X Wire 2 I X

Force between two conductors Force on wire 2 due to B at wire 1: Total force between wires 1 and 2: Force on wire 2 due to B at wire 1: Direction: attractive for I 1, I 2 same direction repulsive for I 1, I 2 opposite direction

Circular Loop x z R R Circular loop of radius R carries current i. Calculate B along the axis of the loop: r BB r z BB   Symmetry  B in z-direction. > > I  At the center (z=0): Note the form the field takes for z>>R: for N coils

Lecture 12, ACT 2 Equal currents I flow in identical circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction. –What is the magnetic field B z (A) at point A, the midpoint between the two loops? (a) B z (A) < 0 (b) B z (A) = 0 (c) B z (A) > 0

Lecture 12, ACT 2 Equal currents I flow in identical circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction. (a) B z (B) < 0 (b) B z (B) = 0 (c) B z (B) > 0 – What is the magnetic field B z (B) at point B, just to the right of the right loop?

B Field of a Solenoid A constant magnetic field can (in principle) be produced by an  sheet of current. In practice, however, a constant magnetic field is often produced by a solenoid. If a << L, the B field is to first order contained within the solenoid, in the axial direction, and of constant magnitude. In this limit, we can calculate the field using Ampere's Law. L A solenoid is defined by a current I flowing through a wire which is wrapped n turns per unit length on a cylinder of radius a and length L. a

B Field of a  Solenoid To calculate the B field of the  solenoid using Ampere's Law, we need to justify the claim that the B field is 0 outside the solenoid. To do this, view the  solenoid from the side as 2  current sheets. x x xxx The fields are in the same direction in the region between the sheets (inside the solenoid) and cancel outside the sheets (outside the solenoid). (n: number of turns per unit length) 

Toroid Toroid defined by N total turns with current i. B=0 outside toroid! B inside the toroid. x x x x x x x x x x x x x x x x r B 

Magnetism in Matter When a substance is placed in an external magnetic field B o, the total magnetic field B is a combination of B o and field due to magnetic moments (Magnetization; M): – B = B o +  o M =  o (H +M) =  o (H +  H) =  o (1+  ) H »where H is magnetic field strength  is magnetic susceptibility Alternatively, total magnetic field B can be expressed as : –B =  m H »where  m is magnetic permeability »  m =  o (1 +  ) All the matter can be classified in terms of their response to applied magnetic field: –Paramagnets  m >  o –Diamagnets  m <  o –Ferromagnets  m >>>  o

Faraday's Law v B N S v B S N n B B 

Induction Effects v v S N v N S N S S N Bar magnet moves through coil  Current induced in coil Change pole that enters  Induced current changes sign Bar magnet stationary inside coil  No current induced in coil Coil moves past fixed bar magnet  Current induced in coil

Faraday's Law Define the flux of the magnetic field B through a surface A=An from: Faraday's Law: The emf induced around a closed circuit is determined by the time rate of change of the magnetic flux through that circuit. The minus sign indicates direction of induced current (given by Lenz's Law). n B B 

Faraday’’s law for many loops Circuit consists of N loops: all same area  B magn. flux through one loop loops in “series” emfs add!

Lenz's Law Lenz's Law: The induced current will appear in such a direction that it opposes the change in flux that produced it. Conservation of energy considerations: Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. »Why??? If current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc.. v B S N v B N S

Lecture 12, ACT 3 A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown. – What is the direction of the induced current in the loop? (c) no induced current (a) ccw (b) cw x y

Lecture 12, ACT 4 A conducting rectangular loop moves with constant velocity v in the -y direction away from a wire with a constant current I as shown. What is the direction of the induced current in the loop? (a) ccw (b) cw (c) no induced current x y