AP Physics Notes: Magnetism
Introduction: For centuries it has been noticed that certain rocks (lodestones) can naturally attract metal, e.g., iron, steel. Pieces of lodestone are called magnets and can create magnetism in the metal objects they attract. The behavior of magnetism is similar to that of the attraction or repulsion of charged particles.
Types of Magnetic Materials Magnetic substances can be classified into three basic groups, according to their response to a magnet. Note the strength and direction of the interaction.
a) Ferromagnetic materials are strongly attracted to magnets
b) Paramagnetic materials are weakly attracted to magnets
c) Diamagnetic materials are weakly repelled by magnets.
Causes of Magnetic Behavior Magnetic properties are thought to result from two types of electron motions within atoms. Electrons orbiting nuclei (like the year long motion of the earth around the sun) cause diamagnetism. Electrons also revolve around their own axes (like the 24 hr motion of the earth around its N – S axis.) The interaction between the two types of motions is thought to cause permanent magnetism.
Magnetic Poles A bar magnet suspended by a thread so that it can twist freely aligns itself with the earth’s magnetic field. The end of the magnet (sometimes colored red) that points to the Arctic is called the N – seeking pole; it is actually the S pole of the magnet. The end of the magnet that points to the Antarctic is called the S – seeking pole; it is really the N pole of the magnet. Just as with electric charges, opposite poles attract and like ones repel.
Induced Magnetism Metals can be made magnetic in a variety of ways. A metal’s permeability indicates how easily it becomes magnetized. If a steel pin is rubbed in one direction with a magnet for a short time it becomes a temporary bar magnet. If a length of hot iron or steel is aligned N–S and hammered until it is cool, it becomes a bar magnet. If a length of steel rail has been lying in one orientation for a few years, it becomes magnetic. A can of food sitting untouched on a shelf at home for just a few weeks will become magnetic.
What happens to a metal that makes it magnetic? Metals like iron contain microscopic areas called domains. Each domain behaves like a miniature bar magnet, all at different angles to each other. If the iron becomes magnetized, most of the domains line up parallel to each other. If any material is heated to its Curie Temperature, the thermal energy causes its particles to jostle about so much the domains will never stay lined up and magnetization is impossible. A permeable material is one in which the domains line up readily and saturation occurs when an external magnetic field has caused most of the domains to line up.
Soft iron is a metal that readily shows induced magnetism – it becomes a temporary magnet when placed next to a magnet: this causes its domains to line up. It gradually loses its induced magnetic properties when taken away from the magnet as the domains lose their common orientation. In contrast, steel would retain its induced magnetism for a long time and is therefore used to make permanent magnets; its domains remain lined up for an extended period of time.
Magnetic Force Fields The shape and behavior of these fields is similar to electric fields.
Electromagnetism Hans Oerstead (Danish, 1819) noticed that current flowing through a wire caused the needle of a nearby compass to swing away from its N– S alignment. His demonstration was a key part of the investigation into the connections between electricity and magnetism.
Some of the basic ideas in electromagnetism are summarized in the four Left Right Rules.
These rules describe how part’s of one’s hand can help one remember the relationships between:
i) conductor shape ii) current direction iii) shape and iv) direction of the magnetic field.
NOTE: a current is just a stream of charged particles. Sometimes charged particles can be found flowing along wires (in our homes) while in other situations they just fly through space or air or force fields (like inside TV’s or in lightning flashes.) A conventional current is a stream of + particles. This idea runs into problems when we speak about current in wires because electrons (e–) are what flow through wire, not + particles. And, the electrons flow in the opposite direction to the motion of the + particles (which don’t really move at all in a wire). Still, the idea of conventional current is still commonly used even though it can be misleading. breathe here
The Right Hand Rules are based on the motion of + particles, i.e., conventional current. So, IF a situation refers simply to “the current”, point the appropriate part of your hand with the flow. IF the situation mentions a flow of electrons, point against the flow.
One way to restate this is to point toward the – end of the power supply (where the electrons come from) because + particles really would move in this direction. But, this idea is not foolproof because the true – end of a power supply is sometimes mislabeled as the + end for historical reasons!! For example, the + end of a battery is the end from which electrons flow but that end is called the + terminal. breathe here
First Right Hand Rule
Situation: current flowing in a straight wire creates a cylindrical magnetic force field around the wire.
The Rule: wrap hand around the wire with thumb pointing in direction of conventional + current flow; fingers wrapped around the wire show the shape of the field; the fingertips point in the direction of the force field (toward N). Since the field is a cylinder, there is no one place that is N. Nonetheless, a compass needle will align itself parallel to the force lines, but with its N – seeking pole pointing in the opposite direction to your fingertips.
