AP Physics C: Electricity and Magnetism · ⁨פיזיקה AP C: חשמל ומגנטיות⁩

Learning Objective 8.2.A: Describe the behavior of a system using conservation of charge.

• 8.2.A.1 The net charge or charge distribution of a system can change in response to the presence of, or changes in, the net charge or charge distribution of other systems.
  • 8.2.A.1.i The net charge of a system can change due to friction or contact between systems.
  • 8.2.A.1.ii Induced charge separation occurs when the electrostatic force between two systems alters the distribution of charges within the systems, resulting in the polarization of one or both systems.
  • 8.2.A.1.iii Induced charge separation can occur in neutral systems.
• 8.2.A.2 Any change to a system's net charge is due to a transfer of charge between the system and its surroundings.
  • 8.2.A.2.i The charging of a system typically involves the transfer of electrons to and from the system.
  • 8.2.A.2.ii The net charge of a system will be constant unless there is a transfer of charge to or from the system.
• 8.2.A.3 Grounding involves electrically connecting a charged object to a much larger and approximately neutral system (e.g., Earth).

Learning Objective 9.1.A: Describe the electric potential energy of a system.

• 9.1.A.1 The electric potential energy of a system of two point charges equals the amount of work required for an external force to bring the point charges to their current positions from infinitely far away.
• 9.1.A.2 The general form for the electric potential energy between two charged objects is given by the equation
  • Equation: $U_E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1 q_2}{r} = k\dfrac{q_1 q_2}{r}$.
• 9.1.A.3 The total electric potential energy of a system can be determined by finding the sum of the electric potential energies of the individual interactions between each pair of charged objects in the system.

Learning Objective 9.2.A: Describe the electric potential due to a configuration of charged objects.

• 9.2.A.1 Electric potential describes the electric potential energy per unit charge at a point in space.
• 9.2.A.2 Expressions for the electric potential of charge distributions can be found using integration and the principle of superposition.
  • Equation: $V = \dfrac{1}{4\pi\varepsilon_0}\displaystyle\int \dfrac{dq}{r}$
  • 9.2.A.2.i The electric potential for single point charge is
    • Equation: $V = \dfrac{q}{4\pi\varepsilon_0 r}$.
  • 9.2.A.2.ii The electric potential due to multiple point charges can be determined by the principle of scalar superposition of the electric potential due to each of the point charges.
    • Equation: $V = \dfrac{1}{4\pi\varepsilon_0}\displaystyle\sum_i \dfrac{q_i}{r_i}$
• 9.2.A.3 The electric potential difference between two points is the change in electric potential energy per unit charge when a test charge is moved between the two points.
  • Equation: $\Delta V = \dfrac{\Delta U_E}{q}$
• 9.2.A.4 Electric potential difference may also result from chemical processes that cause positive and negative charges to separate, such as in a battery.

Boundary statement: AP Physics C: Electricity & Magnetism only expects students to use calculus to find the electric potential resulting from the following charge distributions and locations: an infinitely long, uniformly charged wire or cylinder at a distance from its central axis, a thin ring of charge at a location along the axis of the ring, a semicircular arc or part of a semicircular arc at its center, and a finite wire or line charge at a point collinear with the line charge or at a location along its perpendicular bisector.

Learning Objective 9.2.B: Describe the relationship between electric potential and electric field.

• 9.2.B.1 The value of an electric field component in any direction at a given location is equal to the negative of the spatial rate of change in electric potential at that location.
  • Equation: $E_x = -\dfrac{dV}{dx}$
• 9.2.B.2 The change in electric potential between two points can be determined by integrating the dot product of the electric field and the displacement along the path connecting the points.
  • Equation: $\Delta V = V_b - V_a = -\displaystyle\int_a^b \vec{E}\cdot d\vec{r}$
• 9.2.B.3 Electric field vector maps and equipotential lines are tools to describe the field produced by a charge or configuration of charges and can be used to predict the motion of charged objects in the field.
  • 9.2.B.3.i Equipotential lines represent lines of equal electric potential. These lines are also referred to as isolines of electric potential.
  • 9.2.B.3.ii Isolines are perpendicular to electric field vectors. An isoline map of electric potential can be constructed from an electric field vector map, and an electric field map may be constructed from an isoline map.
  • 9.2.B.3.iii An electric field vector points in the direction of decreasing potential.
  • 9.2.B.3.iv There is no component of an electric field along an isoline.

