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Lecture 14 Chapter 29 Ampere’s law Course website: http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Today we are going to discuss: Chapter 29: Section 29.6 Ampere’s Law Section 29.5 Skip PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Bio-Savart Law The magnetic field of a charged particle q moving with velocity v is given by the Biot-Savart law: Note that the component of B parallel to the line of motion is zero. PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics ConcepTest B field of q A) Into the screen What is the direction of the magnetic field at the position of the dot? B) Out of the screen C) Up D) Down E) Left Oersted Experiment In 1819 Hans Christian Oersted discovered that an electric current in a wire causes a compass to turn. It was a demonstration that an electric current produces magnetic field. This is probably one of the most important experiments ever done. This observation showed that there is connection between electricity and magnetism. https://www.youtube.com/watch?v=Gtp51eZkwoI (History of Electricity, start at 1.04.20) PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics While performing his electric demonstration, Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. Let’s understand what is still missing in our magnetism picture. Ampere’s Law Electric Field Magnetic Field From Coulomb’s law 1 ̂ 4 Bio-Savart law ̂ 4 Gauss’s Law ∙ PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics There must be something similar for B Ampere’s Law The line integral of the magnetic field around the curve is given by Ampère’s law: I1 ∙ + + ∙ I3 ∙ Closed loop (Amperian) B 1) An Amperian loop is imaginary 2) It is a closed loop (any path can be used) 3) Choose direction (up to you). It gives us which current is positive/negative (use a right-hand rule: curl your fingers in a chosen direction and an outstretched thumb shows a positive current direction) Amperian loop ∙ B I2 These currents pass through the area bounded by the loop, so they are enclosed, Iin I3 So I1 is positive; I2 is negative ∙ Amperian loop Ampère’s law is very useful for a problem with a high degree of symmetry. PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics ConcepTest Ampere’s Law A) 0 A The line integral of B around the loop is 0 · 7.0 A. Current I3 is B) 1 A out of the screen C) 1 A into the screen D) 5 A out of the screen E) 5 A into the screen Assume I3 is out of the page ∮ ∙ =7 1 Minus means our original assumption was wrong, it is into the screen Example Magnetic field of a current-carrying wire The wire has cylindrical symmetry so that we can easily use Ampere’s law. One moving charge creates magnetic field lines centered on the motion line: Now we have many moving charges (not just one). The field pattern must be the same. So we’ll take our Amperian loop to be a concentric circles of r. PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Solenoid A solenoid is a helical coil of wire with the same current I passing through each loop in the coil. A uniform magnetic field can be generated with a solenoid. PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Steps to make a solenoid Magnetic field lines produced with a straight wire Let’s bend the wire into a loop Now, let’s add more loops A uniform magnetic field is generated with a solenoid. PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Example The Magnetic Field of a Solenoid Take-home quiz B Along the sides (bc, da), the line integral is zero since the field is perpendicular to the path. ds ds Along the bottom (ab), the line integral is zero since B 0 outside the solenoid. ∙ 0 ∙ B B=0 0 ∙ abcda ds B 0 ∙ ∙ ∙ 0 Amperian loop abcda ∥ There are N loops with current I enclosed by an Amperian loop, so ∙ / Uniform field where n N/l is the number of turns per unit length. PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Application of solenoids This patient is undergoing magnetic resonance imaging (MRI). The large cylinder surrounding the patient contains a solenoid that is wound with superconducting wire to generate a strong uniform magnetic field. B=1.2 T, I=100 A PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics The Magnetic Field Outside a Solenoid • • The magnetic field outside a solenoid looks like that of a bar magnet. Thus a solenoid is an electromagnet PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Electric Field Magnetic Field From Coulomb’s law 1 ̂ 4 Biot-Savart law ̂ 4 Gauss’s Law Ampere’s Law ∙ ∙ So, now we know how to find magnetic fields using Bio-Savart and Ampere’s laws. Now, the question is “how does a magnetic field interact with material (which consists of charges and current)?” Magnetic force on a moving charge PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Magnetic force on current Magnetic force on a moving charge PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics The Magnetic Force on a Moving Charge After Oersted’s discovery, there were many other experiments with magnetic fields, currents, charges, etc. It was found that B exerts a force on a moving charge. The magnetic force on a charge q as it moves through a magnetic field B with velocity v is: where is the angle between v and B. PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics Thank you See you next time PHYS.1440 Lecture14 A.Danylov Department of Physics and Applied Physics