E. ELECTRICITY AND MAGNETISM

Chapter 17 Electromagnetic induction

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

inductance 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

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

B = B · A Units : 1 tesla x m2 = 1 weber

A

B

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

= BA

= BA cos

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

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

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

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)

When B is uniform over A,

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

B = B dA AB

B = BA cos

17.1 Magnetic flux 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

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.

N S

I v

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.

I

dI/dt

S EMF

(right) (left)

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.

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 cosSum the bits:

B N S

dA

B

.

B B dA Bcos dA

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

= - d B /dt

The emf induced around a loop

equals the rate of change of the flux through that loop

N S

i v

1) i

di/dt

S

EMF 2)

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

dsE

i

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

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

d dB=A cosdt dt

Bd dA=B cosdt dt

B d cosd =BAdt dt

B

)cos(BAdtdN

dtdNldE B

Changing Magnetic Field

dB-NA cosdt

A magnet moves toward a loop of wire (N=10 & A is 0.02 m2). During the movement, B 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?

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

).

dA-NB cosdt

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

Changing Orientation

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?

( )

( )

d cos-NABdt

d cos -NAB

r

dt

o

tNAB sin

and therefore the direction of any induced current.

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.

N S

I v

B B

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.

N S

I v

B B

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)

B

Magnetic flux, B

B

Increasing B

i

B

Increasing B

i B

Increasing B

i

B

Increasing B

i

B

Increasing B

i

Lenz's Law 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.

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:

0 emf induced00 emf induced00 emf induced0

dtd B

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 <0 , as shown in figure below.

Lenz's Law In the figure below we illustrate the four possible scenarios of time-varying magnetic

determine the direction of the induced current .

Lenz's Law The situation can be summarized with the following sign convention: The positive and negative signs of I correspond to a counterclockwise and clockwise current, respectively.

B d B/dt I

+ + - - - + +

- + - - - + +

Lenz's Law

may be applied, consider the situation where a bar magnet is moving toward a conducting loop with its north pole down, as shown in figure below.

Lenz's Law With the magnetic field pointing downward and the area vector A pointing upward, the magnetic flux is negative, i.e. B = - B A < 0, where A is the area of the loop. As the magnet moves closer to the loop, the magnetic field at a point on the loop increases (dB/dt>0), producing more flux through the plane of the loop.

Lenz's Law B/dt = - A (dB/dt) < 0, implying

a positive induced emf, > 0, and the induced current flows in the

counterclockwise direction. The current then sets up an induced magnetic field and produces a positive flux to counteract the change. The situation described here corresponds to that illustrated in the slide above position c.

Lenz's Law Alternatively, the direction of the induced current can also be determined from the point of view of magnetic force. that the induced emf must be in the direction that opposes the change. Therefore, as the bar magnet approaches the loop, it experiences a repulsive force due to the induced emf. Since like poles repel, the loop must behave as if it were a bar magnet with its north pole pointing up. Using the right-hand rule, the direction of the induced current is counterclockwise, as view from above. Figure above illustrates how this alternative approach is used.

Motional EMF Consider a conducting bar of length l moving through a uniform magnetic field which points into the page, as shown in Figure below. Particles with charge q>0 inside experience a magnetic force FB = q v x B which tends to push them upward, leaving negative charges on the lower end.

Motional EMF The separation of charge gives rise to an electric field E inside the bar, which in turn produces a downward electric force Fe = qE. At equilibrium where the two forces cancel, we have qvB = qE or E = v B. Between the two ends of the conductor, there exists a potencial difference given by: Vab = Va Vb = = El = Blv Since arises from the motion of the conductor, this potential difference is called the motional emf. In general, motional emf around a closed conducting loop can be written as:

= (v B)ds where ds is a differential length element.

Motional EMF Now suppose the conducting bar moves through a region of uniform magnetic field B = - Bk (pointing into the page) by sliding along two frictionless conducting rails that are at a distance l apart and connected together by a resistor with resistance R, as shown in Figure below.

