E. ELECTRICITY AND MAGNETISM
Chapter 17 Electromagnetic inductionOutline
17.1 Magnetic flux 17.2 17.3 Self-inductance L17.4 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.
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 B A
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
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 cos Sum 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 = iR
coilds
E
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 cos dt dt B d =B cos dA dt dt B d d cos =BA dt dt B ) cos (BA dt d N dt d N l d E B
Changing Magnetic Field
dB
-NA cos
dt
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 (R loop is 1 ).dA
-NB cos
dt
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 (R
loop 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 -NAB dt d cos -NAB r dt o
t
NAB
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, BB
Increasing B iB
Increasing B iB
Increasing B iB
Increasing B iB
Increasing B iLenz'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 induced 0 0 emf induced 0 0 emf induced 0 dt d 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 emfmust be in the direction thatopposes 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, motionalemf 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 frictionlessconducting 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. i R v l B i IlB k B j l I FB ( ) ( ) 2 2
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
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. Arectangular 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 vInduced 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 = BDvIn 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 vConsider 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:
ic EMFR NRd B
dt
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
ic N R d( 0nisAs) dt N R 0nAs dis dt Usedis dt 1.5A 0.16sand As 0.016cm 2 ic 4.72mA ic EMFR NRd B dt 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
o E·dl = - d B/dt
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:
ds
E
NC
)
1
...(
dt
d
Eds
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.e
electric 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 flux 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 amagnetic field out of the page, opposing external field. Therefore, external magnetic field must have been increasing. B. Rate of change is 2.34
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|
dt
d
N
mInduced current Induced field
Potential difference
Potential difference across an inductor
If the inductor current isdecreased, 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
2
Wb T m
1 H 1 1
A 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, I N B 0 B NAB N AI 2 0 A n A N I L B 2 0 2
0 per unit length
turns of number 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
length unit per turns of number the : n A n A N L 0 2 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.
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 shownApply 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
UB 1 2LI 2U
B 1 2 on
2A I
2A
2
o o 2n
2I
2A
2
oB
2Energy 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
2 0
L
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 flowing in the primary coil 1 of N1 turns and this creates flux 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 1 21 2 2 2 L I M I 2 12 1 1 1 LI M I
Typical Transformers
Transformers usually heavy
due to iron core
Step-up Transformer
IRON CORE~
AC POWER SUPPLYNp
Ns
Vs
Vp
V
SV
PN
PN
S=
Np < Ns
PRIMARY COIL SECONDARY COIL
Step-down Transformer
IRON CORE~
AC POWER SUPPLYNs
Np
Vs
Vp
V
SV
PN
PN
S=
Np > Ns
PRIMARY COIL SECONDARY COILTRANSFORMER
TRANSFORM VOLTAGES
CORE COIL COILV
SV
PN
PN
S=
TRANSFORMER
~
AC POWER SUPPLYNp
Ns
Vs
Vp
CORE PRIMARY COIL SECONDARY COILV
SV
PN
PN
S=
=
12
120
DC TRANSFORMER
Step-down transformer
Mutual Inductance
Changing current and
induced emf
Consider two fixed coils
with a varying current
I
1in
coil
1
producing magnetic
field
B
1. The induced emf
in coil 2 due to B
1is
proportional to the
magnetic flux through coil
2:
2 2 2 1 2B
d
A
N
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.
2 2 2 1 2 B dA N
Mutual Inductance
Changing current and induced emf
1 2 1 1 1 2 2 2 ; dI d M dt dI M dt dI dI d dt d
The induced emf is proportional to M and to the rate of change of the current .
1 2 2 1 2 I N I M
Mutual Inductance
ExampleNow 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 1 1 1 1 0 1 nI n N B
The flux through the outer solenoid due to this magnetic field is:
1 2 1 1 2 0 2 1 1 2 1 1 2 ( ) ( ) 2 N BA N B r n n r I B . general in ; ) ( 21 12 2 1 1 2 0 1 21 2 M M M r n n I M 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 -4 1 2 1 1 2 0 1 21 1 12 2 s 10 3 ) ( ; 2 A dt dI r n n I M dt dI M B
V
000
,
6
2Spark 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
B
of 1 through 21 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
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.
cos
BA
B n
circuit is given by the rate of change of magnetic flux.
t
N
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
t
I
L
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.