Mr. P: Ammeters and voltmeters.
An ammeter, you need to put in series,
a voltmeter, you need to put in parallel.
In order to measure the current,
in order to put it in, you need to have it in series, right?
In order to have little change to the circuit,
you need to have as low a resistance as possible.
A voltmeter, your goal is to measure
the electric potential difference
so it needs to be in parallel,
therefore it needs to have
the same electric potential difference there,
so that's why it needs to be in parallel.
I forgot an 'L'.
So a voltmeter needs to have a high resistance
in order to have little change to the circuit
and still have the same electric potential difference.
The magnetic force on a moving charge.
This is how the magnetic field is defined,
believe it or not, the fact that a moving charge
in a magnetic field can experience a magnetic force.
So the magnetic force is equal to qv cross B,
the charge times velocity with the cross-product
of the magnetic field, or qvB sine theta,
or if you have a current-carrying wire,
it is IL cross B, or ILB sine theta.
We do need to talk about the right-hand rule.
In the right-hand rule for figuring out the magnetic force,
your fingers point in the direction of the velocity,
your fingers curl in the direction of the magnetic field,
and your thumb points in the direction of
the magnetic force on a positive charge.
On a negative charge, the magnetic force
is in the opposite direction your thumb is pointing.
Quick example.
In this example, we have a negative charge,
which is moving up.
Point our fingers in the direction of the velocity.
We curl our fingers in the direction of the magnetic field.
You could see the magnetic field is into the board here,
we have Xs, the feathers of the arrow going away from you.
We curl our fingers in the direction of the magnetic field.
Our thumb, then, points to the left.
However, because this is a negative charge,
we're going to end up with a magnetic force
which is to the right.
In other words, this negative charge
will actually describe a circle, which would look like that.
Torque equals NIA cross B.
This is the torque on a current-carrying loop
in a magnetic field.
N is the number of loops.
I is the current.
A is the area vector, and B is the magnetic field.
From the equation we were just talking about,
for the magnetic force on a current-carrying wire,
if we have a wire that is not straight,
you need to break that wire into little pieces, ds,
and if you take the integral,
you end up with a magnetic force that's equal to
I L-prime cross B, where L-prime is the straight-line distance
from where you start to where you stop the wire
or where you start and stop the wire in the magnetic field
if the whole wire isn't in the magnetic field.
Notice this has to be a constant magnetic field
for that to work.
If you have a loop in a constant magnetic field,
notice that's a loop,
which means we're going to have
the closed-loop integral of ds,
or I times the closed-loop integral of ds cross B.
If you have a loop in a constant magnetic field,
then you're going to have the current
times the closed loop integral of ds cross B,
and if it is a loop that's entirely in a constant magnetic field,
that closed loop integral ds will actually work out to be 0,
which means that the net force on a closed loop
in a constant magnetic field is actually equal to 0.
This doesn't mean that the net torque is equal to 0,
but the net force will be equal to 0.
In the example we were just talking about
with the right-hand rule,
that we had the velocity of our charged particle
was perpendicular to the magnetic field,
we had circular motion.
When you have that, you need to draw a free body diagram
sum the forces in the in direction.
It's going to be equal to mass
times the centripetal acceleration,
and remember the centripetal acceleration
is the angular velocity squared times the radius.
In other words, the angular velocity
is equal to 2pi divided by the period.
You could solve for all sorts of stuff
using that basic equation,
the basic idea that the velocity of a charged particle
moving perpendicular to a magnetic field
will move in a circle.
Biot-Savart's Law.
I can't pronounce it, so I just say it terribly.
Mu-not over 4pi times the quantity Ids cross r
over r-squared.
Now notice this is a unit vector r divided by r-squared.
I believe on the equation sheet,
it's actually r divided by r-cubed.
This is the same thing, only I prefer the unit vector.
We use this equation, Biot-Savart, to figure out
the magnetic field at the center of an arc
and along the axis of a loop.
Those are two that are very important
you need to make sure you know how to do.
The magnetic field on the inside of an ideal solenoid
is equal to mu-not times N, the number of loops,
times the current in the solenoid divided by
l, the length of the solenoid.
You use Ampere's Law to derive this equation.
Ampere's Law, the closed loop integral of B dot ds,
where you have to define your Amperian loop,
is equal to Mu-not times the current
inside that Amperian loop.
Again, just like Gauss' Law,
where you have to define your Gaussian surface,
here you have to define your Amperian loop.
Again, you're defining it such that the magnetic field is
constant on your Amperian loop,
and the angle between the magnetic field
and ds is either equal to 0 or 90 degrees.
you have to be able to,
you're responsible for being able to figure out
the magnetic, or derive the magnetic field
inside a solenoid using Ampere's Law.
Let's talk about a couple more.
This is the magnetic field that exists
r distance from a current-carrying wire,
and infinitely long current-carrying wire.
Again, you're responsible for being able to derive this.
You should probably also have it memorized.
