Chapter 21 Lecture Notes
F = BAcosf
1 T = Newton-second/(coulomb-meter) = N/A×m
The main emphasis of this chapter can be summed up in one sentence. A Changing Magnetic Flux induces an emf (electromotive force, or voltage difference). This statement immediately brings up three questions that we need to answer?
Definition: What is Magnetic Flux
Suppose we have a magnetic field going through a surface. Then the perpendicular component of the magnetic field going through the surface times the area of the surface is called the magnetic flux.
F = BAcosf SI Units: T-m2 which is a Weber (Wb) - Wilhelm Weber
Suppose the object in question is a loop of wire. Then the area (A) has a magnitude equal to the area inside of the wire loop which contains the magnetic flux and a direction perpendicular to face of the wire loop.
How can we change F?
So all of the topics in the first half of chapter 21 are simply results of the fact that a changing magnetic flux induces an emf and an emf can produce a current.
What are the consequences of the induced emf?
When we change the magnetic flux we always get an induced emf. In addition, if there is a closed path for charged particles to flow, we also get a current since emf's lead to current. Eureka! We've got electric current simply from a changing magnetic flux. The electric current flows in such a direction as to oppose the change in magnetic flux.
Faraday's Law and Lenz's Law
Now we want to make this more quantitative. What is the magnitude of the emf created when the flux changes? Suppose we have a coil of wire. The emf induced by a changing the magnetic flux is given by Faraday's Law.
E = -NDF/Dt
It is easiest to use the magnitude of the emf from Faraday's law and determine the direction of the induced emf from Lenz's Law.
What is the direction of the induced current? According to Lenz's law, the induced current produces a magnetic field that opposes the change in flux causing the emf. That is the purpose of the minus sign in Faraday's law. It indicates that the induced emf is such that the magnetic force F acts to oppose the change in magnetic flux. We'll see how that works. What matters is the flux inside the wire loop. That is the defined area.
There are three steps to solving problems with Lenz's Law.
Applications and Examples
This is the simplest case of Faraday's Law. A conductor moving in a magnetic field creates an EMF (a potential difference). Explain direction using Lenz's Law.
E = DF/Dt = BDA/Dt = BLd/Dt = Blv
Another way to look at it is that the magnetic field moves the charges
I want you to see two important things about this situation.
First, we know that if we have a current in a magnetic field then a force is created. So the force the charged particles in the rod feel is F=qvB, or in terms of current, F=ILB. The direction of the force is from left to right in this picture. So to create a velocity from right to left, we need an external force, like gravity or a spring or my hand pulling it.
Second, we see from the above force equations that F/q = vB.
Recall from our chapter on electricity, we defined the electric field as E = F/q, so we can write this equation can be written as
E = Bv
This illustrates an extremely important principle. A changing magnetic flux produces an electric field.
(Special Note: Motional emf can actually be explained without introducing magnetic flux and Faraday's Law, so some books use this idea to develop Faraday's law. The idea of a motional emf, then, can be thought of using Faraday's law, or simply using ideas the ideas about the changing magnetic flux.)
Work and Motional EMF
If we have an emf then we have E = IR and we have power dissipated in the resistor. (We may be lighting a light bulb). Where is this power coming from? Energy is supposed to be conserved. Well, the rod doesn't move on its own. It takes energy (or work) to move it. In fact, the RHR shows that the current in the rod creates a force that wants to stop the rod. F = ILB sinq. It takes an external agent to supply the energy/work to light the bulb.
The Electric Generator
Motional emf is one way to create electricity. But the wire would have to be long, so there is a better way. It still uses Faraday's Law. Instead of changing the area, we will change the angle f. How do we do this? We make a generator. A generator changes mechanical energy (of falling water, steam, etc.) into electrical energy. It is exactly the opposite of a motor that changes electrical energy into mechanical energy. Consider the following figure, and look at the two vertical arms of the wire.
The current direction is given by the right hand rule using v and B. The charges in the wire feel this force (in the direction of I) which gives rise to the current. The magnitude of the emf is given by the equation for motional emf that we derived.
E = BLv^ =BLvsinq q is the angle between the velocity and B.
For both wires, and many turns
E = 2NBLvsinq
If the coil is rotating with constant angular velocity, then q= wt.
Also angular equations, v = wr = ww/2
E = 2NBLw(w/2)sinwt
E = NABwsinwt
Recall that the peak current is given by the numbers in front of the sin function, or
The current is going back and forth in the wire. If we use slip rings to make the connection we have an ac generator. With split ring commutators, we have a dc generator.
Countertorque and Eddy Currents
So far we have developed one concept in this chapter and discussed two consequences of that concept. A changing magnetic flux creates an emf. The consequences of that are that an emf can produce a current which creates a magnetic field opposing the change in flux. There is more to the story than this, however. The consequences of this can be quite dramatic. For instance, we know that the magnetic field from the induced current opposes the changing flux. This often produces a force opposing the motion involved. You actually don't even need a loop of wire to feel this opposing force. Any conductor can have localized areas of current loops called Eddy currents. These will create magnetic fields and opposing forces. Let's see the consequence of this opposing force.
