dΦ
E=−
dt
coil
In this demonstration a wire coil is connected
to a current meter. There is no battery in the
circuit. The following observations are made:
B
magnet
current meter
No current when coil is at rest.
Current is observed when the coil is moved
to change the flux through the coil.
The current increases with the rate of change of flux.
The current that occurs while the flux is changing is called an induced
current. This current is created by an induced Emf in the coil.
1
If the magnetic flux through a single loop of wire is changing with
time there will be an induced Emf in the loop:
wire loop
E = - dΦ
dt
Φ - magnetic flux through loop
Faraday’s Law
E - induced Emf in the wire
If the resistance of the loop is R then the induced current created by E is:
I= E
R
The direction of the induced current can be found from Faraday’s Law,
however, it is easier to determine the direction from Lenz’s Law:
The induced current produced by the induced E will flow in the direction
required to produce a magnetic flux that will oppose the change in
magnetic flux which is occurring in the area enclosed by the current loop.
wire loop
B
B
B
I
Bind
Bind
B is increasing
I
B is decreasing
Bind
I
B remains constant but the area of
the current loop is decreasing.
2
sliding
wire
fixed wire
B
v
A0
l
vt
t=0
t
A uniform magnetic field B is directed out
of the page. The green conducting wire,
of length l, slides along the U-shaped
fixed wire with a constant speed v. At a
time t = 0 the area enclosed by the
changing loop is A0.
(a) At any later time t, find the induced Emf, E, in the loop.
At a time t the area of the enclosed loop is:
A(t)= A0 + vlt
The flux through this loop at the time t is: Φ(t)=BA(t)cos0
The magnitude of the induced Emf is: E =
dΦ d (B( A0 + vlt))
=
= Blv
dt
dt
(b) Find the magnitude and direction of the induced current in the loop abcd:
I
d
a
B
If the resistance of the wires in the loop abcd is
negligible then the induced current is:
Bind
x
R
c
b
I=
Blv
E
=
R
R
The direction of the induced current is found using Lenz’s Law. Since the
area enclosed by the loop abcd is increasing the flux out of the page is
increasing. The induced magnetic field, Bind, must be into the page to
oppose this increasing flux. The induced current must flow CW in the loop
abcd to create a field into the page.
3
(c) Find the power generated as heat in the resistor.
2
P = I ind
R=
B 2l 2 v 2
R2
R=
B 2l 2v 2
R
(d) Find the power being produced by the force that we have to apply
to move the wire.
I
d
The magnetic force on the green wire is:
a
B
Fm
l
R
Fwe
Fm = IlBsin90 = IlB
The direction of this force is determined by the
RHR (l x B) and is in the direction shown.
c
b
We must apply a force in the opposite direction to move the wire with a constant
speed. The power produced by this force is:
r r
B 2l 2 v 2
 Blv 
P = Fwe ⋅ v = Fwev = (IlB)v = 
lB v =
R
 R 
Alternate method for finding E using differentials.
dA
d
B
l
R
c
a
b
vdt
In a small time interval dt the wire moves from a to b a distance of vdt.
The change in area of the loop is: dA = lvdt
The flux change is dΦ = BdA = Blvdt
The magnitude of the induced Emf is:
E=
dΦ Blvdt
=
= Blv
dt
dt
4