P
S
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Q
R
Faraday
(© F.Robilliard)
1
Introduction:
In previous topics, we have seen that electric fields can cause
electric currents, that cause magnetic fields.
Thus magnetic and electric fields are fundamentally linked
together. We identify them together as producing one of the
four fundamental forces of nature - the electromagnetic force.
We will now consider the reverse link - how magnetic fields are linked
back to electric currents, and consequently to electric fields.
We will generally be looking at things from the point of view of a
conducting loop in the field – we will take a loop perspective!.
We will consider what happens to a conducting loop if magnetic flux
cuts through it, and, in particular, if changes in that flux take place
To start, we will need to develop the idea of magnetic flux φ,
analogously to that of electric flux.
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Faraday
(© F.Robilliard)
2
Magnetic Flux Lines
As we have seen, magnetic fields can be represented graphically by flux lines.
A magnetic flux line is a line drawn in a magnetic field, with shape such that,
a tangent to the line at any point corresponds to the direction of the magnetic
field vector, B, at that point.
stronger uniform field
N
S
B
flux line
The strength of the field is represented by the
closeness of lines drawn in a given region of the field.
weaker uniform field
A uniform field will have uniformly spaced, parallel flux lines.
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Faraday
(© F.Robilliard)
3
Closeness of Flux lines:
Consider uniformly spaced flux lines cutting, perpendicularly, through a flat
surface of unit area.
We draw the closeness of the flux lines, such that the number of them, that cut a
unit area of surface, is numerically equal to the magnitude of the field vector, B.
A =1 m2
1
2 B=5 T
3
4
5
1
3
5
4
2
(side elevation)
B=5 T
No of flux lines cutting through unit area, here = φ = B A = 5 x 1 = 5
We think of B as the number of perpendicular flux lines cutting per unit surface area.
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Faraday
(© F.Robilliard)
4
Drawing convention for flux direction:
There is a convention that allows us to represent the direction of flux lines,
when they cut through the plane of a diagram.
We think of a flux line as an arrow.
eye
eye
or
or
If the arrow is
viewed from the
tail end, it will
appear thus:
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If the arrow is
viewed from the
head end, it will
appear thus:
Faraday
(© F.Robilliard)
5
Magnetic Flux dφ
φ
We can now develop a more formal quantitative definition of flux.
We use an element of area, dA, to define flux.
Because dA is a tiny element of a surface, it is flat, and consequently has a
definable vector direction. Because it is small, the field will be locally uniform,
over the element, even though it may be non-uniform on a larger scale.
Flux, dφ
φ, should be thought of as the total number of flux
lines, that cut through an element of surface area dA.
Case 1. The vector dA is aligned in the direction of the flux lines:
To quantify flux, dφ, we draw the closeness of flux
lines such that, the number of them cutting, per unit
surface area, numerically equals the magnitude of B
flux
dA
B
Flux and dA parallel
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(Nr flux lines cutting unit area ) = B
Thus (nr flux lines cutting area dA ) = B . dA
Hence we define dφ
Faraday
(© F.Robilliard)
dφ ≡ B.dA
6
dA not Aligned with Flux:
Case 2: dA is not parallel to the flux ( vector B direction)
B sinθ
We resolve the B vector –
in the direction of dA = B cosθ,
and perpendicular to dA = B sinθ.
B
θ
The flux associated with the (B sin θ) component
is along the surface of the dA area, and therefore
does not cut the surface. It contributes zero flux to
the surface.
B cosθ
dA
The flux associated with the (B cos θ) component is perpendicular to the
surface, and works as for the previous Case 1, namely:
dφ = (B cosθ ) dA
or, in vector notation, using the dot product dφ = B.dA
dot product
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Faraday
(© F.Robilliard)
7
General Definition of φ:
Case 3: Define the total flux φ cutting through a surface
of any shape, in a non-uniform magnetic field.
Consider some surface in a magnetic field
dA is an element of the surface
dA
B is the local magnetic field vector, at the
surface element dA
B
(Flux, dφ, cutting single surface element dA) = dφ = B.dA
(Total flux, φ , cutting the entire surface) =
sum of all the fluxes cutting all the surface elements, like dA,
from which the entire surface is composed
φ=
dφ =
over
surface
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B.dA
over
surface
Faraday
(© F.Robilliard)
General
Definition of
flux φ
dot
product
(Case 2)
8
Thought Experiment with a Conducting Loop:
What happens to a conducting loop, when it moves through a region of
uniform magnetic field?
P
S
Q
R
v=const
A flat rectangular conducting loop PQRS is moved through the
region of uniform magnetic field, at a constant velocity, v.