Demonstration: Drop magnet and non-magnet through tube.
This current comes from Lenz's law and acts like a little dipole magnet slowing the progress of the dropped magnet.
Demonstration: Slowing Pendulum and magnet over copper.
All of these things demonstrate that the currents produced from changing magnetic fluxes can actually oppose the force creating the changing flux.
Another example of this is counter-torque in a generator. As I crank a generator, I am rotating a coil and creating a current. We know that a rotating coil with a current passing through it feels a torque. The torque is in the opposite direction of the rotation of the coil. The torque it feels is called counter-torque.
Demonstration: Electric Generator with and without light bulb.
When I unscrew the light bulb, there is no more counter-torque opposing my motion and it becomes easier to rotate the crank. Another way of thinking of this is that when there is current flowing, I am now doing work to turn the crank against friction, and doing work to create electrical energy. Just like it takes work to keep the bar moving in motional emf, it takes work to keep the coil rotating for a generator. This is conservation of energy - mechanical energy (or heat energy) into electrical energy.
Where does the energy come from to drive the motor? From falling water or steam, etc
Counter-emf of a Motor
We talk briefly about motors again. A motor changes electrical energy into mechanical energy. The changing magnetic flux of a motor as it turns, produces an emf that is trying to stop the motor. We call this back-emf or counter-emf. So there are two source of emf on a motor. The emf driving the motor (V) and the back emf (E). The greater the speed of the motor,the greater the back emf.
So as the motor speeds up and E increases, current actually decreases.
We have talked about motional emf and generators as examples of changing magnetic fluxes, changing A and f, respectively. Now we talk about changing B. What happens when we do that. Look at the figure below.
When we change a current in one coil, this changes the magnetic field in that coil. The changing magnetic field in the first coil produces a changing magnetic field in the second coil. That produces an emf in the second coil which causes a current in the second coil. This is called mutual inductance since the changing current in the first coil induces a current in the second coil. The flux in the second coil is proportional to the current in the first coil. The proportionality constant is called the mutual inductance.
N2DF2 = MDI1 SI Units of M are T-m2/A = W/A = V-s/A = Henry (H)
M = (N2DF2)/DI1 (since V = W/s)
From Faraday's Law we know
E 2 = -N2DF2/Dt (substitute above in)
E 2 = -MDI1/Dt
Also, as current increases in a coil itself, the flux changes and an emf is produced. This is called self-inductance. (Same derivation with L used).
E = -LDI/Dt where L =NDF/DI
What is the self-inductance of a solenoid?
L = NDF/DI = ND(BA)/DI
L = ND(monIA)/I n = number turns/length =N/l
L = ND(moNIA)/DI l
L = NmoNADI/DI l = moN2A /l
Where does it all end? A single solenoid has only self-inductance. Two of them have self-inductance and mutual-inductance. Always remember that something has to drive the system. Usually one of the inductors is connected to an ac or dc source and is driving the system. But both are always opposing any change in flux.
Finally, there is energy stored in an inductor. It takes work to establish current in the conductor since Lenz's Law says that the inductor doesn't want to add current. Again, like with capacitors, deriving the equation requires calculus.
Energy = (1/2)LI2. (always true)
For a solenoid:
Energy = (1/2)mon2AlI 2 with B = monI,
Energy = (1/2mo)B2A l
We can generalize this to the energy stored in any magnetic field. We usually write this as the energy density (u) which is the energy per volume.
u = (1/2mo)B2
An emf in 2 is induced from 1 using
E 2 = -N2DF/Dt
In coil 1, we have self-induction:
E 1 = -N1DF/Dt.
Transformers are designed so that the same flux goes through both coils so DF/Dt is the same.
E 2/E 1 = N2/N1 Vs/Vp = Ns/Np
If the secondary has more turns it is a step up transformer, and if it has less turns it is a step down transformer. Transformers change voltage easily. This is one of the principle reasons it is preferred over dc. They only work for ac current.
Energy must be conserved so P1 = P2
I1V1 = I2V2 or Ip/Is =Vs/Vp = Ns/Np
So if the voltage is stepped up, the current is stepped down. Note the inverse relationship here.
Transformers are used to transfer power from the power company to homes. The resistance between the points is constant. Since power lost in a resistor is I2R, we try to minimize I when transferring electricity over a long way. So the power company steps up voltage (lowering the current) at the place where electricity is produced and steps it down (raising the current) when they want to use it. There is a transformer sitting outside of your house.
Capacitors in ac Circuits
There are two important concepts about capacitors in circuits.
Think about what is happening with the charges.