General Physics (PHY 2140) Lecture 8 Electricity and Magnetism 1. Magnetism Application of magnetic forces Ampere s law 2. Induced voltages and induction Magnetic flux http://www.physics.wayne.edu/~alan/2140website/main.htm Chapter 19-20 1
Lightning Review Last lecture: 1. Magnetism Magnetic field Magnetic force on a moving particle Magnetic force on a current Torque on a current loop Motion in a uniform field F qvbsinθ F BIlsinθ τ NBIAsinθ r mv/ qb Review Problem: How does the aurora borealis (the Northern Lights) work? 2
Magnetic Field of the Earth 3
The Aurora compared to a CRT For more info see the aurora home page: http://sec.noaa.gov/pmap/ 4
19.6 Motion of Charged Particle in magnetic field Consider positively charge particle moving in a uniform magnetic field. Suppose the initial velocity of the particle is perpendicular to the direction of the field. Then a magnetic force will be exerted on the particle and make follow a circular path. B in q v F r 5
The magnetic force produces a centripetal acceleration. F qvb 2 mv r The particle travels on a circular trajectory with a radius: r mv qb 6
Example: Proton moving in uniform magnetic field A proton is moving in a circular orbit of radius 14 cm in a uniform magnetic field of magnitude 0.35 T, directed perpendicular to the velocity of the proton. Find the orbital speed of the proton. r 0.14 m B 0.35 T m 1.67x10-27 kg q 1.6 x 10-19 C r mv qb v qbr m ( 19 )( )( 2 1.6 10 C 0.35T 14 10 m) ( 27 1.67 10 kg) 6 4.7 10 m s 7
Application: Mass Spectrometer r mv qb See prob. 30 in text 8
Review Problem 2 How does your credit card work? The stripe on the back of a credit card is a magnetic stripe, often called a magstripe. The magstripe is made up of tiny iron-based magnetic particles in a plastic-like film. Each particle is really a tiny bar magnet about 20-millionths of an inch long. The magstripe can be "written" because the tiny bar magnets can be magnetized in either a north or south pole direction. The magstripe on the back of the card is very similar to a piece of cassette tape. A magstripe reader (you may have seen one hooked to someone's PC at a bazaar or fair) can understand the information on the three-track stripe. 9
Review Example 1: Flying duck A duck flying horizontally due north at 15 m/s passes over Atlanta, where the magnetic field of the Earth is 5.0 10-5 T in a direction 60 below a horizontal line running north and south. The duck has a positive charge of 4.0 10-8 C. What is the magnetic force acting on the duck? B5.0 x 10-5 T. q 4.0 10-8 C v 15 m/s θ 60 F qvbsinθ F qvbsinθ 8 5 (4.0 10 )(15 / )(5.0 10 2.6x10 x C m s x T)sin60 13 N - to the west (into page) B v N Ground 60 10
Review Example 2: Wire in Earth s s B Field A wire carries a current of 22 A from east to west. Assume that at this location the magnetic field of the earth is horizontal and directed from south to north, and has a magnitude of 0.50 x 10-4 T. Find the magnetic force on a 36-m m length of wire. What happens if the direction of the current is reversed? B0.50 x 10-4 T. I 22 A l 36 m F max BIl F max BIl ( 4 0.50 10 T)( 22A)( 36m) 4.0 10 2 N 11
Application: The Velocity Selector F F electric magnetic qe qvb Magnetic force is up But the electric force is down. Since there is no deflection we can set these equal to each other. So we find: qe thus we have v qvb E/ B 12
Example 2: Consider the velocity selector. The electric field between the plates of the velocity selector is 950 V/m, and the magnetic field in the velocity selector has a magnitude of 0.930 T directed at right angles to the electric field. Calculate the speed of an ion that passes undeflected through the velocity selector. E 950 V/m B 0.93 T v? E 950 V/m v B 0.93 T 1021 m/ s v E/ B 13
19.7 Magnetic Field of a long straight wire Danish scientist Hans Oersted (1777-1851) 1851) discovered somewhat by accident that an electric current in a wire deflects a nearby compass needle. In 1820, he performed a simple experiment with many compasses that clearly showed the presence of a magnetic field around a wire carrying a current. I0 I aligned with earth s field aligned in a circular pattern 14
Magnetic Field due to Currents The passage of a steady current in a wire produces a magnetic field around the wire. Field form concentric lines around the wire Direction of the field given by the right hand rule. If the wire is grasped in the right hand with the thumb in the direction of the current, the fingers will curl in the direction of the field. Magnitude of the field I B μ o I 2π r 15
Magnitude of the field I B μ 2 o I π r r B μ o called the permeability of free space μ o 7 4π 10 Tm/ A 16
Ampere s Law Consider a circular path surrounding a current, divided in segments Δl, Ampere showed that the sum of the products of the field by the length of the segment is equal to μ o times the current. Andre-Marie Ampere B l I Δμ o enc I r Δl B 17
Consider a case where B is constant and uniform. B l B l B πr μ I Δ Δ 2 o enc Then one finds: B μ 2 o I π r 18