Motional EMF Let an external force Fext be applied so that the conductor moves to the right with a constant velocity v = vi. The magnetic flux through the closed loop formed by the bar and the rails is given by

B = BA = Blx Thus

the induced emf is: = - d /dt = - d/dt (Blx)

= - Bl dx/dt = - Blv where dx/dt = v is simply the speed of the bar.

Motional EMF The corresponding induced current is : I = l l/R = Blv/R and its direction is

The equivalent circuit diagram is shown in Figure below.

Motional EMF The magnetic force experienced by the bar as it moves to the right is:

which is in opposite direction of v. For the bar to move at a constant velocity, the net force acting on it must be zero. That means that the external agent must supply a force:

Fext = - FB = + ( B² l² v/R )i The power delivered by Fext is equal to the power dissipated in the resistor: P = Fext v = Fext v = ( B² l² v / R) v = (Blv)²/R = ²/R =

I²R as required by energy conservation.

iR

vlBiIlBkBjlIFB )()(22

Motional EMF From the analysis above, in order for the bar to move at a constant speed, an external agent must constantly supply a force Fext. What happens if at t=0 , the speed of the rod is vo, and the external agent stops pushing? In this case, the bar will slow down because of the magnetic force directed to the left. From

FB = - B²l²v / R = ma = m dv/dt or dv/dt = - B²l² / mR dt = - dt/ Where = mR / B²l². Upon integration, we obtain : v (t) = vo exp. t / Thus, we see that the speed decreases exponentially in the absence of an external agent doing work. In principle, the bar never stops moving. However, one may verify that the total distance traveled is finite.

Consider a coil of radius 5 cm with N = 250 turns. A magnetic field B, passing through it, changes in time: B(t)= 0.6 t [T] (t = time in seconds) The total resistance of the coil is 8 W. What is the induced current ?

current.

B

The change in B is increasing the upward flux through the coil. So the induced current will have a magnetic field whose flux (and therefore field) are down.

Hence the induced current must be clockwise when looked at from above.

B

Induced B

I

induced emf and current.

Thus

= - (250) ( 0.0052)(0.6T/s) = -1.18 V (1V=1Tm2 /s)

Current I = / R = (-1.18V) / (8 ) = - 0.147 A

The induced EMF is = - d B/dt Here B = N(BA) = NB ( r2) Therefore = - N ( r2) dB/dt Since B(t) = 0.6t, dB/dt = 0.6 T/s

B

Induced B

I

I

w

l

a

Magnetic Flux in a Nonuniform Field A long, straight wire carries a current I. A rectangular loop (w by l) lies at a distance a, as shown in the figure. What is the magnetic flux through the loop?

I

w

l

a

I

w

l

a

Induced emf Due to Changing Current

A long, straight wire carries a current I = I0 + t. A rectangular loop (w by l) lies at a distance a, as shown in the figure. What is the induced emf in the loop?. What is the direction of the induced current and field?

Motional EMF

B points into screen

x x x x x x x Bx x x x x x x x

R x

D

v

Induced emf Due to Changing Current Up until now we have considered fixed loops. The flux through them changed because the magnetic field changed with time. Now moving the loop in a uniform and constant magnetic field. This changes the flux, too.

The flux is B = B·A = BDx This changes in time:

d B / dt = d(BDx)/dt = BDdx/dt = -BDv

current. What is the direction of the current? : there is less inward flux through the loop.

Hence the induced current gives inward flux. So the induced current is clockwise.

x x x x x x x Bx x x x x x x x

R x

D

v

Motional EMF -

= -d B/dt gives the EMF = BDv In a circuit with a resistor, this gives = BDv = IR I = BDv/R Thus moving a circuit in a magnetic field produces an emf exactly like a battery. This is the principle of an electric generator.