Learning Objective 9.3.A: Describe changes in a system due to a difference in electric potential between two locations.

• 9.3.A.1 When a charged object moves between two locations with different electric potentials, the resulting change in the electric potential energy of the object-field system is given by the following equation.
  • Equation: $\Delta U_E = q\Delta V$
• 9.3.A.2 The movement of a charged object between two points with different electric potentials results in a change in kinetic energy of the object consistent with the conservation of energy.

Learning Objective 10.2.A: Describe the movement of charge and the resulting interactions when conductors physically contact each other.

• 10.2.A.1 When conductors are in electrical contact, charges will be redistributed such that the surfaces of each conductor are at the same electric potential.
• 10.2.A.2 Ground is an idealized reference point that has zero electric potential and can absorb or provide an infinite amount of charge without changing its electric potential.
• 10.2.A.3 Charge can be induced on a conductor by grounding the conductor in the presence of an external electric field.

Learning Objective 10.3.A: Describe the physical properties of a parallel-plate capacitor.

• 10.3.A.1 A parallel-plate capacitor consists of two separated parallel conducting surfaces that can hold equal amounts of charge with opposite signs.
• 10.3.A.2 Capacitance relates the magnitude of the charge stored on each plate to the electric potential difference created by the separation of those charges.
  • Equation: $C = \dfrac{Q}{\Delta V}$
  • 10.3.A.2.i The capacitance of a capacitor depends only on the physical properties of the capacitor, such as the capacitor's shape and the material used to separate the plates.
  • 10.3.A.2.ii The capacitance of a parallel-plate capacitor is proportional to the area of one of its plates and inversely proportional to the distance between its plates. The constant of proportionality is the product of the dielectric constant, $\kappa$, of the material between the plates and the electric permittivity of free space, $\varepsilon_0$.
    • Equation: $C = \dfrac{\kappa \varepsilon_0 A}{d}$
• 10.3.A.3 The electric field between two charged parallel plates with uniformly distributed electric charge, such as in a parallel-plate capacitor, is constant in both magnitude and direction, except near the edges of the plates.
  • 10.3.A.3.i The magnitude of the electric field between two charged parallel plates, where the plate separation is much smaller than the dimensions of the plates, can be determined by applying Gauss's law and the principle of superposition.
    • Equation: $E = \dfrac{Q}{\varepsilon_0 A}$
  • 10.3.A.3.ii The electric field is proportional to the surface charge density on either plate of the capacitor.
  • 10.3.A.3.iii A charged particle between two oppositely charged parallel plates undergoes constant acceleration, and therefore its motion shares characteristics with the projectile motion of an object with mass in the gravitational field near Earth's surface.
• 10.3.A.4 The electric potential energy stored in a capacitor is equal to the work done by an external force to separate that amount of charge on the capacitor.
• 10.3.A.5 The electric potential energy stored in a capacitor is described by the equation $U_C = \dfrac{1}{2} Q \Delta V$.

Boundary statement: While other shapes are also able to separate charges, AP Physics C: Electricity & Magnetism only expects the quantitative analysis and description of parallel-plate capacitors, concentric spherical capacitors, and coaxial cylindrical capacitors.

Learning Objective 10.4.A: Describe how a dielectric inserted between the plates of a capacitor changes the properties of the capacitor.

• 10.4.A.1 In a dielectric material, electric charges are not as free to move as they are in a conductor. Instead, the material becomes polarized in the presence of an external electric field.
• 10.4.A.2 The dielectric constant of a material relates the electric permittivity of that material to the permittivity of free space.
  • Equation: $\kappa = \dfrac{\varepsilon}{\varepsilon_0}$
• 10.4.A.3 The electric field created by a polarized dielectric is opposite in direction to the external field.
• 10.4.A.4 The electric field between the plates of an isolated parallel-plate capacitor decreases when a dielectric is placed between the plates.
  • Equation: $\kappa = \dfrac{E_0}{E}$
• 10.4.A.5 The insertion of a dielectric into a capacitor may change the capacitance of the capacitor.
  • Equation: $C = \kappa C_0$

Learning Objective 11.1.A: Describe the movement of electric charges through a medium.