(Magnetic flux is perpendicular to the plane of the loop.)
Because the loop is conducting, it contains free electrons, that are
carried along as passenger charges, by the conductor. We will
analyse the magnetic forces that act on such passenger charges, as
the loop traverses the magnetic field region.
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Faraday
(© F.Robilliard)
9
Magnetic forces on passenger Charges
Although the passenger charges are actually negative electrons, it is equivalent to
imagine positive passenger charges to be carried along with the loop.
Because they move through a magnetic field, these passenger charges
will experience a magnetic force, F = q v x B (cross product).
The mauve arrows represent the magnetic forces at different stages of the motion
P
S
Q
v
R
No field,
no forces,
no current.
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Faraday
(© F.Robilliard)
10
Magnetic forces on passenger Charges
F=qvxB
P
Q
i
S
R
v
Unbalanced
force from R to Q
- anticlockwise
current.
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Faraday
(© F.Robilliard)
11
Magnetic forces on passenger Charges
F=qvxB
P
S
Q
v
R
Force from S to P
balances force from
R to Q - no current.
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Faraday
(© F.Robilliard)
12
Magnetic forces on passenger Charges
F=qvxB
P
Q
i
S
v
R
Unbalanced force
from S to P –
clockwise current.
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Faraday
(© F.Robilliard)
13
Magnetic forces on passenger Charges
F=qvxB
P
S
Q
v
R
No field,
no forces,
no current
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Faraday
(© F.Robilliard)
14
Loop Current, i:
Consider how the loop current, i, varies as the leading edge of the
loop QR moves through the field region (anticlockwise positive).
P
S
Q
R
i
i=0
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Faraday
(© F.Robilliard)
15
Loop Current, i:
Consider how the loop current, i, varies as the leading edge of the
loop QR moves through the field region (anticlockwise positive).
P
S
Q
R
i
i
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Faraday
(© F.Robilliard)
16
Loop Current, i:
Consider how the loop current, i, varies as the leading edge of the
loop QR moves through the field region (anticlockwise positive).
P
Q
R
S
i
i=0
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Faraday
(© F.Robilliard)
17
Loop Current, i:
Consider how the loop current, i, varies as the leading edge of the
loop QR moves through the field region (anticlockwise positive).
P
S
Q
R
i
i
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Faraday
(© F.Robilliard)
18
Loop Current, i:
Consider how the loop current, i, varies as the leading edge of the
loop QR moves through the field region (anticlockwise positive).
P
S
Q
R
i
i=0
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Faraday
(© F.Robilliard)
19
Loop Current, i:
Consider how the loop current, i, varies as the leading edge of the
loop QR moves through the field region (anticlockwise positive).
P
Q
R
S
i
i=0
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Faraday
(© F.Robilliard)
20
Generalization of Experiment:
From the previous thought experiment, we can draw the following generalization:
If the number of flux lines, φ ,cutting the loop, changes with time, there is a
net magnetic force around the loop ( an “electromotive force” = emf ), which
causes a loop current.
(For a uniform field, this occurs when the loop is either entering, or leaving,
the field region, but not when it is totally outside of, or totally within, the region.)
This loop current is called the “induced current”, and the
emf producing it, the “induced emf”.
Many detailed experimental investigations were carried out into this
phenomenon, which led to a quantitative formulation of the relationship
between the change of flux and the loop current produced.
This relationship is called Faraday’s Law.
The direction in which the current flows is described by the Lenz Law.
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Faraday
(© F.Robilliard)
21
Faraday’s Law:
This law was originally formulated for conducting coils, which are equivalent to
a number of conducting loops in series. N = the number of turns in the coil.
When the number of magnetic flux lines, φ, cutting the loops of a coil,
changes, an emf is induced in the coil. The magnitude of the induced emf,
ε, in each loop is proportional to the time-rate at which φ changes.
Because the coil has
N loops in series -
dφ
ε = −N
dt
Faraday’s Law
where: dφ = the change in the number of flux lines cutting the loops of the
coil that occurred in time interval dt.
Note: the constant of proportionality = 1 in the SI system.
Note: the negative sign is an expression of the Lenz Law.
Note: The current, i, in the coil will be given by the Kirchoff 2 rule, namely ε = ir,
where r = the resistance of the coil.
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Faraday
(© F.Robilliard)
22
Lenz Law:
Faraday gives the magnitude of an induced current, whereas Lenz gives its direction.
The initial flux that changed to produce the induced current is called the
“primary” flux.
The induced current, like all currents, will produce its own magnetic field (flux).
This flux is called the “induced”, or “secondary” flux.
The reasoning of Lenz sees the secondary flux as a reaction
to the change in the primary flux that produced it.