19.8 Magnetic Force between two parallel conductors 19
1 l 2 B 2 I 1 F 1 d I 2 B 2 μ I o 2 2π d μoi2 μoi1i2l F1 B2Il 1 Il 1 2πd 2πd F l 1 μ o I I 2 π 1 2 d Force per unit length ( Attractive ) 20
Definition of the SI unit Ampere F l 1 μ II o 2π d 1 2 Used to define the SI unit of current called Ampere. If two long, parallel wires 1 m apart carry the same current, and d the magnetic force per unit length on each wire is 2x10-7 N/m, then the current is defined to be 1 A. 21
Example 1: Levitating a wire Two wires, each having a weight per units length of 1.0x10-4 N/m, are strung parallel to one another above the surface of the Earth, one o directly above the other. The wires are aligned north-south. When their distance of separation is 0.10 m what must be the current in each h in order for the lower wire to levitate the upper wire. (Assume the two wires carry the same current). 1 l 2 d I 1 I 2 22
1 2 l F 1 B I 2 1 mg/l d I 2 Two wires, each having a weight per units length of 1.0x10-4 N/m, are strung parallel to one another above the surface of the Earth, one directly above the other. The wires are aligned north-south. When their distance of separation is 0.10 m what must be the current in each in order for the lower wire to levitate the upper wire. (Assume the two wires carry the same current). Weight of wire per unit length: mg/l 1.0x10-4 N/m Wire separation: d0.1 m I 1 I 2 F mg 1 μ l l 2π d 4 1.0 10 / I 2 o N m I 7.1A ( 7 )( 2 4π 10 Tm A I ) ( 2π )( 0.10m) 23
Example 2: magnetic field between the wires The two wires in the figure below carry currents of 3.00A and 5.00A in the direction indicated (into the page). Find the direction and magnitude of the magnetic field at a point midway between the wires. 3.00 A 5.00 A X X B i μ I o i 2π d 20.0 cm μo ( ) 6 Bnet 5.00A 3.00A 4 10 T 2π i0.1m Upwards 24
19.9 Magnetic Field of a current loop Magnetic field produced by a wire can be enhanced by having the wire in a loop. Δx 1 B I Δx 2 25
19.10 Magnetic Field of a solenoid Solenoid magnet consists of a wire coil with multiple loops. It is often called an electromagnet. 26
Solenoid Magnet Field lines inside a solenoid magnet are parallel, uniformly spaced and close together. The field inside is uniform and strong. The field outside is non uniform and much weaker. One end of the solenoid acts as a north pole, the other as a south pole. For a long and tightly looped solenoid, the field inside has a value: v B μ ni o 27
Solenoid Magnet B μ ni o n N/l : number of (loop) turns per unit length. I : current in the solenoid. μ o 7 4π 10 Tm/ A 28
Example: Magnetic Field inside a Solenoid. Consider a solenoid consisting of 100 turns of wire and length of o 10.0 cm. Find the magnetic field inside when it carries a current t of 0.500 A. μo N 100 l 0.100 m I 0.500 A 7 4π 10 Tm/A n B N 100 turns 1000 turns/m l 0.10 m μ ni o ( 4π 10 7 Tm/A)( 1000 turns/m)( 0.500A) B 6.28 10 4 T 29
Comparison: Electric Field vs. Magnetic Field Field Lines Electric Magnetic Source Charges Moving Charges Acts on Charges Moving Charges Force F Eq F q v B sin(θ) Direction Parallel E Perpendicular to v,b Opposites Charges Attract Currents Repel 30
Chapter 20 Induced EMF and Induction 31
Introduction Previous chapter: electric currents produce magnetic fields (Oersted( Oersted s experiments) Is the opposite true: can magnetic fields create electric currents? 32
20.1 Induced EMF and magnetic flux Just like in the case of electric flux, consider a situation where the magnetic field is uniform in magnitude and direction. Place a loop in the B-field. B The flux, Φ,, is defined as the product of the field magnitude by the area crossed by the field lines. Φ BA BAcosθ B where is the component of B perpendicular to the loop, θ is the angle between B and the normal to the loop. Units: T mt 2 or Webers (Wb) Definition of Magnetic Flux The value of magnetic flux is proportional to the total number of magnetic field lines passing through the loop. 33
Problem: determining a flux A square loop 2.00m on a side is placed in a magnetic field of strength 0.300T. If the field makes an angle of 50.0 with the normal to the plane of the loop, determine the magnetic flux through the loop. 34
A square loop 2.00m on a side is placed in a magnetic field of strength 0.300T. If the field makes an angle of 50.0 with the normal to the plane of the loop, determine the magnetic flux through the loop. Solution: Given: L 2.00 m B 0.300 T θ 50.0 Find: From what we are given, we use 2 ( )( ) 2 Φ BAcosθ 0.300T 2.00m cos(50.0 ) 0.386 Tm Φ? 35
20.1 Induced EMF and magnetic flux Faraday s experiment Picture Molecular Expressions Two circuits are not connected: no current? However, closing the switch we see that the compass needle moves and then goes back to its previous position Nothing happens when the current in the primary coil is steady But same thing happens when the switch is opened,, except for the needle going in the opposite direction What is going on? 36