Physics Electricity and Magnatism Background

Electricity and Magnetism Background

Magnetic Fields  

When we talked about electric fields, we used the language that it was a measure of the force that one unit of positive charge would feel if placed in that location. The difference with magnets is that, unlike charges, you cannot have an isolated magnetic pole. With electric charge, an electron can be isolated away from a proton. With magnets, an object with a north pole also has a south pole. With this in mind, the north side of the magnetic will align with the direction of the magnetic field. The symbol for the magnetic field is B as an alphabetical accident of history. The unit for the magnetic field strength that we will use the most is the tesla (T). A smaller unit is the gauss (G), where 1 T = 10⁴ G.

Representations of Fields

Unlike when we talked about electric fields, we are going to need to be able to represent vectors in all three spatial dimensions. We don’t want to have to try to draw everything on a three-dimensional axis. For this reason, we introduce two new vector representations. In addition to the normal left-of-page, right-of-page, top-of-page and bottom-of-page, we will now add into the page and out of the page.

Causes of Magnetic Fields

There is only one cause for a magnetic field, moving charge. But there are materials, such as iron, that are more easily magnetized than others. Inside an atom, electrons are constantly moving, thus creating small magnetic fields. In most materials, the motions of the electrons cause fields in opposite directions, so the net field is zero. Substances like iron have electrons that can easily align with one another and therefore are considered magnetic.

The current in a wire will also produce a magnetic field around that wire. The magnitude of the field around a long, straight section of wire with a current is given by the equation (not in your data booklet):

B=0I2 d

μ₀ = permeability of free space

I = current in the wire

d = distance from the wire

As you can see, the strength of the magnetic field will increase if the current increases and decreases as you move away from the wire.

The field is wrapped around the wire. The direction of the field can be described by using your right hand. If you point the thumb of your right hand along the direction of the current, your fingers will wrap around the wire in the direction of the magnetic field. 

Example 1: A current pointing to the right of the page would have a magnetic field out of the page on top and into the page on the bottom:

Example 2: A wire that is carrying a current out of the page would produce a magnetic field in the counterclockwise direction. The central dot represents the direction of the current, out of the page. The circles represent the magnetic field.

Force on a Moving Charge

When a charged particle moves through a magnetic field there is a force on the charge that is perpendicular to both the motion of the charge and the field. The equation in vector form is written F=q(vB) where represents the cross product between the velocity vector and the magnetic field vector. The cross product is a mathematical process to multiply the perpendicular portions of vectors and results in a vector that is perpendicular to the plane containing the velocity and magnetic field vectors. It is generally represented by the right hand rule. A way to visualize it is found at  http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html. For the purposes of this class, the equation in the data booklet is simplified to the method of calculating the size of the force:

F=qvBsin

q = charge of the object

v = speed of the object

B = magnetic field strength

𝜃 = angle between the velocity vector and the magnetic field

There are a couple of outcomes that are important to notice. A charged particle will only experience a force due to a magnetic if it is moving (v=0F=0). This motion must include a component that is perpendicular to the magnetic field (sin(0)=0). The greatest force will occur when a particle is moving fast in a direction that is perpendicular to the field (sin(90)=1). Since the direction of the force is perpendicular to the velocity, the result of the force is to change the direction of the motion, not the speed. This can cause the particle to travel in a circular path or a helical path. You can see simulations of this at https://ophysics.com/em7.html and https://ophysics.com/em8.html.

Some applications include particle accelerators and mass spectrometers. The explanations of these applications are a bit beyond what we are going to do right now in this format. It wouldn’t take much searching to find good animations and videos if you are interested.

Force on a Current-carrying Wire

Current is the motion of charge in a wire. Since a moving charge placed in a magnetic field will experience a force, we can describe the force on a current-carrying wire that is placed in a magnetic field. The vector form of the equation is F=I(LB). The directions we are comparing for the cross product are the length of the wire in the direction of the current and the magnetic field. As before, for calculations, we will concentrate on the equation that gives the magnitude of the force:

F=ILBsin

I = current in the wire

L = Length of the wire in the field

B = magnetic field strength

𝜃 = angle between the length vector and the magnetic field

As before, we can look at some of the implications of this equation. If the wire is along the direction of the magnetic field, or there is no current, the wire will not experience any force. We will get the largest possible force from a wire that has a large current and is placed perpendicular to the field lines.

Example 3: A wire with a current pointed to the right of the page is placed in a magnetic field that is directed out of the page. This wire would experience a force toward the bottom of the page.

If we put the previous two concepts together, we know that a current in a wire will produce a magnetic field in the space around it and a current-carrying wire in a magnetic field will experience a force. This means that two current-carrying wires next to one another will apply forces on each other. 

