磁学与电磁学

• 12.1.A.1 A magnetic field is a vector field that can be used to determine the magnetic force exerted on moving electric charges, electric currents, or magnetic materials.
  • 12.1.A.1.i Magnetic fields can be produced by magnetic dipoles or combinations of dipoles, but never by monopoles.
  • 12.1.A.1.ii Magnetic dipoles have north and south polarity.
• 12.1.A.2 A magnetic field is a vector quantity and can be represented using vector field maps.
  • 12.1.A.2.i Magnetic field lines form closed loops.
  • 12.1.A.2.ii Magnetic fields in a bar magnet form closed loops, with the external magnetic field pointing away from one end (defined as the north pole) and returning to the other end (defined as the south pole).

• 12.1.B.1 Magnetic dipoles result from the circular or rotational motion of electric charges. In magnetic materials, this can be the motion of electrons.
  • 12.1.B.1.i Permanent magnetism and induced magnetism are system properties that both result from the alignment of magnetic dipoles within a system.
  • 12.1.B.1.ii No magnetic north pole is ever found in isolation from a south pole. For example, if a bar magnet is broken in half, both halves are magnetic dipoles.
  • 12.1.B.1.iii Magnetic poles of the same polarity will repel; magnetic poles of opposite polarity will attract.
  • 12.1.B.1.iv The magnitude of the magnetic field from a magnetic dipole decreases with increasing distance from the dipole.
• 12.1.B.2 A magnetic dipole, such as a magnetic compass, placed in a magnetic field will tend to align with the magnetic field.
• 12.1.B.3 A material's composition influences its magnetic behavior in the presence of an external magnetic field.
  • 12.1.B.3.i Ferromagnetic materials such as iron, nickel, and cobalt can be permanently magnetized by an external field that causes the alignment of magnetic domains or atomic magnetic dipoles.
  • 12.1.B.3.ii Paramagnetic materials such as aluminum, titanium, and magnesium interact weakly with an external magnetic field, in that the magnetic dipoles of the material do not remain aligned after the external field is removed.
  • 12.1.B.3.iii All materials have the property of diamagnetism, in that their electronic structure creates a usually weak alignment of the dipole moments of the material opposite the external magnetic field.
• 12.1.B4 Earth's magnetic field may be approximated as a magnetic dipole.

• 12.1.C.1 Magnetic permeability is a measurement of the amount of magnetization in a material in response to an external magnetic field.
• 12.1.C.2 Free space has a constant value of magnetic permeability, known as the vacuum permeability $\mu_0$, that appears in equations representing physical relationships.
• 12.1.C.3 The permeability of matter has values different from that of free space and arises from the matter's composition and arrangement. It is not a constant for a material and varies based on many factors, including temperature, orientation, and strength of the external field.

• 12.2.A.1 A single moving charged object produces a magnetic field.
  • 12.2.A.1.i The magnetic field at a particular point produced by a moving charged object depends on the object's velocity and the distance between the point and the object.
  • 12.2.A.1.ii At a point in space, the direction of the magnetic field produced by a moving charged object is perpendicular to both the velocity of the object and the position vector from the object to that point in space and can be determined using the right-hand rule.
  • 12.2.A.1.iii The magnitude of the magnetic field is a maximum when the velocity vector and the position vector from the object to that point in space are perpendicular.

• 12.2.B.1 Magnetic forces describe interactions between moving charged objects.
• 12.2.B.2 A magnetic field may exert a force on a charged object moving in that field.
  • 12.2.B.2.i The magnitude of the force exerted by a magnetic field on a moving charged object is proportional to the magnitude of the charge, the magnitude of the charged object's velocity, and the magnitude of the magnetic field and also depends on the angle between the velocity and magnetic field vectors.
    • Equation: $F_B = qvB\sin\theta$
  • 12.2.B.2.ii The direction of the force exerted by a magnetic field on a moving charged object is perpendicular to both the direction of the magnetic field and the velocity of the charge, as defined by the right-hand rule.
• 12.2.B.3 In a region containing both a magnetic field and an electric field, a moving charged object will experience independent forces from each field.
• 12.2.B.4 The Hall effect describes the potential difference created in a conductor by an external magnetic field that has a component perpendicular to the direction of charges moving in the conductor.

Boundary statement: Quantitative treatment of the magnitude of the magnetic force exerted by a magnetic field on a moving charge is limited to angles of 0, 90, and 180 degrees between the velocity and the magnetic field. Qualitative analysis of other angles is permitted.

• 12.3.A.1 A current-carrying wire produces a magnetic field.
  • 12.3.A.1.i The magnetic field vectors around a long, straight, current-carrying wire are tangent to concentric circles centered on that wire. The field has no component toward, away from, or parallel to the long, straight, current-carrying wire.
  • 12.3.A.1.ii At a point in space, the magnitude of the magnetic field due to a long, straight, current-carrying wire is proportional to the magnitude of the current in the wire and inversely proportional to the perpendicular distance from the central axis of the wire to the point.
    • Equation: $B = \dfrac{\mu_0}{2\pi}\dfrac{I}{r}$
  • 12.3.A.1.iii The direction of the magnetic field created by a current-carrying wire is determined with the right-hand rule.
  • 12.3.A.1.iv The direction of the magnetic field at the center of a current-carrying loop is directed along the axis of the loop and can be found using the right-hand rule.
  • 12.3.A.1.v The magnetic field at a location near two or more current-carrying wires can be determined using vector addition principles.

• 12.3.B.1 A magnetic field may exert a force on a current-carrying wire.
  • 12.3.B.1.i The magnitude of the force exerted by a magnetic field on a current-carrying wire is proportional to the current, the length of the portion of the wire within the magnetic field, and the magnitude of the magnetic field, and also depends on the angle between the direction of the current in the wire and the direction of the magnetic field.
    • Equation: $F_B = I\ell B\sin\theta$
  • 12.3.B.1.ii The direction of the force exerted by the magnetic field on a current-carrying wire is determined by the right-hand rule.

• 12.4.A.1 Magnetic flux is a description of the amount of the component of a magnetic field that is perpendicular to a cross-sectional area.
• 12.4.A.2 Magnetic flux through a surface is proportional to the magnitude of the component of the magnetic field perpendicular to the surface and to the cross-sectional area of the surface.
  • Equation: $\Phi_B = BA\cos\theta$
  • 12.4.A.2.i The area vector is defined to be perpendicular to the plane of the surface and directed outward from a closed surface.
  • 12.4.A.2.ii The sign of the magnetic flux indicates whether the magnetic field is parallel to or antiparallel to the area vector.
• 12.4.A.3 Faraday’s law describes the relationship between changing magnetic flux and the resulting induced emf in a system.
  • Equation: $|\mathcal{E}| = \left|\dfrac{\Delta\Phi_B}{\Delta t}\right|$
• 12.4.A.4 Lenz’s law is used to determine the direction of an induced emf resulting from a changing magnetic flux.
  • Equation: $\mathcal{E} = -\dfrac{\Delta\Phi_B}{\Delta t} = -\dfrac{\Delta(BA\cos\theta)}{\Delta t}$
  • 12.4.A.4.i An induced emf generates a current that creates a magnetic field that opposes the change in magnetic flux.
  • 12.4.A.4.ii The right-hand rule is used to determine the relationships between current, emf, and magnetic flux.
• 12.4.A.5 A common example of electromagnetic induction is a conducting rod on conducting rails in a region with a uniform magnetic field.
  • Derived equation: $\mathcal{E} = B\ell v$