• 11.1.A.1 Current is the rate at which charge passes through a cross-sectional area of a wire.
  • Equation: $I = \dfrac{dq}{dt}$
  • 11.1.A.1.i Current within a conductor consists of charge carriers traveling through the conductor with an average drift velocity.
    • Equation: $I = nqv_d A$
  • 11.1.A.1.ii Electric charge moves in a circuit in response to an electric potential difference, sometimes referred to as electromotive force, or $\mathrm{emf}$ ($\mathcal{E}$).
  • 11.1.A.1.iii If the current is zero in a section of wire, the net motion of charge carriers in the wire is also zero, although individual charge carriers will not have zero speed.
• 11.1.A.2 Current density is the flow of charge per unit area.
  • Equation: $I = \int \vec{J} \cdot d\vec{A}$
  • 11.1.A.2.i Current density is related to the motion of the charge carriers within a conductor.
    • Equation: $\vec{J} = nq\vec{v}_d$
  • 11.1.A.2.ii Current density is a vector quantity.
  • 11.1.A.2.iii A potential difference across a conductor creates an electric field within the conductor that is proportional to the resistivity of the conductor and the current density.
    • Equation: $\vec{E} = \rho\vec{J}$
• 11.1.A.3 If a function of current density is given, the total current can be determined by integrating the current density over the area.
  • Equation: $I_{\text{tot}} = \int \vec{J}(r) \cdot d\vec{A}$
• 11.1.A.4 Although current is a scalar quantity, it does have a direction. Because its direction is relative to the current carrier and not space, current does not obey the laws of vector addition and has no vector components.
  • 11.1.A.4.i The direction of conventional current is chosen to be the direction in which positive charge would move.
  • 11.1.A.4.ii In common circuits, the current is actually due to the movement of electrons (negative charge carriers).

Learning Objective 11.2.A: Describe the behavior of a circuit.

• 11.2.A.1 A circuit is composed of electrical loops, which can include wires, batteries, resistors, lightbulbs, capacitors, inductors, switches, ammeters, and voltmeters.
• 11.2.A.2 A closed electrical loop is a closed path through which charges may flow.
  • 11.2.A.2.i A closed circuit is one in which charges would be able to flow.
  • 11.2.A.2.ii An open circuit is one in which charges would not be able to flow.
  • 11.2.A.2.iii A short circuit is one in which charges would be able to flow with no change in potential difference.
• 11.2.A.3 A single circuit element may be part of multiple electrical loops.
• 11.2.A.4 Circuit schematics are representations used to describe and analyze electric circuits.
  • 11.2.A.4.i The properties of an electric circuit are dependent on the physical arrangement of its constituent elements.
  • 11.2.A.4.ii Circuit elements have common symbols that are used to create schematic diagrams. Variable elements are indicated by a diagonal strikethrough arrow across the standard symbol for that element. (Symbols: Battery, Bulb, Switch, Capacitor, Resistor, Ammeter, Voltmeter, Inductor.)

Boundary statement: Unless otherwise specified, all circuit schematic diagrams will be drawn using conventional current.

Learning Objective 11.3.A: Describe the resistance of an object using physical properties of that object.

• 11.3.A.1 Resistance is a measure of the degree to which an object opposes the movement of electric charge.
• 11.3.A.2 The resistance of a resistor with uniform geometry is proportional to its resistivity and length and is inversely proportional to its cross-sectional area.
  • Equation: $R = \dfrac{\rho\ell}{A}$
  • 11.3.A.2.i Resistivity is a fundamental property of a material that depends on its atomic and molecular structure and quantifies how strongly the material opposes the motion of electric charge.
  • 11.3.A.2.ii The resistivity of a conductor typically increases with temperature.
  • 11.3.A.2.iii The total resistance of a resistor with uniform geometry, but that is made of a material whose resistivity varies along the length of the resistor, is given by $R = \int \dfrac{\rho(\ell)\,d\ell}{A}$.

Learning Objective 11.3.B: Describe the electrical characteristics of elements of a circuit.

• 11.3.B.1 Ohm's law relates current, resistance, and potential difference across a conductive element of a circuit.
  • Equation: $I = \dfrac{\Delta V}{R}$
  • 11.3.B.1.i Materials that obey Ohm's law have constant resistance for all currents and are called ohmic materials.
  • 11.3.B.1.ii The resistivity of an ohmic material is constant regardless of temperature.
  • 11.3.B.1.iii Resistors can also convert electrical energy to thermal energy, which may change the temperature of both the resistor and the resistor's environment.
  • 11.3.B.1.iv The resistance of an ohmic circuit element can be determined from the slope of a graph of the current in the element as a function of the potential difference across the element.

