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- Title: Week8_1
- Type: Notes
- School: Drexel
- Course: PHYS 102
- Term: Spring
Law Ampere's Ampere's Law r r states that the line integral of B ds around any closed r path ds equals oI where I is the total steady current passing through any surface bounded by the closed path r r B ds r r B ds = o I Ampere's Law, cont Ampere's Law describes the creation of magnetic fields by all continuous current configurations Most useful for this course if the current configuration has a high degree of symmetry Put the thumb of your right hand in the direction of the current through the amperian loop and your figures curl in the direction you should integrate r r around the loop B ds Amperian Loops Each portion of the path satisfies one or more of the following conditions: The value of the magnetic field can be argued by symmetry to be constant over the portion of the path The dot product can be expressed as a simple algebraic product B ds The vectors are parallel r r B ds Amperian Loops, cont Conditions: The dot product is zero The vectors are perpendicular The magnetic field can be argued to be zero at all points on the portion of the path r r B ds Field Due to a Long Straight Wire From Ampere's Law Want to calculate the magnetic field at a distance r from the center of a wire carrying a steady current I The current is uniformly distributed through the cross section of the wire Field Due to a Long Straight Wire Results From Ampere's Law Outside of the wire, r > R r r B ds = B(2 r ) = oI oI B= 2 r Inside the wire, we need I', the current inside the amperian circle r r B ds = B(2 r ) = oI ' I B = o 2 r 2 R r2 I' = 2 I R Field Due to a Long Straight Wire Results Summary The field is proportional to r inside the wire The field varies as 1/r outside the wire Both equations are equal at r = R Magnetic Field of a Toroid Find the field at a point at distance r from the center of the toroid The toroid has N turns of wire o NI B= 2 r r r B ds = B(2 r ) = oNI Weak field exists outside of The toroid Magnetic Field of a Solenoid A solenoid is a long wire wound in the form of a helix A reasonably uniform magnetic field can be produced in the space surrounded by the turns of the wire Each of the turns can be modeled as a circular loop The net magnetic field is the vector sum of all the fields due to all the turns Magnetic Field of a Solenoid, Description The field lines in the interior are Approximately parallel to each other Uniformly distributed Close together This indicates the field is strong and almost uniform Magnetic Field of a Tightly Wound Solenoid The field distribution is similar to that of a bar magnet As the length of the solenoid increases The interior field becomes more uniform The exterior field becomes weaker Ideal Solenoid Characteristics An ideal solenoid is approached when The turns are closely spaced The length is much greater than the radius of the turns For an ideal solenoid, the field outside of solenoid is negligible The field inside is uniform Ampere's Law Applied to a Solenoid Ampere's Law can be used to find the interior magnetic field of the solenoid Consider a rectangle with side l parallel to the interior field and side w perpendicular to the field The side of length l inside the solenoid contributes to the field This is path 1 in the diagram Ampere's Law Applied to a Solenoid, cont Applying Ampere's Law gives r r B ds = path1 r r B ds = B path1 ds = Bl The total current through the rectangular path equals the through current each turn multiplied by the number of turns r r B ds = Bl = oNI Magnetic Field of a Solenoid, final Solving Ampere's Law for the magnetic field is N B = o I = o nI l n = N / l is the number of turns per unit length This is valid only at points near the center of a very long solenoid Consider a solenoid that is very long compared with the radius. Of the following choices, the most effective way to increase the magnetic field in the interior of the solenoid is to 50% 50% 1. 2. 3. double its length, keeping the number of turns per unit length constant, reduce its radius by half, keeping the number of turns per unit length constant, or overwrap the entire solenoid with an additional layer of current-carrying wire. 0% ha ... ... ke en tir e th e ov er w ra p by ... ng th , N B = o I = o nI l its ub le do re du c e its ra di le us Chapter 23 Faraday's Law and Induction Michael Faraday 1791 1867 Great experimental physicist Contributions to early electricity include Invention of motor, generator, and transformer Electromagnetic induction Laws of electrolysis Induction An induced current is produced by a changing magnetic field There is an induced emf associated with the induced current A current can be produced without a battery present in the circuit Faraday's Law of Induction describes the induced emf EMF Produced by a Changing Magnetic Field, 1 A loop of wire is connected to a sensitive ammeter When a magnet is moved toward the loop, the ammeter deflects The deflection indicates a current induced in the wire EMF Produced by a Changing Magnetic Field, 2 When the magnet is held stationary, there is no deflection of the ammeter Therefore, there is no induced current Even though the magnet is inside the loop EMF Produced by a Changing Magnetic Field, 3 The magnet is moved away from the loop The ammeter deflects in the opposite direction EMF Produced by a Changing Magnetic Field, Summary The ammeter deflects when the magnet is moving toward or away from the loop The ammeter also deflects when the loop is moved toward or away from the magnet An electric current is set up in the coil as long as relative motion occurs between the magnet and the coil This is the induced current that is produced by an induced emf Faraday's Experiment Set Up A primary coil is connected to a switch and a battery The wire is wrapped around an iron ring A secondary coil is also wrapped around the iron ring There is no battery present in the secondary coil The secondary coil is not electrically connected to the primary coil Faraday's Experiment Findings At the instant the switch is closed, the galvanometer (ammeter) needle deflects in one direction and then returns to zero When the switch is opened, the galvanometer needle deflects in the opposite direction and then returns to zero The galvanometer reads zero when there is a steady current or when there is no current in the primary circuit Faraday's Experiment Conclusions An electric current can be produced by a timevarying magnetic field This would be the current in the secondary circuit of this experimental set-up The induced current exists only for a short time while the magnetic field is changing This is generally expressed as: an induced emf is produced in the secondary circuit by the changing magnetic field The actual existence of the magnetic field is not sufficient to produce the induced emf, the field must be changing
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