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Magnetic Fields and Magnetic Forces

Properties of magnets:

  1. A magnet has polarity - it has a north and a south pole; you cannot isolate the north or the south pole (there is no magnetic monopole)
  2. Like poles repel; unlike poles attract
  3. A compass is a suspended magnet (its north pole is attracted to a magnetic south pole); the earth’s magnetic south pole is within 200 miles of the earth’s geographic north pole (that is why a compass points "north")
  4. Some metals can be turned in to temporary magnets by bringing them close to a magnet; magnetism is induced by aligning areas called domains within a magnetic field
  5. Permanent magnets are formed of metallic alloys or metals such as iron, nickel, or cobalt.
Every spinning electron is a tiny magnet. Electrons spin about their axis like a top spins around its axis. Thus, the electron is a moving charge. Moving charges create a magnetic field. A pair of electrons spinning in the same direction create a stronger magnetic field; a pair of electrons spinning in opposite directions create a weaker magnetic field. The magnetic fields produced by spinning electrons in ferromagnetic materials do not all cancel each other out.

Magnetic field (symbol is B and SI unit is the Tesla or T)
the environment around a magnet in which the magnetic forces act. Another common unit for magnetic field strength is the gauss (G); 1 G = 1 x 10-4

Magnetic field lines
they represent the area around a magnet; magnetic field lines outside of the magnet flow from the north to the south pole

Domain
Atoms of ferromagnetic materials act in groups called domains; atomic magnets in each domain are aligned so that each domain is a microscopic bar magnet; the domains align themselves with an external magnetic field. Each domain behaves like a tiny magnet and has a north and a south pole. In unmagnetized materials, the domains are randomly arranged. In magnetized materials, the domains are aligned. Anything that randomizes the alignment of the domains destroys the magnetic properties of a material (dropping a magnet or heating it)

Comparing electricity and magnetism:
ElectricityMagnetism
+ and - chargesN and S poles
like charges repellike poles repel
unlike charges attractunlike poles attract
electric monopole existsno magnetic monopole
electric field lines flow from + to -magnetic field lines flow from N to S
density of lines equals strength of Edensity of lines equals strength of B
SI unit: ampere, 1 A = 1 C/secSI unit: Tesla, 1 T = 1 N/Amp meter
E exerts force on a charge, or E = F/qField exerts force on a moving charge, or B = F/(qvsinq)

Curie temperature Important constants to know for this section: The charge of an electron (or proton) is 1.6 x 10-19C and one Volt is equivalent to 1.6 x 10-19 J of energy.

Curie temperature temperature above which a material loses all magnetic properties

Oersted (1820) found that an electric current in a wire produces a magnetic field around it; a stationary charge does not create a magnetic field

Right-hand rules predict the direction of magnetic fields produced by a current. They are used for conventional current flow. Use your left hand to predict the direction an electron or negative charge would follow.

RHR #1 - Straight Wire Conductor

Curl the fingers of the right hand into the shape of a circle. Point the thumb in the direction of the current and the tips of the fingers will point in the direction of the magnetic field.

right hand rule 1 right hand rule 1

RHR #2 - Solenoid

Curl the fingers of the right hand in the direction of the current. Your thumb is the north pole of the electromagnet.

solenoid right hand rule

RHR #3 - Magnetic Force

Extend the right hand so that the fingers point in the direction of the magnetic field and the thumb points in the direction of the current. The palm of the hand then pushes in the direction of the magnetic force.

right hand rule for forces

Forces Due to Magnetic Fields

Ampere found that a force is exerted on a current-carrying wire in a magnetic field

F = B I L sin f
where B is the magnetic field in Teslas (T), I is the curent, L is the length of wire in meters, and f is the angle. Only the perpendicular component of B exerts a force on the wire. If the direction of the current is perpendicular to the field (f=90), then the force is given by
F = B I L

We know how to measure force, current, and length. Thus B can be calculated by using

magnetic field strength

The force produced by a magnetic field on a single charge depends upon the speed of the charge, the strength of the field, and the magnitude of the charge.

F = q v B sin f
where q is the charge in Coulombs and v is the velocity of the charge. If f=90, then F = q v B

How speed affects the force on a charged particle moving in a magnetic field. Effects of speed of particle in a Magnetic Field

If the charged particle moves parallel to the field lines (f=0), then the magnetic force on the particle is zero. If a charged particle is moving perpendicular to a uniform magnetic field, the path of the charged particle is an arc (or circle). The magnetic force is the source of the centripetal force on the charged particle. This relationship can be used to find the radius of the arc.
(m v2)/ r = q v B

Since the magnetic force is perpendicular to the velocity of the charged particle, the force does not cause the speed of the particle to change, only its direction. Thus, no work is done by the magnetic force on the charged particle.