Calculations: B = k I Magnetic flux is the imaginary flow of magnetic force through a magnetic D force field. This equation describes how the density of magnetic flux (the strength of the magnetic force field) changes with distance out from the wire.
B = the magnetic field’s strength (its magnetic induction or magnetic flux density ) at some point out
away from the wire, measured in Tesla’s (T) or Webers / m2.
k = a constant, 2 * 10 – 7 N/amp2
I = current in wire, in A
D = distance out from the wire to a selected spot, in m
Q. Find the magnetic induction at a point 1 m out from a straight wire carrying 5 A of current.
B = 2 * 10 – 7 N/amp2 * 5 A = 1 * 10 – 6 T
1 m
Note: A more accurate equation for this 1st Right Hand Rule accounts for the fact the field is cylindrical and from the end, is a circle. This equation refers also to the permeability of the air (μ = 4π * 10 –7 N/A2) out through which the field operates. The equation is:
B = μ I which simplifies to B = 2 * 10 –7 I
2π r r
Q. At what distance out from a wire carrying a 3 A current is the magnetic field strength 4 * 10 – 5 T?
r = 2 * 10 –7 I = 2 * 10 –7 * 3 = 0.015 m.
B 4 * 10 – 5 T
Second Right Hand Rule
Situation: current flowing through a coil creates an invisible bar magnet within the coil’s core.
The Rule: wrap fingers around coil, fingertips follow the current flow; the thumb along the coil’s axis points to the magnet’s N–seeking pole.
Calculations: B = k π I This equation relates the magnetic induction at the center of a loop to the r radius of the loop.
B = the magnetic flux density at the center of a loop, in T or Webers / m2
k = as before
I = current running through loops of the coil, in A
r = the radius of the coil’s loops, in m
Q. Find the magnetic induction at the center of a coil, radius = 0.16 m, and carrying 4 A of current.
B = 2 * 10 – 7 N/amp2 * π * 4 A = 1.57 * 10 – 5 T
0.16 A
Note: another coil equation accounts for the number of loops (N) in the coil is: B = μ NI / 2r.
Q. What magnetic field strength is created in a 20 cm diameter coil carrying 2.5 A with 30 loops?
B = 4π * 10 –7 * 30 * 2.5 / .02 = .0047 T
Applications: Electromagnets These are a direct tie-in with electrified coils. A piece of soft iron (high permeability) placed within the coil’s core concentrates the magnetic force lines and so creates a stronger magnet. An electromagnet’s strength is related to: the amount of current in the coil, the number of loops (turns) in the coil, and the permeability of the metal in its core.
Third Right Hand Rule
Situation: an electrified straight wire in a magnetic field in a gap between opposite poles (in the jaws of a
horseshoe magnet) will try to jump out of the gap. The wire creates an invisible magnet which behaves just like a solid magnet placed next to another magnet – it is attracted or repelled.
The Rule: hand flat, fingers point across the gap toward S; thumb follows current flow; the palm shows direction in which wire moves.
Calculations: F = B I L This equation describes the force exerted on the wire by the magnetic field in
the gap between the magnet’s N and S poles.
F = the push or pull felt by the wire, in N
B = the magnetic flux density in the gap between the magnet poles, in T or Webers / m2
I = current carried in wire, in A
L = the length of the wire in the gap, in m
Q. Find the force exerted on a 0.2 m long wire carrying 20 A of current when it is in a 0.4 T magnetic
field.
F = 0.4 T * 20 A * 0.2 m = 1.6 N.
This 3rd Right Hand Rule also applies to single charges moving through a magnetic force field. The magnetic field gives the moving charge a push, an impulse, causing its path to change. And, because a single charge is SO much lighter than a piece of wire, the magnetic force field has a huge effect on the motion of the charge (compared to a wire jumping up or down which is still a very neat thing in itself). Investigations show that the magnetic field exerts the greatest force on the charge (and changes its path the most) when it enters the field at 900. Imagine the magnetic field is looks like a sheet of looseleaf: entering at 900 means going into the face of the page, not into an edge.
Now, just what path does the magnetic field cause the charge take? If it enters at exactly 900, the straight line path becomes an arc! Each instant the charge moves ahead a tiny bit, the magnetic field forces it into a new path at a slight angle to the previous one. As this process keeps up, the charge’s original straight path is bent into an arc of a certain radius, R. Remember how gravity progressively bends a satellite’s straight (tangential) path into an orbit. Note: if the charged particle enters the field at an angle ≠ 900, the straight line path becomes “distorted” into a spiral rather than a simple arc.
The equation describing the force of the magnetic field on the electric charge is: F = BQ v sin θ.