.

x x x x x x x Bx x x x x x x x

R x

D

v

Consider a loop of area A in a uniform magnetic field B. Rotate the loop with an angular frequency .

A

B

The flux changes because angle changes with time: = t. Hence d B/dt = d(B · A)/dt = d(BA cos )/dt = BA d(cos( t))/dt = - BA sin( t)

A

Rotating Loop - The Electric Generator

= - d B /dt = BA sin( t) This is an AC (alternating current) generator.

B A

d B/dt = - BA sin( t)

Rotating Loop - The Electric Generator

Consider a stationary wire in a time-varying magnetic field. A current starts to flow.

x dB/dt

So the electrons must feel a force F.

It is not F = qvxB, because the charges started stationary. Instead it must be the force F=qE due to an induced electric field E. That is:

A time-varying magnetic field B causes an electric field E to appear!

Induced Electric Fields A New Source of EMF If we have a conducting loop in a magnetic field, we can create an EMF (like a battery) by changing the value of B · A. This can be done by changing the area, by changing the magnetic field, or the angle between them. We can use this source of EMF in electrical circuits in the same way we used batteries. Remember we have to do work to move the loop or to change B, to generate the EMF (Nothing is for free!)

Example: a 120 turn coil (r= 1.8 cm, R = 5.3 ) is placed outside a solenoid (r=1.6cm, n=170/cm, i=1.5A). The current in the solenoid is reduced to 0 in 0.16s. What current appears in the coil ?

Current induced in coil:

icEMF

RNR

d Bdt

B B A 0nis As Only field in coil is inside solenoid

Example: a 120 turn coil (r= 1.8 cm, R = 5.3 ) is placed outside a solenoid (r=1.6cm, n=170/cm, i=1.5A). The current in the solenoid is reduced to 0 in 0.16s. What current appears in the coil ?

Current induced in coil:

Only field in coil is inside solenoid

icNR

d( 0nisAs)dt

NR 0nAs

dis

dt

Use dis

dt1.5A0.16s

and As 0.016cm 2 ic 4.72mA

icEMF

RNR

d Bdt

B B A 0nis As

Consider a stationary conductor in a time-varying magnetic field. A current starts to flow.

x B

So the electrons must feel a force F.

It is not F = qvxB, because the charges started stationary. Instead it must be the force F=qE due to an induced electric field E.

That is: A time-varying magnetic field B

causes an electric field E to appear!

Induced Electric Fields

E·dl = - d B/dt o

A technical detail: The electrostatic field E is conservative: E·dl = 0. Consequently we can write E = - V. The induced electric field E is NOT a conservative field. We can NOT write E = - V for an induced field.

o

Induced Electric Fields

Electrostatic Field Induced Electric Field

F = q E F = q E

Vab = - E·dl

E·dl = 0 and Ee = V

E · dl = - d B/dt

E·dl 0

Conservative Nonconservative

Work or energy difference does NOT depend on path

Work or energy difference DOES depend on path

Caused by stationary charges

Caused by changing magnetic fields

o

o

o

x B E · dl = - d B/dt o

Now suppose there is no conductor: Is there still an electric field?

YES! The field does not depend on the presence of the conductor.

For a dB/dt with axial or cylindrical symmetry, the field lines of E are circles. dB/dt

E

Induced Electric Fields

Induced Electric Field We have seen that the electric potential difference between two points A and B in an electric field E can be written as

V = VB VA = - E ds

B

A

time, an induced current begins to flow. What causes the charges to move? It is the induced emf which is the work done per unit charge. However, since magnetic field can do not work, the work done on the mobile charges must be electric, and the electric field in this situation cannot be conservative because the line integral of a conservative field must vanish.

Induced Electric Field Therefore, we conclude that there is a non-conservative electric field ENC associated with an induced emf:

dsENC

)1...(dt

dEds B

Induced Electric Field The above expression implies that a changing magnetic flux will induce a non-conservative electric field which can vary with time.

field which points into the page and is confined to a circular region with radius R, as shown in Figure right.

