Electromagnetic Induction

• 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: $\Phi_B = \displaystyle\int \vec{B} \cdot d\vec{A}$

• 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.

• 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.

• 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: $\mathcal{E}_i = -L\dfrac{dI}{dt}$

• 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.

• 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): $\omega = \dfrac{1}{\sqrt{LC}}$