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ELECTROMAGNETIC INDUCTION
(INCLUDING FARADAY’S LAW AND LENZ’S LAW)
Faraday’s law is stated in two steps. Refer to the loop and magnetic field shown below. The magnetic flux through a loop of area, A, is defined as the product φ= AB cosθ.
The angle, θ, is the angle between the magnetic field and the direction perpendicular (normal) to the loop area. (Calculus users, note that the formal definition of flux is φ =
∫
B• dAAREA OF LOOP
.)
The experiment shows that, for a loop of wire as shown, a potential difference is produced when the magnetic flux through the loop changes. If the flux stops changing, the potential drops to zero. Since this potential is not passive (it is capable of delivering power), it is called an EMF, following the practice for a battery.
The EMF is said to be induced by a changing flux and is simply
ε
= (rate of change of) φ.•
If a plot of flux versus time makes a straight line, the rate of change of flux (and the induced EMF) is simply the slope of the straight line.•
Calculus users, note that the EMF is given byε
= −( )dφdt . The minus sign is associated with Lenz’s law; see below.
•
Calculus users note that the potential difference between the location of the two terminals of the loop of wire can be calculated even if the wire is not present. The potential difference is the integral along a pathfollow-ing the loop path: ( )− dtd
∫
B dA• = −( )ddt = =∫
E ds•AREA OF LOOP
LOOP
φ ε
. Theminus sign now makes sense: Placing the fingers of the right hand along the direction of ds, the right thumb points in the direction of dA
. Lenz’s law shows how to determine which terminal of the loop will be positive.
•
The law depends on the current that flows in the wire to produce the EMF. The positive terminal will be made positive by current flowing toward it. This current is called the induced current.•
The induced current creates, as if flows, its own magnetic field near the loop. This field adds with the original magnetic field, but it is useful to think of it separately.•
Lenz’s law states that the induced magnetic field acts to oppose the change in flux of the original field.The flux that induces the EMF can change in several interesting ways:
•
The original magnetic field changes magnitude. This is the basis of the electrical transformer.•
The original magnetic field changes direction. This is used in some magnetic navigation sensors.•
The loop changes direction. Also used in magnetic direction sensors and in magnetic field strength sensors.•
The loop changes size. If the sides of the loop are square, we can imagine the loop growing so that one straight side moves perpendicular to its length. The work done by the EMF created this way is the same as would be calculated using the force calculated from F =qv ×B.Together, Faraday’s law and Lenz’s law take the place of the simple force equation above, for the magnetic force on a moving charge.
INDUCTANCE (INCLUDING LR AND LC CIRCUITS)
A coil of wire with N turns (loops) of cross sectional area, A, experiences a flux φ = NAB when a uniform field, B, is applied parallel to the axis of the coil.
If the magnetic field is changing with time, an EMF is induced in each loop. The net EMF from one end of the coil to the other is the sum of the individual EMF’s.
ε
= dφ =dt NAdB
COIL dt
.
It is possible that the magnetic field above is caused by the same coil that is producing the induced EMF. To make this happen, a changing current must be passed through the coil. The magnetic field is proportional to the current flowing through the coil. The rate of change of flux is proportional to the rate of change of current.
In this case, the induced EMF is proportional to the rate of change of the current, i, in the coil. The constant of proportionality is called the induc-tance, L:
|ε|
=L didt . The unit of inductance is the Henry,Henry Volt second Ampere
= .
The energy stored in an inductor is UL = 1Li 2
2.
IRCUITS
When a coil is used in a circuit whose current can vary, the potential differ-ence across the coil is a passive response to the changing current. Because the coil is passive and not a source of energy, the potential difference is represented by ∆Vor V, just as for a resistor or a capacitor. Lenz’s law is used to determine the sense of the voltage. The potential across the coil acts in a sense to oppose the change in current: If the current is increasing, the coil potential acts to prevent the increase.
In the circuit, the experiment begins when the switch is thrown so that current must flow in the resistor-inductor circuit, without going through the battery.
An EMF is induced in the coil that opposes the tendency of the current to fall.
Kirchhoff’s loop law for this circuit is –Ldi
dt−iR=0. The solution to this equation is i=i e0 −t
/τ, where
i
0
= ε R
is the initial current and the decay time is τ = LR. If at t = 0 the switch is thrown the other way, connecting to the battery, the current is given by i=i∞
(
1−e−t/τ)
, wherei
∞
= ε R
is the final steady state current.LC C
IRCUITSIn the figure on the following page, a capacitor starts with an initial charge, Q0, and the switch is closed, allowing current to flow. Kirchhoff’s law for this loop is Ldi
dt Q
−C =0.
The relation between current and charge is i dQ
= − dt , making the Kirchhoff equation d Q
dt
Q LC
2
2 = − . This has the form of the equation for simple harmonic motion. The solution for this problem is Q=Q0cosω0t , where
Q0is the initial charge on the capacitor and ω0
= 1
LC is the natural frequency of the LC oscillator.
The current for this solution is i dQ
dt iMAX t
= − = sinω0 , where iMAX = ω0Q0.
The energy stored in this oscillator is U L i Q
L = 12