Deflection of electron in a Magnetic Field due to Magnetic Force

The magnetic field near a long straight wire is directly proportional to the current I in the wire and inversely proportional to the distance r from the wire. The magnetic field at any point a distance R away from a straight-wire conductor can be calculated using,

magnetic field around a straight wire conductor

or, it can be written in its true form (This is an important formula for the AP B exam.)

magnetic field around a straight wire conductor
where mo is a constant called the permeability of free space and has a value of 4p x 10-7 T m/A

Since a wire carrying a current produces a magnetic field and the wire experiences a force when placed in a magnetic field, two current-carrying wires exert a force on each other. The force exerted on the second wire is only due to the magnetic field exerted by the first wire. Parallel currents in the same directions attract each other and parallel currents in opposite directions repel each other.

Force on a Loop of Wire (represented by multiple choice questions on the AP B exam in which you predict the direction of current, etc.)At the center of the loop, the magnetic field is perpendicular to the plane of the loop. If there are N loops, the strength of the magnetic field at the center of the loop is given by multiplying the following by N. The direction of the magnetic field at the center of the loop can be determined using a RHR (the thumb is pointed in the direction of the current and the curled fingers are placed at the center of the loop, then the palm pushes in the direction of the magnetic field.)

Magnetic field at the center of a loop of wire

Electromagnetic Induction

There are two ways that electricity and magnetism are related: an electric current produces a magnetic field and a magnetic field exerts a force on an electric current or moving charged particle. Henry and Faraday independently found found that a current could be induced in a wire by moving it in a magnetic field. An electric current is generated in a wire when the wire cuts across magnetic field lines.

Faraday found that a steady magnetic field does not produce any current, only a changing magnetic field produces an electric current.

Hints for the AP B exam:

  1. Approximately, two out of every three years one of the free response questions involves a situation where a charge is accelerated through two charged plates. The charge then enters a magnetic field whose direction causes the charge to move in a circle. These are common questions that are asked:
    • Calculate the speed of the charge as it exits the region between the two charged plates.
    • Draw the direction of the electric field between the two charged plates.
    • If there is also a magnetic field between the two charged plates in addition to the electric field, explain the relationship between the two fields that allows the charge to pass through undeflected.
    • Calculate the radius of the path of the charge in the magnetic field.
  2. Remember these three formulas for regions where both electric and magnetic fields exist: V=Ed, qE=F, and F=qvB. Manipulating these formulas allows you to express the velocity of the charge in terms of E and B. Manipulating these formulas allow you to write an expression for the accelerating voltage in terms of v, B, and d.
  3. Remember, if the charge is moving in a circle, the magnetic force provides the centripetal force. This allows you to calculate the radius.
  4. Remember, if the charge is moving in a circle and the magnetic field is perpendicular to it, it does no work on the charge. It only changes its direction.
  5. Sometimes they ask you to calculate the thermal energy dissipated by the accelerated charge if it is allowed to strike a target. You know its speed; calculate its energy.
  6. A mass spectrometer is also representative of this type of problem. In a mass spectrometer, the radius of the path of a particle is proportional to its mass. If you have several particles, you can set up a proportion between their masses and radii to determine the mass of an unknown particle.
  7. If the charged particle moves in a circular path, the centripetal force equals the magnetic force. This equality can be solved for the ratio of charge to mass of the particle (q/m).
  8. More points are awarded when they ask for a direction if you express it in terms of positive or negative x, y, or z.

Electromagnetic induction
process of generating a current by using a magnetic field. This is sometimes called motional emf.
emf = B L v sin f
where emf is the potential difference measured in volts, v is the velocity with which the wire is moved through the magnetic field B, p is the angle at which the wire is moved in the magnetic field, and L is the length of the wire

electromotive force (emf)
a potential difference, measured in volts, that can cause an induced current to flow in a wire. It is not a force, but is a historical term coined before electricity was understood.

An induced emf is produced by a changing magnetic field.

Electromagnetic Induction

the process where current is produced when either a wire or a magnetic field move relative to one another; as long as the wire cuts across magnetic field lines during the motion, a current is produced. A current is induced in a coil of wire if it is moved into or out of a magnetic field; a current is induced in a coil of wire if a magnet is inserted or removed from the coil of wire. It doesn't matter if the magnet or the coil moves-motion or change is required to induce an emf.