The v = charge’s velocity in m/s, Q = charge magnitude in Coulombs, C, θ = charge’s entrance angle into magnetic field. Note: for simplicity’s sake, the situation is usually discussed with the charge entering the field at 900 and since sin 900 = 1, the equation simplifies to F = BQv.
Look at this equation chain: because the force on the charge acts like a centripetal force, we can write F = mac = mv2 / R = BQv or mv / R = BQ or R = mv / BQ. And, an equation describing the force exerted on a charge by an electric force field (in electrostatic investigations) is F = Eq. The two F’s mean the same thing. So, F = F or Eq = BQv or v = E/B.
Variations: Interacting Wires A related situation is the interaction of two parallel electrified wires. Each creates an its own invisible magnet and the two interact just like two solid magnets, attracting or repelling each other. Note: When the currents in the two wires are parallel, the wires attract. Currents flowing in opposite directions along the wires cause repulsion. Diagrams from the First Right Hand Rule help us see why.
Calculations: F = k I1 I2 L
d
F = force of interaction between the two electrified wires, in N
k = as above
I1, I2 = the currents in the two wires, in A
L = the common straight length of the two wires, in m
d = the distance between the wires, in m
Q. Find the interactive force between two parallel wires, one carrying 1 A, the other 2 A, if their
common straight length is 1 m and they lie 0.1 m apart. If the currents are parallel, what is the
direction of the interaction?
F = 2 * 10 – 7 N/amp2 * 1 A * 2 A * 1 m = 4 * 10 – 6 N, attraction.
0.1 m
Note: another equation for this situation is: F = μ I I ℓ / 2π r = 2 I I ℓ / r.
Q. Find the force between two 15 cm long wires separated by 5 cm and each carrying 1.6 A.
F = 2 * 1.6 * 1.6 * .15 / .05 = 15.36 N
Applications: Electric motors These are a direct off shoot of the Third Right Hand Rule. A single straight wire in a magnetic field will move in just one direction. If this linear motion can become rotational, an electric motor is the result. If the straight wire is shaped like a fly swatter or loop it becomes an armature. The current flows up one side, across the end and down the other side. The current flowing in opposite directions along the two sides of the armature causes an upward motion on one side and a downward motion on the other – the result is rotation.
Galvanometer A galvanometer is a sensitive DC meter that is basically an electric motor with a limited amount of rotation. A spring attached to the armature creates a back force to counterbalance the rotational force created by current flowing through the armature. A pointer attached to the back end of the armature moves across a faceplate with appropriate quantities. Should the spring not be strong enough to prevent over– rotation, a small pin at either end of the scale prevents the pointer (and the armature) from rotating out of sight. The placement and size of a protective resistor inside the galvanometer case allows it to be used as either an ammeter or a voltmeter.
Ammeter An ammeter is hooked up in series to a circuit so that all the current will have to pass through it. Inside the ammeter’s case however, a small resistor is hooked up in parallel to allow most of the current to bypass the delicate galvanometer. By knowing how much current is diverted through the small resistor, appropriately large values can be placed on the meter’s scale.
Q. A galvanometer has an armature resistance of 4 Ω and can tolerate a voltage of 8 mV. A
protective resistor of what size will allow it to read up to 7.5 A?
a) the first step is to find the tiny amount of current that can safely pass through the galvanometer:
IG = VG / RG = 0.008 V / 4 Ω = 0.002 A
b) the second step is to find how much current must be diverted through the protective resistor.
Because of the parallel setup inside the ammeter’s case, the IT = IR + IG and so:
IT = IR + IG or IR = IT – IG = 7.5 A – 0.002 A = 7.498 A
c) the last step is to find the rating of the protective resistor. Also because of the parallel setup
inside the voltmeter’s case, the VG = VR. And so:
RR = VR / IR = 0.008 V / 7.498 A = 0.00107 Ω
Voltmeter
A voltmeter can be hooked up in parallel to a circuit because, although it creates a new branch, the voltage is the same in all branches. Unlike an ammeter which must have all the current flowing through it to measure the number of electrons in motion, a voltmeter works by holding back the current and measuring the electrical pressure, the voltage. Inside its case, a large resistor is hooked up in series to the sensitive galvanometer to keep most of the current from entering it. By knowing how much current has been held back, appropriately large values can be printed on the scale.
