Chapter17 Electromagnetic Induction STPM phsyics

Outline

E. ELECTRICITY AND MAGNETISM

17.1 Magnetic flux 17.2 17.3 Self-inductance L 17.4 Energy stored in an inductor 17.5 Mutual induction

Chapter 17 Electromagnetic induction

Objectives (a) define magnetic flux = B A BA (b (c) derive and use the equation for induced e.m.f. in linear conductors and plane coils in uniform magnetic fields (d) explain the phenomenon of self-induction, and define self-inductance (e) use the formulae E Ldl/dt, LI=N (f) derive and use the equation for selfinductance of a solenoid L = 0N2A/l

Objectives (g) use the formula for the energy stored in an Inductor U = ½ LI2 (j) explain the phenomenon of mutual Induction, and define mutual inductance; (i) derive an expression for the mutual inductance between two coaxial solenoids of the same cross-sectional area M = 0NpNsA/lp

17.1 Magnetic Flux In the easiest case, with a constant magnetic field B, and a flat surface of area A, the magnetic flux is

17.1 Magnetic flux

A B B=

B·A

Units : 1 tesla x m2 = 1 weber

17.1 Magnetic flux Definition: Number of magnetic field lines that pass through an area (usually a loop) = BAcos ; Units: Weber (Wb) = Flux measured in Webers (Wb); 1 Wb = 1Tm2 B = Magnetic Field (T) A = area of region that the flux is passing through (m2) = angle formed between the magnetic field lines and the area. A changing magnetic flux creates an induced EMF

17.1 Magnetic flux Magnetic flux: is defined as the product of the magnetic field B (also called magnetic flux density) and the area A of the plane of the loop through which it passes, where is the angle between the direction of B and a line drawn perpendicular to the plane of the loop. If

= BA

= BA cos

17.1 Magnetic flux

17.1 Magnetic flux

A change in flux can occur in two ways: 1. By changing the flux density B going through a constant loop area A:

17.1 Magnetic flux

17.1 Magnetic flux

Faraday referred to changes in B field, area and orientation as changes in magnetic flux inside the closed loop The formal definition of magnetic flux ( B (analogous to electric flux) B

= B dA

When B is uniform over A, B

2. By changing the effective area A in a magnetic field of constant flux density B:

B

A

The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field Magnetic flux is defined in a manner similar to that of electrical flux

= BA cos

Magnetic flux is a measure of the # of B field lines within a closed area (or in this case a loop or coil of wire) Changes in B, A and/or change the magnetic flux electromotive force (& thus current) in a closed wire loop

Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop

(Electromagnetic Induction) In 1831, Michael Faraday discovered that when a conductor cuts magnetic flux lines, an emf is produced. The induced emf in a circuit is proportional to the rate of change of magnetic flux, through any surface bounded by that circuit. e = - d B / dt

17.1 Magnetic flux Flux through coil changes because bar magnet is moved up and down.

Moving the magnet induces a current I. Reversing the direction reverses the current. Moving the loop induces a current. The induced current is set up by an induced EMF.

S

N v I

Changing the current in the right-hand coil induces a current in the left-hand coil. The induced current does not depend on the size of the current in the right-hand coil. The induced current depends on dI/dt. (right) dI/dt EMF

(left)

S

I

When B is not constant, or the surface is not flat, one must do an integral. Break the surface into bits dA. The flux through one bit is d B = B · dA = B dA cos Sum the bits: B B dA Bcos dA

Relative motion between a conductor and a magnetic field induces an emf in the conductor. The direction of the induced emf depends upon the direction of motion of the conductor with respect to the field. The magnitude of the emf is directly proportional to the rate at which the conductor cuts magnetic flux lines. The magnitude of the emf is directly proportional to the number of turns of the conductor crossing the flux lines.

Moving the magnet changes the flux B (1). Changing the current changes the flux B (2). Faraday: changing the flux induces an emf. N

1) i

S

N

=-d

.

dA B

When no voltage source is present, current will flow around a closed loop or coil when an electric field is present parallel to the current flow. Charge flows due to the presence of electromotive force, or emf ( ) on charge carriers in the coil. The emf is given by: = · dl = iRcoil

ds

i

E Changing Magnetic Field

di/dt

B

/dt equals the rate of change of the flux through that loop

The emf induced around a loop

An E-field is induced along a coil when the magnetic flux changes, producing an emf (e). The induced emf is related to: The number of loops (N) in the coil The rate at which the magnetic flux is changing inside the loop(s), or E dl

N

d B dt

N

d (BA cos ) dt

Note: magnetic flux changes when either the magnetic field (B), the area (A) or the orientation (cos f) of the loop changes: d B dB d B =B cos dA =A cos dt dt dt dt

d cos d B =BA dt dt

Changing Area A loop of wire (N=10) contracts from 0.03 m2 to 0.01 m2 in 0.5 s, where B is 0.5 T and is 0o (Rloop is 1 ).

A magnet moves toward a loop of wire (N=10 & A is 0.02 m2). dB During the movement, B -NA cos dt changes from is 0.0 T to 1.5 T in 3 s (Rloop is 2 ). 1) What is the induced in the loop? 2) What is the induced current in the loop?

EMF S

i

B S

2)

v

-NB cos

dA dt

1) What is the induced in the loop? 2) What is the induced current in the loop?

Changing Orientation -NAB

d(cos ) dt

-NAB

d(cos dt

and therefore the direction of any induced current.

or

NAB

)

sin t

A loop of wire (N=10) rotates from 0o to 90o in 1.5 s, B is 0.5 T and A is 0.02 m2 (Rloop is 2 ). 1)What is the average angular frequency, ? 2)What is the induced in the loop? 3)What is the induced current in the loop?

If we move the magnet towards the loop the flux of B will increase. the current induced in the loop will generate a field B opposed to B.

straight, with less effort. The induced emf is directed so that any induced current flow will oppose the change in magnetic flux (which causes the induced emf). This is easier to use than to say ... Decreasing magnetic flux emf creates additional magnetic field Increasing flux emf creates opposed magnetic field

If we move the magnet towards the loop the flux of B will increase. the current induced in the loop will generate a field B opposed to B.

B

B B

N

S

N

v

I

B

S

v

I

Lenz's Law When the magnetic flux changes within a loop of wire, the induced current resists the changing flux The direction of the induced current always produces a magnetic field that resists the change in magnetic flux (blue arrows) i

B Magnetic flux,

B

B Increasing

B

i

B Increasing

When an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant.

B

Lenz's Law In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant.

Lenz's Law The induced current produces magnetic fields which tend to oppose the change in magnetic flux that induces such currents. conducting loop placed in a magnetic field. We follow the procedure below: 1. Define a positive direction for the area vector A. 2. Assuming that B is uniform, take the dot product of B and A. This allows for the determination of the sign of the magnetic flux B. 3. Obtain the rate of flux change d B/dt by differentiation. There are three possibilities: d

B

dt

0

induced emf

0

0 0

induced emf induced emf

0 0

Lenz's Law

Lenz's Law

4. Determine the direction of the induced current using the right-hand rule. With your thumb pointing in the direction of A, curl the fingers around the closed loop. The induced current flows in the same direction as the way your fingers curl if >0, and the opposite direction if