Lenz's Law

The direction of the induced current is such that the magnetic field resulting from the induced current opposes the change in the flow (or flux) that causes the induced current. It is the change in the flow or flux that causes the induced current, not the flux itself.

How I predict the direction of the induced current using Lenz's Law:

  1. Determine whether the magnetic field strength is increasing or decreasing.
  2. Determine the direction in which the original field enters the coil.
  3. Determine the direction of the induced magnetic field so that it opposes the change in the magnetic flux.
  4. Use RHR to predict the direction of the current knowing the direction of the induced magnetic field.

AP B Multiple Choice Questions Hints:There are always questions asked in which you much predict the direction of an induced current (or emf, e)

  1. Know the situations when a current or emf is induced in a coil or wire (Look under the explanation for Faraday's law). Remember - there must be relative motion or something that causes a changing flux!
  2. Be able to write an expression (or calculate) for induced current or emf. In other words, this can be easily done using a combination of e = Blv sin f and e = iR.
  3. KNOW your RHR that enables you to predict the current through a single loop or coil. Remember, your thumb points in the direction of the current, clockwise or counterclockwise. Your fingers enter the loop or coil in the direction of the magnetic field.
  4. Be able to interpret directions in terms of the x, y, and z axes.
  5. Be able to calculate (or write an expression) induced voltage using Faraday's Law. Faraday's law can also be used to calculate the rate of change of the magnetic field in a moving coil (in other words, it can calcualte B/t).
  6. The faster something is moved, the greater the induced voltage (current) because emf is directly proportional to the velocity.

Self-inductance

induced emf produced in a coil by a changing current

Mutual inductance

a changing current in one coil induces an emf in another coil

Transformer

an electrical device that increases or decreases AC voltage; a step-up transformer has more turns in the secondary than in the primary; a step-down transformer has more turns in the primary than in the secondary. We will call the primary the incoming voltage or current and the secondary the outgoing voltage or current.

transformer equation

where N is number of turns, V is the voltage, and I is the current. s and p stand for secondary and primary, respectively.

Magnetic flux (FB and SI unit is the Weber, Wb)the number of magnetic field lines that pass through a surface of area A. A changing magnetic flux produces an electric field. This is true not only of wires and conductors, but also applies to any region in space.

FB = Bperpendicular A
where Bperpendicular is the component of B perpendicular to the face of the coil

Faraday's Law of Induction Faraday found that the amount of emf induced in a coil of wire depended upon how rapidly the magnetic field changes in the coil of wire. The faster the magnetic field changes, the greater the induced emf. If the flux through a coil of N loops of wire chagnes by an amount FB during a time Dt.

e = - N (FB/Dt)
The negative sign indicates the direction in which the induced emf acts. For our purposes, we will use Faraday's law to calculate the magnitude of the induced emf and apply right hand rules for Lenz's Law to determine the direction of the induced emf.

An emf can be induced three ways:

  1. By a changing magnetic field
  2. By changing the area of the loop in the field
  3. By changing the loop's orientation with respect to the field

Electric Motors and Generators

Electric motor
uses electrical energy to produce mechanical energy. In a motor, there must be a source of a magnetic field; brushes serve as a connection to the split-ring commutator, allowing current flow from the motor to an outside source. In order to continue rotating, current direction must be reversed. This is achieved by the use of the split-ring commutator and the brushes. The force on a current-carrying wire in a magnetic field causes an electric motor to rotate

The Electric Motor

Electric generator
uses mechanical energy to create electrical energy; rotation of wire loop in a magnetic field causes current to be induced. This current changes direction every 180 degrees, producing alternating current (AC current).

The Electric Generator

Magnetic Moment When an electric current flows in a closed loop of wire placed in a magnetic field, the magnetic force on the current can cause a torque. This is the basic principle behind meters and motors. If the coil consists of N loops of wire carrying current I with area A, the torque is given by

t = NIAB sin f
where NIA is the vector quantity called the magnetic dipole moment of the coil. Its direction is perpendicular to the plane of the coil.

Magnetism Sample Problems

Magnetism Homework

AP Magnetostatics Hand Rule Class Problems

AP Magnetostatics Sample Problems

AP Electromagnetic Induction Sample Problems

AP Lenz's Law Class Problems

AP Magnetostatics Objectives

AP Electromagnetism Objectives