Learning Objective 11.4.A: Describe the transfer of energy into, out of, or within an electric circuit, in terms of power.

• 11.4.A.1 The rate at which energy is transferred, converted, or dissipated by a circuit element depends on the current in the element and the electric potential difference across it.
  • Equation: $P = I\Delta V$
  • Equation: $P = I^2 R = \dfrac{\Delta V^2}{R}$
• 11.4.A.2 The brightness of a lightbulb increases with power, so power can be used to qualitatively predict the brightness of lightbulbs in a circuit.

Boundary statement: AP Physics C: Electricity & Magnetism only expects students to analyze the transfer of mechanical and electrical energy, although students should be aware that electrical energy can also be dissipated in the form of thermal energy.

Learning Objective 11.5.A: Describe the equivalent resistance of multiple resistors connected in a circuit.

• 11.5.A.1 Circuit elements may be connected in series and/or in parallel.
  • 11.5.A.1.i A series connection is one in which any charge passing through one circuit element must proceed through all elements in that connection and has no other path available. The current in each element in series must be the same.
  • 11.5.A.1.ii A parallel connection is one in which charges may pass through one of two or more paths. Across each path, the potential difference is the same.
• 11.5.A.2 A collection of resistors in a circuit may be analyzed as though it were a single resistor with an equivalent resistance $R_{\text{eq}}$.
  • 11.5.A.2.i The equivalent resistance of a set of resistors in series is the sum of the individual resistances.
    • Equation: $R_{\text{eq},s} = \sum_{i} R_i$
  • 11.5.A.2.ii The inverse of the equivalent resistance of a set of resistors connected in parallel is equal to the sum of the inverses of the individual resistances.
    • Equation: $\dfrac{1}{R_{\text{eq},p}} = \sum_{i} \dfrac{1}{R_i}$
  • 11.5.A.2.iii When resistors are connected in parallel, the number of paths available to charges increases, and the equivalent resistance of the group of resistors decreases.

Learning Objective 11.5.B: Describe a circuit with resistive wires and a battery with internal resistance.

• 11.5.B.1 Ideal batteries have negligible internal resistance. Ideal wires have negligible resistance.
  • 11.5.B.1.i The resistance of wires that are good conductors may normally be neglected, because their resistance is much smaller than that of other elements of a circuit.
  • 11.5.B.1.ii The resistance of wires may only be neglected if the circuit contains other elements that do have resistance.
  • 11.5.B.1.iii The potential difference a battery would supply if it were ideal is the potential difference measured across the terminals when there is no current in the battery and is sometimes referred to as its $\mathrm{emf}$ ($\mathcal{E}$).
• 11.5.B.2 The internal resistance of a nonideal battery may be treated as the resistance of a resistor in series with an ideal battery and the remainder of the circuit.
• 11.5.B.3 When there is current in a nonideal battery with internal resistance $r$, the potential difference across the terminals of the battery is reduced relative to the potential difference when there is no current in the battery.
  • Equation: $\Delta V_{\text{terminal}} = \mathcal{E} - Ir$

Learning Objective 11.5.C: Describe the measurement of current and potential difference in a circuit.

• 11.5.C.1 Ammeters are used to measure current at a specific point in a circuit.
  • 11.5.C.1.i Ammeters must be connected in series with the element in which current is being measured.
  • 11.5.C.1.ii Ideal ammeters have zero resistance so that they do not affect the current in the element that they are in series with.
• 11.5.C.2 Voltmeters are used to measure electric potential difference between two points in a circuit.
  • 11.5.C.2.i Voltmeters must be connected in parallel with the element across which potential difference is being measured.
  • 11.5.C.2.ii Ideal voltmeters have infinite resistance so that no charge flows through them.
• 11.5.C.3 Nonideal ammeters and voltmeters will change the properties of the circuit being measured.

Boundary statement: Unless otherwise stated, all batteries, wires, and meters are assumed to be ideal. Circuits with batteries of different potential differences connected in parallel will not be assessed.

Learning Objective 11.6.A: Describe a circuit or elements of a circuit by applying Kirchhoff's loop rule.

