Magnetic Fields and Electromagnetism

• 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.3 Magnetic field lines must form closed loops, as described by Gauss's law for magnetism.
  • 12.1.A.3.i Maxwell's equations are the collection of equations that fully describe electromagnetism. Gauss's law for magnetism is Maxwell's second equation.
    • Equation: $\oint \vec{B} \cdot d\vec{A} = 0$
  • 12.1.A.3.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.B.4 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 A magnetic field will exert a force on a charged object moving within that field, with magnitude and direction that depend on the cross-product of the charge's velocity and the magnetic field.
  • Equation: $\vec{F}_B = q\left(\vec{v} \times \vec{B}\right)$
• 12.2.B.2 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.3 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.

• 12.3.A.1 The Biot-Savart law defines the magnitude and direction of a magnetic field created by an electrical current.
  • Equation: $d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I(d\vec{\ell} \times \hat{r})}{r^2}$
• 12.3.A.2 The magnetic field vectors around a small segment of a current-carrying wire are tangent to concentric circles centered on that wire. The field has no component toward, away from, or parallel to the segment of the current-carrying wire.
• 12.3.A.3 The Biot-Savart law can be used to derive the magnitudes and directions of magnetic fields around segments of current-carrying wires, for example at the center of a circular loop of wire.
  • Equation: $B_{\text{center of loop}} = \dfrac{\mu_0 I}{2R}$

• 12.3.B.1 A magnetic field will exert a force on a current-carrying wire.
  • Equation: $\vec{F}_B = \int I\left(d\vec{\ell} \times \vec{B}\right)$

Boundary statement: AP Physics C: Electricity & Magnetism only expects students to perform quantitative analysis of certain cases of current-carrying conductors using the Biot-Savart law, such as at a location along the perpendicular bisector of a straight conductor, at a location along the central axis of a circular loop, or at the center of a segment of a circular loop.

• 12.4.A.1 Ampère's law relates the magnitude of the magnetic field to the current enclosed by a closed imaginary path called an Amperian loop.
  • Equation: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}$
  • 12.4.A.1.i Ampère's law can be used to determine the magnetic field near a long, straight current-carrying wire.
    • Equation: $B_{\text{wire}} = \dfrac{\mu_0}{2\pi} \dfrac{I}{r}$
  • 12.4.A.1.ii Unless otherwise stated, all solenoids are assumed to be very long, with uniform magnetic fields inside the solenoids and negligible magnetic fields outside the solenoids.
  • 12.4.A.1.iii Ampère's law can be used to determine the magnetic field inside of a long solenoid.
    • Equation: $B_{\text{sol}} = \mu_0 n I$
• 12.4.A.2 An Amperian loop is a closed path around a current-carrying conductor.
• 12.4.A.3 The principle of superposition can be used to determine the net magnetic field at a point in space created by various combinations of current-carrying conductors, or conducting loops, segments, or cylinders.
• 12.4.A.4 Maxwell's equations are the collection of equations that fully describe electromagnetism. Maxwell's fourth equation is Ampère's law with Maxwell's addition; it states that magnetic fields can be generated by electric current (Ampère's law) and that a changing electric field creates a magnetic field, similar to the way a moving charge creates a magnetic field (Maxwell's addition).
  • Equation: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I + \mu_0 \varepsilon_0 \dfrac{d\Phi_E}{dt}$

Boundary statement: AP Physics C: Electricity & Magnetism only expects quantitative application of Ampère's law limited to situations involving symmetrical magnetic fields. Long straight wires, long solenoids carrying currents, as well as conductive slabs or cylindrical conductors carrying a current density, are the types of shapes to which Ampère's law will be applied on the AP Physics C: Electricity & Magnetism Exam.

Boundary statement: AP Physics C: Electricity & Magnetism does not expect students to use Maxwell's fourth equation with a changing electric field. However, students should understand that a changing electric field generates a magnetic field.