Induced electric field due to changing magnetic flux

Induced Electric Field Suppose the magnitude of B increases with time, i.eelectric field everywhere due to the changing magnetic field. Since the magnetic field is confined to a circular region, from symmetry arguments we choose the integration path to be a circle of radius r. The magnitude of the induced field Enc at all points on a circle is the same.

Induced Electric Field Enc

must be such that it would drive the induced current to produce a magnetic field opposing the change in magnetic flux. With the area vector A pointing out of the page, the magnetic flux is negative or inward. With dB/dt > 0 , the inward magnetic flux is increasing. Therefore, to counteract this change the induced current must flow counterclockwise to produce more outward flux. The direction of Enc is shown in Figure above.

Induced Electric Field

nc. In the region r < R , the rate of changing of magnetic f lux is:

Using equation (1) we obtain:

which implies: Enc = r / 2 (dB / dt)

Induced Electric Field Similarly, for r > R, the induced electric field may be obtained as:

Enc (2 r) = - d B/dt = dB/dt R² or

Enc = R²/2r dB/dt

Numerical Problem A. Induced current is

shown moving ccw. RH rule indicates a magnetic field out of the page, opposing external field. Therefore, external magnetic field must have been increasing.

B. Rate of change is 2.34 T/s

17.3 Self-inductance L

Inductors

An inductor is a device that produces a uniform magnetic field when a current passes through it. A solenoid is an inductor. The magnetic flux of an inductor is proportional to the current. For each coil (turn) of the solenoid: per coil sol 0(Au0N2

sol This is actually a self-inductance

Inductors

The proportionality constant is defined as L, the inductance:

Lsol = sol /I = Au0N2 Note that the inductance, L depends only on the geometry of the inductor, not on the current. The unit of inductance is the henry

1 H = 1 Wb/Ampere The circuit symbol for an inductor:

Potential difference across an inductor For the ideal inductor, R = 0, therefore potential difference across the inductor also equals zero, as long as the current is constant. What happens if we increase the current?

Potential difference across an inductor

Increasing the current increases the flux. An induced magnetic field will oppose the increase by pointing to the right. The induced current is opposite the solenoid current. The induced current carries positive charge to the left and establishes a potential difference across the inductor.

Induced current Induced field

Potential difference

Potential difference across an inductor

The potential difference across the inductor can be

Where m = per coil

sol = N per coil We defined = LI d sol/dt = L |dI/dt|

dtdN m

Induced current Induced field

Potential difference

Potential difference across an inductor

If the inductor current is decreased, the induced magnetic field, the induced current and the potential difference all change direction. Note that whether you increase or decrease the current, the inductor

with an induced current.

The sign of potential difference across an inductor

L = -L dI/dt

L decreases in the direction of current flow if current is increasing.

L increases in the direction of current flow if current is decreasing.

L is measured in the direction of current in the circuit

The potential always decreases

The potential decreases if the current is increasing The potential increases if the current is decreasing

Self Inductance: When a current flows in a circuit, it creates a magnetic flux which links its own circuit. This is called self-for the flux linkage B). The strength of B is everywhere proportional to the I

B = LI, Where L = self-inductance of the circuit L depends on shape and size of the circuit. It may

B when I = 1 amp. The unit of inductance is the henry

2Wb T m1 H 1 1A A

Self Inductance Calculation of self inductance : A solenoid Accurate

calculations of L are generally difficult. Often the answer depends even on the thickness of the wire, since B becomes strong close to a wire. In the important case of the solenoid, the first approximation result for L is quite easy to obtain: earlier we had

Hence

Then,

INB 0 IANNABB

2

0

AnANI

L B 20

2

0 lengthunit per turnsofnumber the: n

So L is proportional to n2 and the volume of the solenoid

Self Inductance

Example: the L of a solenoid of length 10 cm, area 5 cm2, with a total of 100 turns is L = 6.28 10 H 0.5 mm diameter wire would achieve 100 turns in a single layer. Going to 10 layers would increase L by a factor of 100. Adding an iron or ferrite core would also increase L by about a factor of 100.