• 11.6.A.1 Energy changes in simple electrical circuits may be represented in terms of charges moving through electric potential differences within circuit elements.
  • Equation: $\Delta U_E = q\Delta V$
• 11.6.A.2 Kirchhoff's loop rule is a consequence of the conservation of energy.
  • 11.6.A.2.i Kirchhoff's loop rule states that the sum of potential differences across all circuit elements in a single closed loop must equal zero.
    • Equation: $\sum \Delta V = 0$
  • 11.6.A.2.ii The values of electric potential at points in a circuit can be represented by a graph of electric potential as a function of position within a loop.

Learning Objective 11.7.A: Describe a circuit or elements of a circuit by applying Kirchhoff's junction rule.

• 11.7.A.1 Kirchhoff's junction rule is a consequence of the conservation of electric charge.
• 11.7.A.2 Kirchhoff's junction rule states that the total amount of charge entering a junction per unit time must equal the total amount of charge exiting that junction per unit time.
  • Equation: $\sum I_{\text{in}} = \sum I_{\text{out}}$

Learning Objective 11.8.A: Describe the equivalent capacitance of multiple capacitors.

• 11.8.A.1 A collection of capacitors in a circuit may be analyzed as though it was a single capacitor with an equivalent capacitance $C_{\text{eq}}$.
  • 11.8.A.1.i The inverse of the equivalent capacitance of a set of capacitors connected in series is equal to the sum of the inverses of the individual capacitances.
    • Equation: $\dfrac{1}{C_{\text{eq},s}} = \sum_{i} \dfrac{1}{C_i}$
  • 11.8.A.1.ii The equivalent capacitance of a set of capacitors in series is less than the capacitance of the smallest capacitor.
  • 11.8.A.1.iii The equivalent capacitance of a set of capacitors in parallel is the sum of the individual capacitances.
    • Equation: $C_{\text{eq},p} = \sum_{i} C_i$
• 11.8.A.2 As a result of conservation of charge, each of the capacitors in series must have the same magnitude of charge on each plate.

Learning Objective 11.8.B: Describe the behavior of a circuit containing combinations of resistors and capacitors.

• 11.8.B.1 The charge on a capacitor or the current in a resistor in an RC circuit can be described by a fundamental differential equation derived from Kirchhoff's loop rule.
  • Equation: $\mathcal{E} = \dfrac{dq}{dt}R + \dfrac{q}{C}$
• 11.8.B.2 The time constant ($\tau$) is a significant feature of an RC circuit.
  • 11.8.B.2.i The time constant of an RC circuit is a measure of how quickly the capacitor will charge or discharge and is defined as $\tau = R_{\text{eq}}C_{\text{eq}}$.
  • 11.8.B.2.ii For a charging capacitor, the time constant represents the time required for the capacitor's charge to increase from zero to approximately 63 percent of its final asymptotic value.
  • 11.8.B.2.iii For a discharging capacitor, the time constant represents the time required for the capacitor's charge to decrease from fully charged to approximately 37 percent of its initial value.
• 11.8.B.3 The potential difference across a capacitor and the current in the branch of the circuit containing the capacitor each change over time as the capacitor charges and discharges, but both will reach a steady state after a long time interval.
  • 11.8.B.3.i Immediately after being placed in a circuit, an uncharged capacitor acts like a wire, and charge can easily flow to or from the plates of the capacitor.
  • 11.8.B.3.ii As a capacitor charges, changes to the potential difference across the capacitor affect the charge on the plates of the capacitor, the current in the circuit branch in which the capacitor is located, and the electric potential energy stored in the capacitor.
  • 11.8.B.3.iii The potential difference across a capacitor, the current in the circuit branch in which the capacitor is located, and the electric potential energy stored in the capacitor all change with respect to time and asymptotically approach steady state conditions.
  • 11.8.B.3.iv After a long time, a charging capacitor approaches a state of being fully charged, reaching a maximum potential difference at which there is zero current in the circuit branch in which the capacitor is located.
  • 11.8.B.3.v Immediately after a charged capacitor begins discharging, the amount of charge on the capacitor and the energy stored in the capacitor begin to decrease.
  • 11.8.B.3.vi As a capacitor discharges, the amount of charge on the capacitor, the potential difference across the capacitor, and the current in the circuit branch in which the capacitor is located all decrease until a steady state is reached.
  • 11.8.B.3.vii After either charging or discharging for times much greater than the time constant, the capacitor and the relevant circuit branch may be modeled using steady-state conditions.

Learning Objective 12.1.A: Describe the properties of a magnetic field.