Remember the Field Produced By a Loop Current:
Right Hand Grip Rule (RHR):
B
i
Clasp the loop in the right hand, with the fingers
in the direction of the loop current.
i
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The thumb then points in the direction of the
magnetic field produced through the centre of
the loop.
Faraday
(© F.Robilliard)
23
Statement of Lenz Law:
If a conducting loop is threaded by a primary magnetic flux, and the number
of primary flux lines cutting the loop changes, then a current will be induced
in the loop.
This current will produce its own secondary flux through the loop, in a
direction as given by the RHR.
The direction of the induced current will be consistent with the following:
the direction of the secondary flux, will oppose the change in the
primary flux that created it.
Note: This statement is from the point of view of the loop. The secondary
current is unaware of its own field.
Note: the secondary flux opposes the change in the primary flux, not the
primary flux itself.
Note: this opposition is in terms of the number of flux lines. If the number of
primary flux lines through the loop is increasing, then the secondary flux will
be in the opposite direction to the primary flux, thereby opposing the
increase; if the number of primary flux lines is decreasing, then the
secondary flux will be in the same direction, thereby opposing the decrease.
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Faraday
(© F.Robilliard)
24
Faraday & the Rectangular Loop:
Rectangular loop PQRS enters a region of uniform magnetic field with
field vector B, at constant velocity v. B is perpendicular to the plane of
the loop. PQ = b, PS = a. Resistance of loop = r.
v
P
B
Q
a
S
t=0
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b
R
Faraday
(© F.Robilliard)
25
Faraday & the Rectangular Loop:
Rectangular loop PQRS enters a region of uniform magnetic field with
field vector B, at constant velocity v. B is perpendicular to the plane of
the loop. PQ = b, PS = a. Resistance of loop = r.
Loop Area in field = A=a.vt
P
a
v
B
Q
Flux cutting loop at time t
= φ = B.A = BA = B.a.vt
i
R
S t
Faraday
(N=1)
vt
Faraday:
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dφ
d
loop emf = = ir = − N
= − (Bavt) = −Bav
dt
dt
Bav
∴ loop current = i = r
Faraday
(© F.Robilliard)
The loop
current is
proportional to
the velocity of
the loop.
26
Lenz & the Rectangular Loop:
The primary flux through the loop is downward and increasing.
To oppose this downward increase, the secondary flux will need to be upward.
Thus, using the RHR, the loop current, i, will need to be anticlockwise.
v
P
Q
a
i
S
R
t
The direction of the secondary current can also be got from Faraday using a sign
convention. The direction of the primary flux is taken as positive
The negative in i = - Bav/r indicates that the induced current must produce a
secondary field in the opposite direction to that of the primary field – must be
anticlockwise! (using the RHR).
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Faraday
(© F.Robilliard)
27
Find the Emf of a Rotating Loop in a Uniform Field:
y
A
ω
z
V
B is a uniform magnetic field,
pointing in the (+x)-direction
A rectangular loop is shown,
with vector area, A.
θ
θ is the angle from the B vector,
B
to the A vector.
x
Looking edge-on at the
loop (in the (-z)-direction)
Rectangular loop.
– View V.
ω
A
θ
B
View V
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Faraday
(© F.Robilliard)
The rectangular loop is
rotated, at a constant
angular velocity, ω, with
the z-axis as its spin
axis. The ω vector points
in the (+z)-direction.
28
ω
Find emf in loop:
A
θ
B
View V
Assume that A is parallel to B,
( θ = 0 ) at time t = 0 .
d
≡
= const. ∴ =
t
dt
= t
Transposing :
Find an expression for the emf, ε, induced in the loop.
Total flux through the loop, at time t:
dot product
≡ φ ≡ B .dA = B.A = BA cos = BA cos t
Thus, from Faraday, the emf induced in the loop, at time t
dφ
d
= = −N
= − (BA cos t) = BA sin t
dt
dt
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N=1
Faraday
(© F.Robilliard)
B, A & ω const.
29
= BA
sin
t
Graphically:
Thus loop current, i, is given by
(r = loop resistance)
BA
sin t
i= =
r
r
The emf (and loop current) are sinusoidal functions of time. Firstly, the
current goes anticlockwise round the loop, then reverses and goes
clockwise. Such a current is called an alternating current.
Amplitude
= BAω
ε
Period =2π/ω
t
A rotating loop in a uniform magnetic field is the basis for almost all electrical
power generation. In Australia, mains power is alternating, with an amplitude of
339 V (240 V RMS), and a period of 1/50 s.
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Faraday
(© F.Robilliard)
30
ω
B
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Faraday
(© F.Robilliard)
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