The expression for L shows that 0 has units H/m, c.f, Tm/A obtained earlier

lengthunit per turnsofnumber the: n

AnANL 20

2

0

Self Inductance Self Inductance

If the current is steady, the coil acts like an ordinary piece of wire. But if the current changes, B changes and so then does , and Faraday tells us there will be an induced emf.

such a direction as to produce a current which makes a magnetic field opposing the change.

I B

A changing current in a coil can induce an emf in itself

Self Inductance

The self inductance of a circuit element (a coil, wire, resistor or whatever) is L = B/I. Then exactly as with mutual inductance = - L dI/dt. Since this emf opposes changes in the current (in

-

inductance.

L = 0n2Ad

Example: Finding Inductance What is the (self) inductance of a solenoid with area A, length d, and n turns per unit length?

In the solenoid B = 0nI, so the flux through one turn is B = BA = 0nIA The total flux in the solenoid is (nd) B Therefore, B = 0n2IAd and so L = B/I gives

(only geometry)

Inductance Affects Circuits and Stores Energy

First an observation: Since cannot be infinite neither can dI/dt. Therefore, current cannot change instantaneously. We will see that inductance in a circuit affects current in somewhat the same way that capacitance in a circuit affects voltage.

circuit is called an inductor.

17.4 Energy stored in an inductor

Energy Stored in an Inductor

Recall the original circuit when current was changing (building up). The loop method gave: e0 - IR + eL = 0

Multiply by I and use eL = - L dI/dt Then: Ie0 - I2R - ILdI/dt = 0

or: Ie0 - I2R d[(1/2)LI2]/dt = 0 {d[(1/2)LI2]/dt=ILdI/dt}

R + -

S I

0 L

UB = (1/2) LI2

Think about I 0 - I2R - d((1/2)LI2)/dt = 0 I 0 is the power (energy per unit time) delivered by the battery. I2R is the power dissipated in the resistor.

2]/dt as the rate

at which energy is stored in the inductor. In creating the magnetic field in the

inductor, we are storing energy The amount of energy in the magnetic field is:

Energy Density in a Magnetic Field We have shown Apply this to a solenoid: Dividing by the volume of the solenoid, the stored energy density is: uB = B2/(2 0) This turns out to be the energy density in a magnetic field

UB1

2LI2

U B1

2 on 2A I 2 A2 o

o2n 2I 2 A

2 o

B 2

Energy Stored in a Magnetic Field

The left side of Eq. represents the rate at which the emf device delivers energy to the rest of the circuit. The rightmost term represents the

rate at which energy appears as thermal energy in the resistor. Energy that is delivered to the circuit

but does not appear as thermal energy must, by the conservation-of-energy, be stored in the magnetic field of the inductor.

Energy Density of a Magnetic Field Consider a length l near the middle of a long solenoid of cross-sectional area A carrying current i; the volume associated with this length is Al. The energy stored per unit volume of the field is

20L n lA

17.5 Mutual induction

17.5 Mutual Inductance Transformer and mutual

inductance The classic examples of mutual inductance are transformers for power conversion and for making high voltages as in gasoline engine ignition.

