Fundamental Physics II PETROVIETNAM UNIVERSITY FACULTY OF FUNDAMENTAL SCIENCES Vungtau, 2013 Pham Hong Quang

2 CHAPTER 3 Pham Hong Quang 2 PetroVietnam University Magnetic Fields

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

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 4 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.

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 5 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. The magnetic field line

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

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 7 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. Right-hand rule

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 8 The Definition of magnitude Magnetic 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 q 0, v is the velocity of the charge which makes an angle  with the direction of the magnetic field. SI Unit of Magnetic Field:

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 9 Differences of Electric and magnetic Fields 1.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. 2.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.

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 10 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.

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

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 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.) Crossed Fields: Discovery of the Electron

3.1 The Magnetic Field Pham Hong Quang Faculty of Fundamental Sciences 13 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!!

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

3.2 Magnetic Force on a Current-Carrying Wire Pham Hong Quang Faculty of Fundamental Sciences 15 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. Check Your Understanding

3.2 Magnetic Force on a Current-Carrying Wire Pham Hong Quang Faculty of Fundamental Sciences 16 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: Or, since A = hw, τ =I.A.B Τ= (IhB) (w/2) +(IhB)(w/2) =I (hw)B

3.2 Magnetic Force on a Current-Carrying Wire Pham Hong Quang Faculty of Fundamental Sciences 17 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

3.2 Magnetic Force on a Current-Carrying Wire Pham Hong Quang Faculty of Fundamental Sciences 18 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:

3.2 Magnetic Force on a Current-Carrying Wire Pham Hong Quang Faculty of Fundamental Sciences 19 The magnitude of is given by The Magnetic Moment of Coil 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.

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

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 21 If ds and ř are parallel, the contribution is zero! BSL is also an inverse square law!! Assume a small segment of wire ds causing a field dB:  o = 4  T m/A

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 22 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.

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 23 The integral is over the entire current distribution To find the total field, sum up the contributions from all the current elements I ds

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

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 25

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 26 If the conductor is an infinitely long, straight wire,    = 0 and    = 

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 27 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:  s/R, will be in radians

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 28 B at the Center of a Circular Loop of Wire Consider the previous result, with  = 2 

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

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 30 The field B 2 due to the current in wire 2 exerts a force on wire 1 of F 1 = I 1 ℓ B 2 Magnetic Force Between Two Parallel Conductors

3.3 Biot-Savart law Pham Hong Quang Faculty of Fundamental Sciences 31 Define Ampere as the quantity of current that produces a force per unit length of 2 x N/m for separation of 1 m Then μ 0 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

3.4 Ampere’s law Pham Hong Quang Faculty of Fundamental Sciences 32 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

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

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

3.4 Ampere’s law Pham Hong Quang Faculty of Fundamental Sciences 35 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 Magnetic Field in a Solenoid

36 Pham Hong Quang 36 PetroVietnam University Thank you!