Example 4: In order to visualize this, we’ll break it apart. In this, we are going to imagine two wires that are sitting parallel to one another with the current running in the same direction. We’ll use two wires that run parallel to one another for a length L and are separated by a distance r.

Part 1: Looking at the force on the top wire (1) due to the magnetic field of the bottom wire (2).

The magnetic field at 1 due to 2 would be: B=0I22 r

This means the size of the force would be:

F21=I1LBsin

Since = 90 degrees, sin()=1

F21=I1L(0I22 r)=0L I1I22 r

Part 2: Looking at the force on the bottom wire (2) due to the magnetic field of the top wire (1).

The magnetic field at 2 due to 1 would be: B=0I12 r

This means the size of the force would be:

F12=I2LBsin

Since = 90 degrees, sin()=1

F12=I2L(0I12 r)=0L I1I22 r

Notice that the magnitudes of the forces are the same, but the directions are opposite. These are, of course, an action-reaction pair. Two wires that are parallel to one another with currents running the same direction will experience attractive forces. 

The result of the calculation is dependent on the length of the wires in question. It is sometimes more useful to talk about the amount of force per unit length between two parallel wires. A bit of algebra will give us the data booklet equation:

FL=0 I1I22 r

Flux

Magnetic flux is how we describe the magnetic field through an area. Think flow. The equation for magnetic flux is:

=BA=BA cos

= Magnetic Flux

B=Magnetic field

A=Area

=Angle between the field vector and the area vector

The dot represents the vector dot product. This is a way to multiply parallel parts of vector quantities and results in a scalar (non-directional) result. The area vector might be the most foreign concept of this equation. It is helpful to describe the area of a surface by the vector that points perpendicular to the surface of the object and whose length is determined by the size of the area. The area vector for your table would be a vector pointing straight up (assuming your table is not tilted). The area vector for your wall would be pointing straight out at you.

Example 5: In the picture below, the magnetic field is parallel to the area vector. All of the field “flows” through the area.The magnetic flux is =BA.

Example 6: In the picture below, the magnetic field is perpendicular to the area vector. None of the field “flows” through the area. The magnetic flux is zero.

Example 7: In the picture below, the magnetic field and the area vector make a 45° angle with one another. The flux is determined by the component of the field that is in the direction of the area. The magnetic flux is =BA cos 45=0.707BA.

Lenz’s Law

The reason we developed the concept of magnetic flux is that it is key to describing the electromagnetic effects we see in circuits. One of these effects is described by Lenz’s Law. When the magnetic flux in a closed loop changes, a current is induced in the loop in a direction that will generate a magnetic field to oppose the change in flux. 

Example 8: Suppose we have a loop of wire that is sitting in a magnetic field directed out of the page. If the magnetic field decreases, a current will be induced in the wire that will create a magnetic field out of the page to oppose the change in the flux. This current would be counterclockwise.

Example 9: If we take the same set-up and increase the magnetic field, it will induce a current in the opposite direction. 

Example 10: If the field is directed into the page and the field increases, the current induced will be counterclockwise.

A demonstration of this effect is available at https://www.youtube.com/watch?v=k2RzSs4_Ur0.

Faraday’s Law

Lenz’s Law is a descriptive rule that describes the direction of the flow, but not the reason. It is contained in a formal mathematical representation of Faraday’s Law. The calculus version is written as: =-Bt. The version for this course is:

=-Nt

Let’s look at the parts of the equation: 

• is the magnetic flux, so tis the rate of change of the magnetic flux through a loop. This can happen due to a magnetic field increasing or decreasing in the loop. It could also be due to the loop turning to change the relationship between the area vector and the magnetic field vector.

• N represents the number of loops that are connected together. This result is called flux linkage. The flux linkage through N connected loops is N𝚽. 

• is the electromotive force (emf) induced in the loop. EMF, or voltage, is the motivation for the charges to move in a particular direction.

• - is in the equation as a connection to Lenz’s Law. The emf is going to be established in the loop in a way to produce a current that will oppose the change in the magnetic flux.

Faraday’s Law gives us mathematical instructions for how to convert kinetic energy into electrical energy. The kinetic energy input must result in a change to the flux. Standard approaches are to change the area of the loop, move the loop into or out of a field or rotate the loop in the field. There are others, but we are going to concentrate on these scenarios.

Bar Moving in a Magnetic Field

Imagine we have a U-shaped conductor with a straight bar, length l, resting across the wires as shown. The entire object is placed inside a magnetic field. If we pull the bar along the wires at a constant speed, we will generate a potential difference due to the change in the flux.