17.5 Mutual Inductance A current I1 is f lowing in the primary coil 1 of N1 turns and this creates f lux B which then links coil 2 of N2 turns. The mutual inductance M2 1 is defined such that the induction 2 is given by

Also M2 1: Mutual Inductance of the coils Generally, M 1 2 = M 2 1

121222 IMIL

212111 IMIL

Typical Transformers

Transformers usually heavy due to iron core

Step-up Transformer

IRON CORE

~ AC POWER

SUPPLY Np

Ns

Vs

Vp

VS

VP NP

NS =

Np < Ns PRIMARY COIL SECONDARY COIL

Step-down Transformer

IRON CORE

~ AC POWER

SUPPLY

Ns Np

Vs

Vp

VS

VP NP

NS =

Np > Ns PRIMARY COIL SECONDARY COIL

TRANSFORMER TRANSFORM VOLTAGES

CORE

COIL COIL

VS

VP NP

NS =

TRANSFORMER

~ AC POWER

SUPPLY

NpNs

Vs Vp

CORE

PRIMARY COIL

SECONDARY COIL

VS

VP NP

NS = = 12 120

DC TRANSFORMER

Step-down transformer Mutual Inductance Changing current and

induced emf Consider two fixed coils with a varying current I1 in coil 1 producing magnetic field B1. The induced emf in coil 2 due to B1 is proportional to the magnetic flux through coil 2:

22212 NAdB

Mutual Inductance Changing current and

induced emf f2 is the flux through a single loop in coil 2 and N2 is the number of loops in coil 2. But we know that B1 is proportional to I1 which means that F2 is proportional to I1. The mutual inductance M is defined to be the constant of proportionality between F2 and I1 and depends on the geometry of the situation.

22212 NAdBMutual Inductance

Changing current and induced emf

1

2

11

1

222 ;

dIdM

dtdIM

dtdI

dId

dtd

The induced emf is proportional to M and to the rate of change of the current .

1

22

1

2

IN

IM

Mutual Inductance Example

Now consider a tightly wound concentric solenoids. Assume that the inner solenoid carries current I1 and the magnetic flux on the outer solenoid FB2 is created due to this current. Now the flux produced by the inner solenoid is:

/ where 111101 NnInB

The flux through the outer solenoid due to this magnetic field is:

12

11202

112112 )()(2

IrnnrBNABNB

. generalin ; )( 12212

11201

212 MMMrnn

IM B

Mutual Inductance Example of inductor: Car ignition coil

Two ignition coils, N1=16,000 turns, N2=400 turns wound over each other. l=10 cm, r=3 cm. A current through the primary coil I1=3 A is broken in 10-4 sec. What is the induced emf ?

1-41

21120

121

1122

s 103

)(; 2

AdtdI

rnnI

MdtdIM B

V 000,62

Spark jumps across gap in a spark plug and ignites a gasoline-air mixture

Mutual Inductance Two coils, 1 & 2, are arranged such that flux from one passes through the other. We already know that changing the current in 1 changes the flux (in the other) and so induces an emf in 2. This is known as mutual inductance.

I Bof 1 through 2

1 2

Mutual Inductance

The mutual inductance M is the proportionality constant between 2 and I1:

2 = M I1 so d 2 /dt = M dI1 /dt

2= - d 2 /dt = - M dI1 /dt Hence M is also the proportionality constant between 2 and dI1 /dt.

Bof 1 through 2 I

1 2

Mutual Inductance M arises from the way flux from one coil passes through the other: that is from the geometry and arrangement of the coils. Mutual means mutual. Note there is no subscript on M: the effect of 2 on 1 is identical to the effect of 1 on 2. The unit of inductance is the Henry (H).

1 H = 1Weber/Amp = 1 V-s/A

Summary

Magnetic Flux Defined Magnetic flux depends on field strength, area and angle to the field.

cosBA

B

n

circuit is given by the rate of change of magnetic flux.

tN

Lenz: the minus sign in

the polarity of the induced emf opposes the applied change. Application: circuit breakers.

Motional Emf Conducting bar moves through a magnetic field perpendicular to bar. Emf depends on field, speed and bar length. Application: voltages across aircraft wings.

Blv

Self-inductance Inductors are devices where a changing current induces an emf voltage. Application: electronic circuits

tIL

Summing up The magnetic force on a moving charge helps us define magnetic field strength. The magnetic field strength can be readily calculated for a current-carrying wire. A changing magnetic field and flux can induce voltages.