Download Lenz` Law, Motional emf, Induced emf and Electric Field Script Lenz

Lenz’ Law, Motional emf, Induced emf and Electric Field Script
Lenz’ Law
Now that we can determine the induced emf or current we need to find the
direction of the current. Not long after Faraday proposed his law of induction
Heinrich Friedrich Lenz devised what came to be known as Lenz’ Law. Lenz’ Law is
used to determine the direction of the induced current in a loop.
Lenz’ Law
Lenz’ Law states that an induced current has a direction such that the magnetic
field due to the current opposes the change in flux that induces the current. The
direction of the induced emf is the direction of the induced current.
Applying Lenz’ Law
The current i induced in a loop has the direction such that the current’s magnetic
field opposes the change in magnetic field causing it. The field of I is always
opposite an increasing magnetic field within the loop. The field of i is always in the
same direction of a decreasing field.
Curled Right Hand Rule
The thumb points in the direction of the induced current, I when the fingers curl in
the direction of the magnetic field of the induced current. We first determine the
magnetic field causing the induced current and how it is changing. Then the
magnetic field of the induced current must oppose the change in field that caused
it. This tells us which way to place the fingers and the thumb points in the direction
of the induced current.
Try it out!
What is the direction of the induced current in the figure below if the magnetic field
is increasing into the loop? Decreasing into the loop?
Solving for Direction
Lenz’ Law states that induced current’s magnetic field must oppose the change in
flux that caused it. Here the change in flux is an increase of field into the loop. We
oppose that by making field out of the loop. The fingers must curl out of the loop of
wire, so the thumb points counterclockwise. This is the direction of the induced
current.
Using Lenz’ Law
What is the direction of the induced current in the loop in each of the three
situations?
The loop moves to the right into the field.
The loop is moving across the B field.
The loop is moving out of the B field.
The Loop Enters the B Field
Here the loop is moving into the field and the field is increasing out of the loop. We
oppose this by making a B field into the loop so our thumb points in the clockwise
direction. This is the direction of the induced current.
The loop is Within the Field
The loop stays within the B field so there is no change in flux and no induced
current.
Loop Moving Out of the B Field
As the loop moves out of the B field the field is decreasing out of the loop. To
oppose this change in flux we must put more field lines out of the loop. This puts
the thumb in the counter clockwise direction, which is the direction of the induced
current.
Motional EMF
Now let’s consider a conductor moving in a magnetic field. The conducting rod
below is moved to the right in the magnetic field. What happens to the charges in
the conducting rod?
Charges in the rod
The mobile charges in the conducting rod will experience a force equal to qvB. We
use the right hand rule to determine the direction of the force. The charges
experience a force toward the bottom of the rod building up a positive charge on
that end. The other end of the rod will be negative.
Electric Potential
With positive charge built up on the bottom of the wire and negative charge built up
at the top of the conducting rod a potential difference is created between the ends
of the rod. If we connect a circuit to the rod a current will flow through the circuit.
Electrostatic force = magnetic force
qE = qvB
E = vB
Vab = EL
Vab = vBL
Motional EMF
Motional EMF is directly proportional to the velocity of the bar, the length of the bar
and the magnetic field strength. EMF = B L v. Use Lenz’ law to determine the
direction of the current.
Problem
The length of the rod is 0.10 m and the velocity is 2.1 m/s. The resistance of the
loop is 0.040 Ohm and the magnetic field is 0.70 T. What is the induced emf, the
induced current and the force on the rod?
Solution
Emf = BLv
Emf = (0.70) (0.10m) (2.1 m/s)
Emf = 0.15 V
Emf = IR
0.15 V = I (0.040 Ohms)
I = 3.8 Amps
F=IlB
F = 3.8 A (0.10 m) (0.70)
F =0.27 N
The force is opposite the velocity by the right hand rule.
Induced EMF and Electric Field
A changing magnetic flux in a conductor induces an electric field or in other words a
changing magnetic field produces an electric field.
Electric Field in the Wire
Consider a loop of wire, radius r, in a magnetic field perpendicular to the plane of
the loop. The magnetic field changes with time by Faraday’s law and emf would be
induced in the loop of wire so the emf = - change in flux/ change in time which
causes a flow of current. The induction of the current implies there is an electric
field tangent to the loop of wire because that is the direction of the current.
Work Done on the Charge
The work done by the electric field on charges is qE. The work done by the electric
force to move the charges around the loop is qE (2 pi r). These two must be equal
to each other so we can set them equal to each other as I do in the next slide.
q! = qE(2" r)
!
2" r
1 d$
E=#
2" r dt
$ = BA = B" r 2
r dB
E=#
2 dt
E=
Electric Field Created by a Changing Magnetic Field
Faraday’s Law in General Format
—
∫ E ⋅ ds = −
dΦ
dt
Problem
Suppose a long solenoid is wound with 500 turns per meter and the current in the
coil is increasing at a rate of 100 A/s. The cross-sectional area of the coil is 4.0 cm2.
Find the magnitude of the induced electric field within the loop if its radius is 2.0
cm.
Solution
dΦ
dI
= − µo nA
dt
dt
−7
ε = −(4π x10 Wb / Am)(500 / m)(4.0 x10−4 m 2 )(100 A / s)
ε =−
ε = −25 x10−6Wb / s = −25µV
ε
25 x10−6V
E=
=
= 2.0 x10−4V / m
2π r 2π (0.02m)