Q. A galvanometer has an armature resistance of 90 Ω and can tolerate a current of 90 mA. A
protective resistor of what size will allow it to function as a 300 V voltmeter?
a) the first step is to find the maximum voltage that can be tolerated by the galvanometer:
VG = IG RG = 0.09 A * 90 Ω = 8.1 V
b) the second step is to find what voltage must be tolerated by the protective resistor. Because of the
series set up inside the voltmeter’s case, the VT = VR + VG and so:
VR = VT – VG = 300 V – 8.1 V = 291.9 V
c) the last step is to find the rating of the protective resistor. Also because of the series setup inside the
voltmeter’s case, IG = IR and so:
RR = VR / IR = 291.9 V / 0.09 A = 3243 Ω
Fourth Right Hand Rule
Situation: a wire moved through the magnetic field in the gap between opposite poles releases current.
The process of getting current from wires moving through magnetic fields is electromagnetic
induction. Michael Faraday discovered the process in 1821.
The Rule: hand flat, fingers point to S across the gap; thumb points in direction in which wire is moved;
palm faces in the direction of current flow.
Calculations: EMF = V B L This equation describes the factors affecting the amount of potential
difference is created in the moving wire.
EMF = the electromotive force created in the moving wire by electromagnetic induction, in V
V = the velocity of the wire through the magnetic field, in m/s
B = the magnetic flux density, in T or Webers / m2
L = the length of the wire exposed to the magnetic field, in m
Q. Find the potential difference in a 0.4 m wire moving at 5 m/s through a 2 * 10 – 2 T magnetic flux.
EMF = 5 m/s * 2 * 10 – 2 T * 0.4 m = 0.04 V
Paradox: Lenz’s Law The ironic thing about the Fourth Right Hand Rule is that it tries to stop itself
from happening! This Rule tells us that, as soon as the wire bathed in the magnetic field between the
magnet’s jaws is caused to move, current begins to flow from it. The RHR 1 tells us the wire immediately
uses this “induced” current to create a magnetic field around itself – no surprise. But, the direction in
which the induced current flows in the wire causes its magnetic force field to be opposite in polarity to
that of the field between the magnet’s jaws. Since the polarity of the two fields are opposite, they grab at
each other: the large magnet tries to keep the small magnet (the wire) from moving. And, if the wire does
not move, it releases no current. This idea is called Lenz’s Law.
Applications: Transformers This device is a combination of the 2nd and 4th Right Hand Rules. It is just a metal donut with insulated wire wrapped around opposite sides. Its main function is to change AC voltage and as a consequence it also changes
the current, but in the reverse fashion. Current
flowing through the primary coils creates a
magnetic field, as in the 2nd Right Hand Rule.
The metal concentrates and channels the field
and as the force lines flow through the center
of the secondary coils, the 4th Right Hand Rule
generates current in these coils.
Q. A transformer has 150 primary and 3 000 secondary coil turns. If 110 V are applied to the primary
side and the secondary current is 0.1 A, find: secondary voltage, primary current, input power, output
power, and efficiency.
a) secondary voltage
VS = NS or VS = NS * VP = 3 000 * 110 V = 2 200 V.
VP NP NP 150
b) primary current
IS = NP or IP = IS * NS = 0.1 A * 3 000 = 2 A.
IP NS NP 150
c) input power
PIN = IP VP = 2 A * 100 V = 220 W.
d) output power
POUT = IS VS = 0.1 A * 2 200 V = 220 W.
e) efficiency
Eff = POUT = 220 W = 100%
PIN 220 W
Note: in reality, the efficiency is always less than 100% because some energy is lost as heat and perhaps a bit of electrical buzz.
Power ratings Power is a measure of the rate at which energy (electrical in this case) is transferred. Its unit is the Watt or W. A kilowatt = 1000 W. A kilowatt hour = the number of watts * the number of hours of energy use. A kilowatt hour also = 1000 W * 3600 s = 3.6 * 10 6 Joules or 3.6 M J.
Transformer History: Their Role as Safety Devices Many electrical devices used to have metal cases and if the internal wiring became frayed, current might leak onto the metal case. When you touched it, there was a danger of serious shock because the leaked current, trying to complete the circuit, might use you as a path to try to reach “ground”, i.e., to get back to the wall socket or actually into the Earth! Transformers reduced this hazard because the current from a transformer starts in the transformer and has no impulse to reach the wall socket or the Earth. So, if your hand touched a bare wire coming from the transformer, the current would tend not go through you since this was not a path back to its source, the transformer. The current might enter your hand and create a tingle but it would not travel down through your body and out your legs, causing major shock damage. This is sort of like a bird perched on a wire – it gets no shock because its body is not a route for the electricity trying to get to ground and also because the bird’s body is a less favorable path for the electricity to flow through than just along the wire itself. Today, nearly everything has a plastic case so the shock hazard is reduced to start with. So, why are transformers still so common? To allow manufacturers to use circuits requiring any voltage they find convenient – the transformer can change the line voltage of 110 V into whatever is called for.