Fundamental Physics II

1. PETROVIETNAM UNIVERSITY FACULTY OF FUNDAMENTAL SCIENCES Fundamental Physics II • Pham Hong Quang • E-mail: quangph@pvu.edu.vn Vungtau, 2013

2. CHAPTER 3 Magnetic Fields • Pham Hong Quang2PetroVietnam University

3. 3.1 The Magnetic Field Permanent bar magnets have opposite poles on each end, called north and south. Like poles repel; opposites attract. No Magnetic monopole available in nature. Pham Hong QuangFaculty of Fundamental Sciences

4. 3.1 The Magnetic Field A magnetic field exists in the region around a magnet. The magnetic field is a vector that has both magnitude and direction. The direction of the magnetic field at any point in space is the direction indicated by the north pole of a small compass needle placed at that point. Pham Hong QuangFaculty of Fundamental Sciences

5. 3.1 The Magnetic Field The magnetic field line The lines originate from the north pole and end on the south pole; they do not start or stop in midspace. The magnetic field at any point is tangent to the magnetic field line at that point. The strength of the field is proportional to the number of lines per unit area that passes through a surface oriented perpendicular to the lines. The magnetic field lines will never come to cross each other. Pham Hong QuangFaculty of Fundamental Sciences

6. 3.1 The Magnetic Field The Magnetic Force on Moving Charges -Lorentz force This is an experimental result –we observe it to be true. It is not a consequence of anything we’ve learned so far. Pham Hong QuangFaculty of Fundamental Sciences

7. 3.1 The Magnetic Field Right-hand rule In order to figure out which direction the force is on a moving charge, you can use a right-hand rule. This gives the direction of the force on a positive charge; the force on a negative charge would be in the opposite direction. Pham Hong QuangFaculty of Fundamental Sciences

8. 3.1 The Magnetic Field The Definition of magnitudeMagnetic Field • The magnitude B of the magnetic field at any point in space is defined as where F is the magnitude of the magnetic force on a positive test charge q0 , v is the velocity of the charge which makes an angle  with the direction of the magnetic field. SI Unit of Magnetic Field: Pham Hong QuangFaculty of Fundamental Sciences

9. 3.1 The Magnetic Field Differences of Electric and magnetic Fields • Direction of forces • The electric force on a charged particle (both moving and stationary) is always parallel (or anti-parallel) to the electric field direction. • The magnetic force on a moving charged particle is always perpendicular to both magnetic field and velocity of the particle. No magnetic force on a stationary charged particle. • The work done on a charged particle: • The electric force can do work on the particle. • The magnetic force cannot do work and change the kinetic energy of the charged particle. Pham Hong QuangFaculty of Fundamental Sciences

10. 3.1 The Magnetic Field Hall effect When a current-carrying conductor is placed in a magnetic field, a voltage is generated in a direction perpendicular to both the current and the magnetic field. The Hall Effect results from the deflection of the charge carriers to one side of the conductor as a result of the Lorentz force experienced by the charge carriers. Pham Hong QuangFaculty of Fundamental Sciences

11. 3.1 The Magnetic Field When the electric and magnetic forces are in balance Because Then Pham Hong QuangFaculty of Fundamental Sciences

12. 3.1 The Magnetic Field Crossed Fields: Discovery of the Electron • 1. Set E and B to zero and note the position of the spot on screen S due to the undeflected beam. • 2. Turn on E and measure the resulting beam deflection y. • 3.  Maintaining E , now turn on B and adjust its value until the beam returns to the undeflected position. (With the forces in opposition, they can be made to cancel.) Pham Hong QuangFaculty of Fundamental Sciences

13. 3.1 The Magnetic Field With B ≠ 0, there is no deflection when v = E/B But Therefore This is how one can measure the e-/m ratio for electrons!! Pham Hong QuangFaculty of Fundamental Sciences

14. 3.2 Magnetic Force on a Current-Carrying Wire The force acting on one electron The force acting on the wire of length L and But Therefore Pham Hong QuangFaculty of Fundamental Sciences

15. 3.2 Magnetic Force on a Current-Carrying Wire Check Your Understanding The same current-carrying wire is placed in the same magnetic field B in four different orientations (see the drawing). Rank the orientations according to the magnitude of the magnetic force exerted on the wire, largest to smallest. Pham Hong QuangFaculty of Fundamental Sciences

16. 3.2 Magnetic Force on a Current-Carrying Wire In the current loop shown, the vertical sides experience forces that are equal in magnitude and opposite in direction. They do not operate at the same point, so they create a torque around the vertical axis of the loop. The total torque is the sum of the torques from each force: Τ= (IhB) (w/2) +(IhB)(w/2) =I (hw)B Or, since A = hw, τ =I.A.B Pham Hong QuangFaculty of Fundamental Sciences