It's a very important one, just like the one for a solenoid.
And you need to be able to figure out,
use Gauss' Law to figure out the magnetic field
in the center of a loop or an arc, a part of a loop.
Those are very important things to be able to do as well.
If you have two parallel current-carrying wires,
there is a magnetic force that exists
between the two of them.
The magnetic force that exists between
two current-carrying wires is going to be
the current of the first one times the length of the first one
with the cross-product of the magnetic field
caused by the second one.
Notice that the magnetic force on the current-carrying wire
includes the current in the wire itself,
and that wire is in the magnetic field from the other wire,
wire 2, so you can then solve for,
substituting into the equation we just talked about
for the magnetic field caused by
a current-carrying wire,
r distance from that current carrying wire.
That's going to be the current in wire 2.
You end up with the equation for the magnetic force
caused on one wire by another wire.
This equation right here.
You can use the right-hand rule to go through
and figure out the directions for the magnetic forces
on the two wires,
and it's something you should be able to do.
It's also important to realize
that it's actually the reverse of the law of charges,
where if the currents are in the same direction, they attract.
If they're in opposite directions, they actually repel.
Gauss' Law for Magnetic Fields.
The closed-surface integral of B dot dA is equal to 0.
The reason it's equal to 0 is because
magnetic fields, as far as we know are always loops
because we've never found a magnetic monopole.
They're always existing a north and a south together.
Therefore, the closed surface integral of the magnetic field
of dot product dA is always equal to 0.
If you want to talk about the magnetic flux
not for a closed surface,
it's just going to be these, integral of B dot dA, or BA cosine theta.
emf.
It turns out that a changing magnetic flux
as a function of time causes an emf.
It can cause a current to flow.
Therefore, we know that the emf
is equal to negative of N, the number of loops,
times the derivative of the magnetic flux as a function of time.
Magnetic flux is BA cosine theta.
In other words, the emf, you could induce an emf
by changing the magnetic field,
by changing the area of the loop,
or by changing the angle between the loop
and the magnetic field.
There are all sorts of different ways,
or all three at once.
There are all sorts of ways to induce a magnetic field
by changing the magnetic flux.
That is called Faraday's Law.
Lenz' Law has to do with the direction of the magnetic,
or the direction of the emf caused by
a changing magnetic flux.
Let's just walk through a quick example.
Let's say we have a loop initially
with no magnetic field inside it,
and finally we have a magnetic field
which is into the board.
We have a change in the magnetic field.
Notice the area is not changing,
the angle's not changing.
The only thing that's changing is the magnetic field.
The magnetic field is going from zero to into the board.
The magnetic flux, it starts out as 0,
and it's increasing into the board.
According to Lenz' Law,
or the concept of electromagnetic inertia
or the idea that it wants to keep the status quo,
it does not want to change the magnetic flux,
the induced magnetic flux is going to be out of the page,
or out of the board.
If the induced magnetic flux is out of the board,
we're going to use the other right-hand rule,
which is to point the fingers in the direction
of the induced magnetic field,
and point our thumb in the direction of the current,
and you will see that the induced current on this wire
is going to be counter-clockwise.
One more time, let's just walk through it.
Lenz' Law.
We have initially no magnetic flux.
We have a magnetic field that's increasing into the boards.
We have a magnetic flux into the board, which is increasing.
According to Lenz' Law, or electromagnetic inertia,
we're going to have an induced magnetic flux,
or in this case, induced magnetic field,
which is opposite the direction
of the increasing magnetic field here
in order to try to keep that magnetic flux equal to 0,
or keep it constant.
In other words, the magnetic,
induced magnetic field is going to be out of the board.
We point our fingers in the direction,
we curl our fingers in the direction of that magnetic field,
and we point our thumb in the direction
of the induced current,
and that induced current is going to be counter-clockwise.
Motional emf.
The idea that a moving metal object in a magnetic field
will actually induce an emf,
and induce an electric potential difference in that object,
which means, you can go through and derive it,
I'm not going to do that here.
There are actually two different ways
to derive the motional emf.
You should know both of them.
That electric potential difference is equal to
the velocity times the magnetic field
times the length of the object,
times the sine of the theta,
where theta is the angle between the velocity
and the magnetic field.
Often, we have derived it actually where
the theta was equal to 90 degrees,
and it just ended up being velocity times magnetic field
times the length.
It is important for you to realize that is not always the case.
It is also important for you to realize
that this is one of the reasons that I always put a delta
in front of my electric potential difference
because if this said V equals V times B times L,
you'd be confused.
So it is the electric potential difference
equals the velocity times magnetic field,
times the length times the sine of the angle between
the velocity and the magnetic field.
The emf of a generator.
you're responsible for being able to derive this,
but it's also good for you to have it memorized.
It is WNBA, easy to remember.
The angular velocity of the motor itself
times the number of turns,
times the magnetic field,
times the cross-sectional area of the loops,
times the sine of omega t.