• 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).

Learning Objective 12.1.B: Describe the magnetic behavior of a material as a result of the configuration of magnetic dipoles in the material.

• 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.

Learning Objective 12.1.C: Describe the magnetic permeability of a material.

• 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.

Learning Objective 12.2.A: Describe the magnetic field produced by moving charged objects.

• 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.

Learning Objective 12.2.B: Describe the force exerted on moving charged objects by a magnetic field.

• 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.

Learning Objective 12.3.A: Describe the magnetic field produced by a current-carrying wire.

• 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: $B_{\text{center of loop}} = \dfrac{\mu_0 I}{2R}$

Learning Objective 12.3.B: Describe the force exerted on current-carrying wires by a magnetic field.

• 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.

Learning Objective 12.4.A: Use Ampère's law to describe the magnetic field created by a moving charge carrier.

• 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.

Learning Objective 13.1.A: Describe the magnetic flux through an arbitrary area or geometric shape.

• 13.1.A.1 For a magnetic field $\vec{B}$ that is constant across an area $\vec{A}$, the magnetic flux through the area is defined as $\Phi_B = \vec{B} \cdot \vec{A}$ .
  • 13.1.A.1.i The area vector is defined as perpendicular to the plane of the surface and outward from a closed surface.
  • 13.1.A.1.ii The sign of flux is given by the dot product of the magnetic field vector and the area vector.
• 13.1.A.2 The total magnetic flux passing through a surface is defined by the surface integral of the magnetic field over the surface area.
  • Equation: $\Phi_B = \displaystyle\int \vec{B} \cdot d\vec{A}$

Learning Objective 13.2.A: Describe the induced electric potential difference resulting from a change in magnetic flux.

• 13.2.A.1 Faraday's law describes the relationship between changing magnetic flux and the resulting induced emf in a system.
  • Equation: $\mathcal{E} = -\dfrac{d\Phi_B}{dt} = -\dfrac{d\left(\vec{B} \cdot \vec{A}\right)}{dt}$
  • 13.2.A.1.i When the area of the surface being considered is constant, the induced emf is equal to the area multiplied by the rate of change in the component of the magnetic field perpendicular to the surface.
  • 13.2.A.1.ii When the magnetic field is constant, the induced emf is equal to the magnetic field multiplied by the rate of change in area perpendicular to the magnetic field.
  • 13.2.A.1.iii When an emf is induced in a long solenoid, the total induced emf is equal to the induced emf in a single loop multiplied by the number of loops in the solenoid.
    • Equation: $\left|\mathcal{E}_{\text{sol}}\right| = N\left|\dfrac{d\Phi_B}{dt}\right|$
• 13.2.A.2 Lenz's law is used to determine the direction of an induced emf resulting from a changing magnetic flux.
  • 13.2.A.2.i An induced emf generates a current that creates a magnetic field that opposes the change in magnetic flux.
  • 13.2.A.2.ii The right-hand rule is used to determine the relationships between current, emf, and magnetic flux.
• 13.2.A.3 Maxwell's equations are the collection of equations that fully describe electromagnetism. Maxwell's third equation is Faraday's law of induction, which describes the relationship between a changing magnetic flux and an induced electric field.
  • Equation: $\mathcal{E} = \oint \vec{E} \cdot d\vec{\ell} = -\dfrac{d\Phi_B}{dt}$
• 13.2.A.4 Maxwell's equations can be used to show that electric and magnetic fields obey wave equations and that electromagnetic waves travel at a constant speed in free space.
  • Equation (derived): $c = \dfrac{1}{\sqrt{\varepsilon_0 \mu_0}}$

Boundary statement: AP Physics C: Electricity & Magnetism does not expect students to mathematically derive the speed of light in free space from Maxwell's equations. This relationship is included above solely as an indication of the further applications, implications, and connections to physical phenomena that students may study in more advanced physics courses.

Learning Objective 13.3.A: Describe the force exerted on a conductor due to the interaction between an external magnetic field and an induced current within that conductor.