17. 3.2 Magnetic Force on a Current-Carrying Wire If the plane of the loop is at an angle to the magnetic field, τ =I.A.B sinθ To increase the torque, a long wire may be wrapped in a loop many times, or “turns.” If the number of turns is N, we have τ =N.I.A.B sinθ Pham Hong QuangFaculty of Fundamental Sciences

18. 3.2 Magnetic Force on a Current-Carrying Wire The torque on a current loop is proportional to the current in it, which forms the basis of a variety of useful electrical instruments. Here is a galvanometer: Pham Hong QuangFaculty of Fundamental Sciences

19. 3.2 Magnetic Force on a Current-Carrying Wire The Magnetic Moment of Coil The magnitude of is given by N is the number of turns in the coil, i is the current through the coil, and A is the area enclosed by each turn of the coil. Pham Hong QuangFaculty of Fundamental Sciences

20. 3.3 Biot-Savart law Experimental observation: Electric currents can create magnetic fields. Biot-Savart law Pham Hong QuangFaculty of Fundamental Sciences

21. 3.3 Biot-Savart law Assume a small segment of wire ds causing a field dB: mo = 4p 10-7 T m/A BSL is also an inverse square law!! If ds and ř are parallel, the contribution is zero! Pham Hong QuangFaculty of Fundamental Sciences

22. 3.3 Biot-Savart law To find the direction of the magnetic field due to a current-carrying wire, point the thumb of your right hand along the wire in the direction of the current I. Your fingers are now curling around the wire in the direction of the magnetic field. Pham Hong QuangFaculty of Fundamental Sciences

23. 3.3 Biot-Savart law To find the total field, sum up the contributions from all the current elements Ids • The integral is over the entire current distribution Pham Hong QuangFaculty of Fundamental Sciences

24. 3.3 Biot-Savart law Derivation of B for a Long, Straight Current-Carrying Wire Integrating over all the current elements gives Pham Hong QuangFaculty of Fundamental Sciences

25. 3.3 Biot-Savart law Pham Hong QuangFaculty of Fundamental Sciences

26. 3.3 Biot-Savart law If the conductor is an infinitely long, straight wire, q1= 0 and q2= p Pham Hong QuangFaculty of Fundamental Sciences

27. 3.3 Biot-Savart law B for a Curved Wire Segment • Find the field at point O due to the wire segment A’ACC’: B=0 due to AA’ and CC’ Due to the circular arc: • q=s/R, will be in radians Pham Hong QuangFaculty of Fundamental Sciences

28. 3.3 Biot-Savart law B at the Center of a Circular Loop of Wire Consider the previous result, with q = 2p Pham Hong QuangFaculty of Fundamental Sciences

29. 3.3 Biot-Savart law B along the axis of a Circular Current Loop Pham Hong QuangFaculty of Fundamental Sciences

30. 3.3 Biot-Savart law Magnetic Force Between Two Parallel Conductors • The field B2 due to the current in wire 2 exerts a force on wire 1 of F1 = I1ℓB2 Pham Hong QuangFaculty of Fundamental Sciences

31. 3.3 Biot-Savart law Define Ampere as the quantity of current that produces a force per unit length of 2 x 10-7 N/m for separation of 1 m • Thenμ0 Permeability of free space The SI unit of charge, the Coulomb (C), can be defined in terms of the Ampere When a conductor carries a steady current of 1 A, the quantity of charge that flows through a cross section of the conductor in 1 s is 1 C Pham Hong QuangFaculty of Fundamental Sciences

32. 3.4 Ampere’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 • Sum over the closed path Pham Hong QuangFaculty of Fundamental Sciences

33. 3.4 Ampere’s law Choose an arbitrary closed path around the current Sum all the products of B|| around the closed path Pham Hong QuangFaculty of Fundamental Sciences

34. 3.4 Ampere’s law Ampère’s Law to Find B for a Long Straight Wire Use a closed circular path The circumference of the circle is 2  r • This is identical to the result previously obtained Pham Hong QuangFaculty of Fundamental Sciences

35. 3.4 Ampere’s law Magnetic Field in a Solenoid • A cross-sectional view of a tightly wound solenoid • If the solenoid is long compared to its radius, we assume the field inside is uniform and outside is zero • Apply Ampère’s Law to the blue dashed rectangle Pham Hong QuangFaculty of Fundamental Sciences

36. Thank you! • Pham Hong Quang36PetroVietnam University