The magnetic flux would be determined by the strength of the magnetic field and the area enclosed by the circuit. We want to determine the induced ε, so we will look to Faraday’s Law. There is only one loop, so N = 1 and we don’t have to worry about it. Remember that magnetic flux =BA cos and =0, so =BA. 

=-t=-(BA)t

The negative is a description of Lenz’s Law. We only care about the magnitude of the induced ε, so I’m going to ignore the sign. The magnetic field is not changing, so the rate of change in the flux is completely dependent on the rate of change in the area. 

=(BA)t=B(A)t

The change in the area is equal to the small rectangle of added enclosed space that is s wide and l long.

=B(ls)t=Blst

The displacement of the bar over the time interval is equal to the speed of the bar. This brings us to the induced potential difference across a conducting bar that is moving in a magnetic field. This equation is in your data booklet.

=Blv

If we want to know the magnitude of the induced current, we can apply Ohm’s Law:

I=VR=BlvR.

The direction of the induced current would be determined by Lenz’s Law. In the case above, we are adding a field out of the page, so we must induce a clockwise current.

Coil of Wire Entering a Magnetic Field

Let’s look at the result of a square coil consisting of N loops and sides of length l entering a magnetic field as shown. 

Once again, we are only going to worry about the magnitude of the induced ε, so I am going to drop the negative sign from the equation. The magnetic field is not changing, so the change in the flux is based on the change in the area of the loop that contains the field.

=-Nt=N(BA)t=NBAt

The change in the area is given by the rectangle defined by the length of the leading edge (l) and the distance the edge moved (s) in a small time interval.

=NBlstNBlst

The displacement of the front edge over time is equal to the speed of the coil entering. This gives an equation that is not in the data booklet, but is useful to think about:

=BvlN

If the loop is completely inside the field, the amount of field gained at the front edge would equal the amount of field lost at the back edge, so the total change in flux would be zero. There would be zero induced ε and no current in the loop.

If the loop is leaving the field, you can make a very similar argument as above and see the magnitude of the induced ε would have the same form. The current would be in the opposite direction.

Spinning Coil in a Field - AC Generators

Alternating-current (AC) generators exploit Faraday’s Law to convert rotational kinetic energy into electrical energy. Later in the year, we will return to energy generation for our national energy system. As we look at the details of how we get the electrical energy, you will notice that most of the methods end in “which spins a turbine, generating a potential difference.”

A simple version of a generator is a single wire that is placed in a magnetic field and spun on its axis. The two ends of the wire are connected to different conducting loops by brushes. This maintains contact to the same parts of the circuit without having to twist wires as the loop turns. We are only going to concentrate on the part of the circuit with the loop. As always, the circuit must be complete to have current flow.

This is not a full cycle. I would suggest that you find a good animation that will represent it more clearly than I have here. If you watch the current that would pass through point 1, you will see that it changes direction through the cycle. Analysis of the rate of change of the flux throughout the cycle will change like a sine function. This means the EMF induced and therefore the current induced will also oscillate. This is the reason for the name alternating current (AC).

The graphs below are taken from a simulation. As you can see, the coil was turning with a period of four seconds. The size of the current is the highest when the flux is zero. This is due to the fact that the flux is changing at the highest rate at that point. If you took the slope of the graph at the point where flux is zero, you would find it to be its maximum. At the extreme ends of the flux graph the slope is momentarily zero before changing signs, so the current goes through zero before starting to increase in the other direction.

( http://physics.bu.edu/~duffy/HTML5/electric_generator.html )

The electrical energy coming into our houses and buildings is 60 Hz AC. This means there are 60 cycles every second. The standards vary country by country, but primarily 60 Hz is the result of strong American influence. Others use 50 Hz.

(Optional)

The question then arises of how to describe the voltage or the current if they are constantly changing. We can talk about the peak voltage or peak current, but that only allows us to describe the behavior of the circuit for very small portions of the cycle. If we calculate the average voltage, we will get zero because we have equal positive and negative values for each cycle. We solve this by calculating the root-mean-square (RMS) value. This is calculated by squaring all of the values, finding the mean and taking the square root of that value. You don’t need to worry about doing this calculation, because there is an equation that relates the peak values of a sinusoidal (sine or cosine) function to its RMS value.

IRMS=I02

VRMS=V02

Notice that the description of the resistance of a circuit element will still work for either version of voltage and resistance.

R=V0I0=VRMSIRMS

Be careful to make sure you know whether you are given the peak or RMS values when you use this equation.

When we talk about appliances in our house, we often care about the power usage. Remember that power is the rate of energy conversion. For a DC circuit, we used the equation: P=IV. This brings up a couple of options of how we describe the performance of devices. The peak power is given by:

P=I0V0

The average power takes the RMS values into account:

P=IRMSVRMS=12I0V0