• 13.3.A.1 When an induced current is created in a conductive loop, the already-present magnetic field will exert a magnetic force on the moving charge carriers within the loop.
  • Equation: $\vec{F}_B = \displaystyle\int I\left(d\vec{\ell} \times \vec{B}\right)$
• 13.3.A.2 When current is induced in a conducting loop, magnetic forces are only exerted on the segments of the loop that are within the external magnetic field. These magnetic forces may cause translational or rotational acceleration.
• 13.3.A.3 The force on a conducting loop is proportional to the induced current in the loop, which depends on the rate of change of magnetic flux, the resistance of the loop, and the velocity of the loop.
• 13.3.A.4 Newton's second law can be applied to a conducting loop moving in a magnetic field as it experiences an induced emf.

Learning Objective 13.4.A: Describe the physical and electrical properties of an inductor.

• 13.4.A.1 Inductance is the tendency of a conductor to oppose a change in electrical current.
  • 13.4.A.1.i Inductance of a conductor depends on the physical properties of the conductor. Straight wires are typically modeled as having zero inductance.
  • 13.4.A.1.ii An inductor, such as a solenoid, is a circuit element that has significant inductance.
  • 13.4.A.1.iii The inductance of a solenoid is dependent on the total number of turns, the length of the solenoid, the cross-sectional area of the solenoid, and magnetic permeability of the solenoid's core.
    • Equation: $L_{\text{sol}} = \dfrac{\mu_{\text{core}} N^2 A}{\ell}$
• 13.4.A.2 Inductors store energy in the magnetic field that is generated by current in the inductor.
  • Equation: $U_L = \dfrac{1}{2} L I^2$
  • 13.4.A.2.i The energy stored in the magnetic field generated by an inductor in which current is flowing can be dissipated through a resistor or used to charge a capacitor.
  • 13.4.A.2.ii The transfer of energy generated in an inductor to other forms of energy obeys conservation laws.
• 13.4.A.3 By applying Faraday's law to an inductor and using the definition of inductance, induced emf can be related to inductance and the rate of change of current.
  • Equation: $\mathcal{E}_i = -L\dfrac{dI}{dt}$

Learning Objective 13.5.A: Describe the physical and electrical properties of a circuit containing a combination of resistors and a single inductor.

• 13.5.A.1 A resistor will dissipate energy that was stored in an inductor as the current changes.
• 13.5.A.2 Kirchhoff's loop rule can be applied to a series LR circuit with a battery of emf $\mathcal{E}$, resulting in a differential equation that describes the current in the loop.
  • Equation (derived): $\mathcal{E} = IR + L\dfrac{dI}{dt}$
• 13.5.A.3 The time constant is a significant feature of the behavior of an LR circuit.
  • 13.5.A.3.i The time constant of a circuit is a measure of how quickly an LR circuit will reach a steady state and is described with the equation $\tau = \dfrac{L}{R_{\text{eq}}}$ .
  • 13.5.A.3.ii The time constant represents the time an LR circuit would take to reach a steady state if the system continued to change at the initial rate of change.
  • 13.5.A.3.iii For an inductor that has zero initial current, the time constant represents the time required for the current in the inductor to reach approximately 63 percent of its final asymptotic value.
  • 13.5.A.3.iv For an inductor with an initial current, the time constant represents the time required for the current in the inductor to reach approximately 37 percent of its initial value.
• 13.5.A.4 The electric properties of inductors change during the time interval in which the current in the inductor changes, but will exhibit steady state behavior after a long time interval.
  • 13.5.A.4.i When a switch is initially closed or opened in a circuit containing an inductor, the induced emf will be equal in magnitude and opposite in direction to the applied potential difference across the branch containing the inductor.
  • 13.5.A.4.ii The potential difference across an inductor, the current in the inductor, and the energy stored in the inductor are exponential with respect to time and have asymptotes that are determined by the initial conditions of the circuit.
  • 13.5.A.4.iii After a time much greater than the time constant of the circuit, an inductor will behave as a conducting wire with zero resistance.

Learning Objective 13.6.A: Describe the physical and electrical properties of a circuit containing a combination of capacitors and a single inductor.

• 13.6.A.1 In circuits containing only a charged capacitor and an inductor (LC circuits), the maximum current in the inductor can be determined using conservation of energy within the circuit.
• 13.6.A.2 In LC circuits, the time dependence of the charge stored in the capacitor can be modeled as simple harmonic motion.
  • Equation (derived): $\dfrac{d^2 q}{dt^2} = -\dfrac{1}{LC} q$
• 13.6.A.3 The angular frequency of an oscillating LC circuit can be derived from the differential equation that describes an LC circuit.
  • Equation (derived): $\omega = \dfrac{1}{\sqrt{LC}}$