Now, this does assume that the initial time is equal to 0,
and it is also equal to the back emf in a motor,
if you're talking about a motor.
Again, the back emf of a motor actually only occurs
if you have the motor actually turning.
This is one of the reasons why if you have
a motor that stops suddenly,
the current will increase because there is
a back emf caused in the motor as it's moving,
opposite the direction of the current
being put into the electric motor.
Self-induced emf.
Negative LdI/dt, where L is the inductance,
dI/dt is the change in current,
and that's also equal to the negative of the number of loops
times the derivative of the magnetic flux
as a function of time.
You'll notice that's the same emf.
If you have those two are equal,
this is going to give you the L,
the inductance and self-inductance is equal to
N times the magnetic flux divided by the current.
So, the basic idea of an inductor.
The emf of an inductor is equal to the negative L,
the inductance times dI/dt.
I am going to put down the equation for
electric potential difference equals current times resistance,
just to talk about the relationship here.
Resistance is the resistance to a current,
whereas inductance is a resistance to a change in the current,
so they're both kind of resistances,
but an inductor is a resistance to a change in the current.
It's important to realize that small piece of information.
An RL circuit,
a circuit with a resistor and an inductor and a battery.
What you're doing is you're putting energy
into the inductor.
Now, energy in a capacitor stored in the electric field,
energy in an inductor is stored in the magnetic field
of the inductor.
In other words, you have to have current passing through it
in order to have energy stored in an inductor.
You are responsible for being able to
derive these equations,
current as a function of time
and the derivative of current as a function of time.
In other words, how the current is changing
as a function of time.
Now, what you're doing is you're putting energy
into the inductor.
You're responsible for not only knowing how to derive these.
You should also be responsible,
but you should also know the limits.
In other words, the current at the very beginning,
remember an inductor is a resistance to a change
in the current,
so a current before you flip the switch is equal to 0,
so the current right after you flip the switch,
or right at that very beginning is also equal to 0
because it's resisting that change.
However, the current after a very long time
is just going to be the emf divided by the resistance
because the inductor after a little while
is no longer resisting the change in current,
because the change in current is equal to 0.
It's not longer changing at the very end,
so at the very beginning, it's changing the most.
It's E over L, the emf divided by the inductance,
but after a very long time,
that current is just going to be constant,
therefore the derivative of the current as a function of time
is going to be equal to 0.
Now, if instead we're taking energy from the inductor,
in other words, we take out the battery
and we just have a circuit with an RL circuit,
we're taking that energy from the magnetic field.
We have the equations for the current
as a function of time
and how that current is changing as a function of time.
Again, you have to be able to derive these
and you need to know the limits.
Initially, that current is just going to be equal to E over R.
Remember, if it's going to be at a maximum,
remember the current has, in order to have energy in there,
we have to have current going through,
going through the inductor
in order to have a magnetic field.
In other words, as a function of time,
that current is going to decrease all the way down to 0.
At the very beginning, that current is changing the most.
It's going to be E over L,
and after a very long time,
that current is no longer changing 'cause it's equal to 0.
The limits for an RL circuit,
taking the energy from an RL circuit.
The energy stored in an inductor,
one-half LI squared.
The inductance times the current squared.
Again, it's stored in the magnetic field of an inductor,
and you have to have current passing through it.
An LC circuit, so we have a circuit with an inductor
and a capacitor.
We have energy stored in the capacitor
that's then being transferred to the inductor,
that's then being transferred to the capacitor,
that's then being transferred to the inductor
and so on and so forth.
It is simple harmonic motion.
You can use the equations to figure out
that the second derivative of charge
is a function of time squared,
is equal to the -1 over LC times Q.
This is the same format as simple harmonic motion
from mechanics.
Then the angular velocity,
this is the position of angular velocity squared,
therefore, I'm sorry, the angular frequency.
Therefore, the angular frequency,
you can see this is a position where we have
the angular frequency squared.
That angular frequency is going to be equal to
the square root of one over LC,
and that's equal to 2pi divided by the period
again because it's equal to change in position
over change in time.
Therefore, you could figure out the period
for simple harmonic motion for an LC circuit
from this basic equation.
In an LC circuit, you also have
the simple harmonic motion equations
which look like this.
The charge as a function of time,
is equal to the charge maximum
times the cosine of the quantity omega t plus phi.
Note the similarity here to mechanics.
It's just the charge as a function of time,
and you could figure out the current
as a function of time.
You just take the derivative of this equation.
Again, because this current is a derivative of charge
as a function of time.
Simple harmonic motion, an LC circuit,
simple harmonic motion,
you have conservation of energy.
So you have energy in the inductor that's being converted
into energy in the capacitor so on and so forth,
and some points you are going to have some energy
in the capacitor, some energy in the inductor,
and it's just going to go back and forth, back and forth,
and that is it.
Ladies and gentlepeople,
thank you very much for learning with me today.
I enjoyed learning with you.
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