Induction Last week we learned that a current carrying loop in a magnetic field experiences a torque If we start with a loop with no current in a magnetic field and force the loop to rotate we find that a current is induced in the loop Physics for Scientists Engineers 2 Further if we start with a loop with no current and turn on a magnetic field without moving the coil again a current is induced in the loop Spring Semester 2005 Lecture 25 These effects are described by Faraday s Law of Induction and are the basis of electric motors and electric power generation from mechanical motion February 23 2005 Physics for Scientists Engineers 2 1 Faraday s Experiments February 23 2005 Physics for Scientists Engineers 2 2 Faraday s Experiments 2 Consider the situation in which we have a wire loop connected to an ammeter so that we can measure current flowing in the loop When we move the magnet toward the loop we induce a positive current in the loop We hold a bar magnet some distance from the loop pointing the north pole of the magnet toward the loop When we move the magnet toward the loop we induce a negative current in the loop Now we turn the magnet around so that the south pole points toward the loop While the magnet is stationary there is no current flowing in the loop What happens if we move the magnet February 23 2005 Physics for Scientists Engineers 2 3 February 23 2005 Physics for Scientists Engineers 2 4 Faraday s Experiments 3 Faraday s Experiments 4 Now let s point the north pole toward the loop but move away from the loop We can create similar effects by placing a second loop near the first loop but with a more quantitative result as shown below We get a negative current If a constant current is flowing through loop 1 no current will be induced in loop 2 We turn the magnet around so that the south pole points toward the loop and move away from the loop If we increase the current in the loop 1 we observe that a current is induced in the loop 2 in the opposite direction We get a positive current Thus not only does the changing current in the first loop induce a current in the first loop the induced current is in the opposite direction February 23 2005 Physics for Scientists Engineers 2 5 February 23 2005 Faraday s Experiments 5 Physics for Scientists Engineers 2 6 Law of Induction From these observations we see that a changing magnetic field induces a current in a loop Now if we have the current flowing in loop 1 in the same direction as before and decrease the current as shown below we induce a current flowing loop 2 in the same direction as the current in loop 1 We can visualize the change in magnetic field as a change in the number of magnetic field lines passing through the loop Faraday s Law of Induction states that An emf is induced in a loop when the number of magnetic field lines passing through the loop changes with time The rate of change of magnetic field lines determines the induced emf February 23 2005 Physics for Scientists Engineers 2 7 February 23 2005 Physics for Scientists Engineers 2 8 Magnetic Flux Magnetic Flux Special Case To quantify the amount of magnetic field lines we define the magnetic flux in analogy to the electric flux Consider the special case of a flat loop of area A in a constant magnetic field B When we introduced Gauss Law for the electric field we defined the electric flux as E E dA In this case we can re write the magnetic flux as B BA cos is the angle between the surface normal vector of the plane of the loop and the magnetic field For the magnetic field we can define magnetic flux in analogy as B B dA If the magnetic field is perpendicular to the plane of the loop 0 B BA The unit of magnetic flux is the weber Wb If the magnetic field is parallel to the plane of the loop 1 Wb 1 Tm 2 February 23 2005 90 B 0 Physics for Scientists Engineers 2 9 February 23 2005 Faraday s Law of Induction The magnitude of the Vemf induced in a conducting loop is equal to the time rate of change of the magnetic flux from the loop This induced emf tends to oppose the flux change The magnetic flux for this case is B BA cos d B dt We can change the magnetic flux in several ways including changing the magnitude of the magnetic field changing the area of the loop or by changing the angle the loop with respect to the magnetic field Physics for Scientists Engineers 2 Let s explore Faraday s Law for the special case of a flat loop inside a magnetic field that is constant in space but that can vary in time The induced emf is then Faraday s Law of Induction is thus contained in the equation February 23 2005 10 Induction in a Flat Loop We can then recast Faraday s Law of Induction in terms of the magnetic flux as Vemf Physics for Scientists Engineers 2 11 d B d BA cos dt dt Carrying out the derivative we get Vemf Vemf A cos dB dA d B cos ABsin dt dt dt Taking d dt we get Vemf A cos February 23 2005 dB dA B cos ABsin dt dt Physics for Scientists Engineers 2 12 Induction in a Flat Loop Special Cases Example Changing Magnetic Field If we leave two of the three variables A B constant then we can have the following three special cases A direct current of 600 mA is delivered to an ideal solenoid resulting in a magnetic field of 0 025 T We leave the area of the loop and its orientation relative to the magnetic field constant but vary the magnetic field in time A constant Vemf Then the current is increased according to dB A cos dt i t i0 1 2 4s 2t 2 We leave the magnetic field as well as the orientation of the loop relative to the magnetic field constant but change the area of the loop that is exposed to the magnetic field B constant Vemf B cos Question dA dt We leave the magnetic field constant and keep the area of the loop fixed as well but allow the angle between the two to change as a function of time If a circular loop of radius 3 4 cm with 200 windings is located inside the solenoid and perpendicular to the magnetic field what is the induced voltage at t 2 0 s in this loop A B constant Vemf ABsin February 23 2005 Physics for Scientists Engineers 2 13 Example Changing Magnetic Field 2 Physics for Scientists Engineers 2 14 Example Changing Magnetic Field 3 Answer Answer First the area of the loop is computed the number of winding acts as …
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