Electromagnetism: Faraday's Law, Lenz's Law, and Induced EMF, Slides of Electromagnetic Engineering

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CHAPTER 28

Electromagnetic Induction Magnetic Flux Induced EMF and Faraday’s Law Lenz’s Law Motional EMF Eddy Currents Inductance Magnetic Energy RL Circuits

2 The flux of any vector field through a open surface is calculated in the same way as the flux of an electric field through a open surface.

MAGNETIC FLUX

S n S m B n ˆ dA B dA

 Magnetic flux trough the open surface

The unit of magnetic flux is that of magnetic field strength multiplied by area, namely, the tesla-meter-squared, which is called a weber ( Wb ): If the surface is flat and has an area A , and if B is uniform We will consider the flux through a surface bounded by a coil that has several turns of wire. If the coil has N turns, the flux through the surface is N multiplied by the flux through each turn. where A is the area of the flat surface bounded by a single turn. 2 1 Wb  1 T  m  (^) m  B  n ˆ A  BA cos   Bn A   NBA cos m

4 A square loop of sides a lies in the yz plane with one corner at the origin. A varying magnetic field B = ky passes through the loop and points in the + x direction. The magnetic flux through the loop is A. ka 2 B. ka 2 / C. ka 3 / D. ka 3 / E. None of these is correct. 2 ˆ 3 0 0 ka B ndA Bady ak ydy a a s     

 

x

y

z

a

a

B

dy

In the early 1830s, Michael Faraday in England discovered that in a changing magnetic field a changing magnetic flux through a surface bounded by a closed stationary loop of wire induces a current in the wire. The emf and current caused by such changing magnetic flux are called induced emf and induced current. The process itself is referred to as induction. Faraday also discovered that in a static magnetic field, at changing magnetic flux through a surface bounded by a moving loop of wire, induces an emf in the wire. 1 2 B

The electric fields that we studied in earlier chapters resulted from static electric charges. Such electric fields are conservative. The electric field in conductor loop associated with a changing magnetic field is nonconservative. The circulation electrical field about close loop C is equal to the induced emf in the loop of wire with resistance R. dt d B n dA dt d E d l m ind C nc S          ˆ    induced emf for a stationary circuit in changing magnetic field Faraday’s Law I R ind ind  

LENZ’S LAW

The minus sign in Faraday’s law has to do with the direction of the induced emf. The induced emf is in such a direction as to oppose the change that produced it. Lenz’s Law Lenz’s Law is reflection of fundamental principal of conservation energy. The magnetic moment of the loop due to the induced current is such as to oppose the motion of the bar magnet. When the current changes in the coil, there is a large emf induced in the coil opposing the change.

 BAcos  m

dt

d

m ind  here   0

A copper ring lies in the yz plane as shown. The magnet's long axis lies along the x axis. Induced current flows through the ring as indicated. The magnet A. must be moving away from the ring. B. must be moving toward the ring. C. must remain stationary to keep the current flowing.

According to Faraday's law, a necessary and sufficient condition for an electromotive force to be induced in a closed circuit loop is the presence in the loop of A.a magnetic field. B.magnetic materials. C.an electric current. D.a time-varying magnetic flux. E. a time-varying magnetic field.

The magnetic flux through a loop is made to vary according to the relation where the units are SI. The magnitude of emf induced in the loop when t = 2 s is A.38 V B.39 V C.40 V D.31 V E. 19 V 6 7 1 2 m   t  t 

16

MOTIONAL EMF

Motional emf is any emf induced by the motion of a close loop conductor in a region in which there exists a magnetic field. A conducting rod sliding on conducting rails in a magnetic field. As the rod moves to the right, the area of the surface S increases, so the magnetic flux through S into the slide increases. Taking the time derivative of both sides gives A conducting rod with current that is moving through a magnetic field experiences a magnetic force (Lorenz force) that has a leftward component. Therefore, moving rod to the right, we produce positive work to overcame Lorenz force. The power we produce: P  F  v F  Il  B P  IlBv   I or   lBv      We derive same equation without use Faraday law (except correct sign). m  B^^  n ˆ A  BnA ^ Blx   Blv dt dx Bl dt d (^) m    Blv dt d (^) m ind    

A rectangular coil moving at a constant speed v enters a region of uniform magnetic field from the left. While the coil is entering the field, which arrow shows the direction of the magnetic force?

EDDY CURRENTS

Changing flux often induces circulating currents, which are called eddy currents , in a piece of bulk metal like the core of a transformer. The heat produced by such current constitutes a power loss in the transformer. Eddy currents are frequently undesirable because power is lost due to Joule heating by the current, and this dissipated energy must be transferred to the environment. The power loss can be reduced by increasing the resistance of the possible paths for the eddy currents, Eddy currents are not always undesirable. For example, eddy currents are often used to damp unwanted oscillations (example sensitive mechanical balance) or for brake.

Eddy currents A.are a consequence of changing magnetic flux. B.generate heat and result in power loss. C.can be used for damping and braking purposes. D.are described by both Faraday's and Lenz's laws. E. All of these are correct.