SMU PHYS 1304 - The Laws of Biot-Savart & Ampere - D841605

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SMU PHYS 1304 - The Laws of Biot-Savart & Ampere

School name Southern Methodist University

Course Phys 1304- Introductory Electricity and Magnetism

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PowerPoint PresentationOverview of LectureCalculation of Electric FieldCalculation of Magnetic FieldBiot-Savart Law…bits and piecesMagnetic Field of ¥ Straight WireSlide 7Lecture 14, ACT 1Slide 9Slide 10Lecture 14, ACT 2Slide 12Slide 13Force on 2 Parallel Current-Carrying ConductorsLecture 14, ACT 3Slide 17Slide 19Slide 20B Field Inside a Long WireB Field of a Long WireLecture 15, ACT 1Slide 24Slide 25B Field of ¥ Current SheetB Field of a SolenoidB Field of a ¥ SolenoidToroidCircular LoopSlide 31Lecture 15, ACT 2Slide 33Slide 34Slide 35Physics 1304: Lecture 12, Pg 1The Laws of Biot-Savart & AmperedBI dx rr 034B dl I- 0dlIPhysics 1304: Lecture 12, Pg 2Overview of LectureOverview of LectureFundamental Law for Calculating Magnetic Field•Biot-Savart Law (“brute force”)•Ampere’s Law (“high symmetry”)Example: Calculate Magnetic Field of  Straight Wire•from Biot-Savart Law•from Ampere’s LawCalculate Force on Two Parallel Current-Carrying ConductorsText Reference: Chapter 30.1-4Physics 1304: Lecture 12, Pg 3Calculation of Electric FieldCalculation of Electric Field"Brute force"Eqrr1402"High symmetry"0 E dS q- Two ways to calculate the Electric Field:•Coulomb's Law:•Gauss' LawWhat are the analogous equations for the Magnetic Field?Physics 1304: Lecture 12, Pg 4Calculation of Magnetic FieldCalculation of Magnetic Field"High symmetry"B dl I- 0"Brute force"dBI dl rr034I Two ways to calculate the Magnetic Field:•Biot-Savart Law:•Ampere's LawThese are the analogous equations for the Magnetic Field!Physics 1304: Lecture 12, Pg 5Biot-Savart Law…Biot-Savart Law…bits and piecesbits and piecesIdldBXrdBI dl rr03470104So, the magnetic field “circulates” around the wirePhysics 1304: Lecture 12, Pg 6Magnetic Field of Magnetic Field of  Straight Wire Straight Wire•Calculate field at point P using Biot-Savart Law:Rewrite in terms of R,:xRrPIdxB dBI dx rr   034( ) sinrRsintan Rxx R cot dx R d 12sin dxrdR(sin ) sin  2dBI dx rr 034Which way is B?Physics 1304: Lecture 12, Pg 7Magnetic Field of Magnetic Field of  Straight Wire Straight WireBI dR004sinBIRd 004sin BIR 004cosBIR02xRrPIdxPhysics 1304: Lecture 12, Pg 8Lecture 14, ACT 1Lecture 14, ACT 1What is the magnitude of the magnetic field at the center of a loop of radius R, carrying current I? (a) B = 0(b) BR(c) BRRIPhysics 1304: Lecture 12, Pg 9Lecture 14, ACT 1Lecture 14, ACT 1What is the magnitude of the magnetic field at the center of a loop of radius R, carrying current I? (a) B = 0(b) BR(c) BR• To calculate the magnetic field at the center, we must use the Biot-Savart Law:dBI dx rr 034Idxr• Two nice things about calculating B at the center of the loop:• Idx is always perpendicular to r• r is a constant (=R) R2IμπR2πR4IμRR(dx)π4IμdBB02030RIPhysics 1304: Lecture 12, Pg 10Magnetic Field of Magnetic Field of  Straight Wire Straight Wire•Calculate field at distance R from wire using Ampere's Law:•Ampere's Law simplifies the calculation thanks to symmetry of the current! ( axial/cylindrical ) BIR02dlRI Choose loop to be circle of radius R centered on the wire in a plane  to wire. Why?»Magnitude of B is constant (fcn of R only)»Direction of B is parallel to the path.B dl B R- ( )2Evaluate line integral in Ampere’s Law:Current enclosed by path = I20 RB IApply Ampere’s Law:B dl I- 0Physics 1304: Lecture 12, Pg 11Lecture 14, ACT 2Lecture 14, ACT 2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. What is the magnetic field Bx(a) at point a, just outside the cylinder as shown?(a) Bx(b) < 0(b) Bx(b) = 0(c) Bx(b) > 0– What is the magnetic field Bx(b) at point b, just inside the cylinder as shown?(a) Bx(a) < 0(b) Bx(a) = 0(c) Bx(a) > 0xxxxxxxx2IIabxyPhysics 1304: Lecture 12, Pg 12Lecture 14, ACT 2Lecture 14, ACT 2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. What is the magnetic field Bx(a) at point a, just outside the cylinder as shown?• This situation has massive cylindrical symmetry!• Applying Ampere’s Law, we see that the field at point a must just be the field from an infinite wire with current I flowing in the -z direction!xIBBBBxxxxxxxx2IIabxy(a) Bx(a) < 0(b) Bx(a) = 0(c) Bx(a) > 0Physics 1304: Lecture 12, Pg 13Lecture 14, ACT 2Lecture 14, ACT 2• Just inside the cylinder, the total current enclosed by the Ampere loop will be I in the +z direction!• Therefore, the magnetic field at b will just be minus the magnetic field at a!!xxxxxxxx2IIabxy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. What is the magnetic field Bx(a) at point a, just outside the cylinder as shown?(a) Bx(a) < 0(b) Bx(a) = 0(c) Bx(a) > 0(a) Bx(b) < 0(b) Bx(b) = 0(c) Bx(b) > 0 What is the magnetic field Bx(b) at point b, just inside the cylinder as shown?Physics 1304: Lecture 12, Pg 15Force on 2 ParallelForce on 2 ParallelCurrent-Carrying ConductorsCurrent-Carrying Conductors•Calculate force on length L of wire b due to field of wire a:The field at b due to a is given by:Calculate force on length L of wire a due to field of wire b:The field at a due to b is given by:FFBIdaa02LdIbIaLdIbIaForce on b = dLIIBLIFbaabb20BIdbb02Force on a = dLIIBLIFbabaa20Physics 1304: Lecture 12, Pg 16Lecture 14, ACT 3Lecture 14, ACT 3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. What is Fx, net force on the loop in the x-direction? (a) Fx <

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