AP Physics C: E&M:: Electromagnetism

What is FARADAY'S LAW OF INDUCTION, EMF=-dPhi_B/dt (the induced EMF around a loop equals the NEGATIVE derivative of magnetic flux with respect to time), and how does it show that an EMF can be induced by ANY of THREE distinct means -- CHANGING B, CHANGING the loop's AREA, or CHANGING the ANGLE between B and the loop?

- **A)** This concept has no actual relationship between the derivative of magnetic flux and the different physical ways an EMF can be induced
- **B)** This derivation requires the loop's AREA to remain PERMANENTLY constant, contradicting the actual GENERAL validity of Faraday's law even when area changes over time
- **C)** Faraday's law applies ONLY when the magnetic field B itself changes with time, contradicting its actual GENERAL applicability to changes in B, changes in area, OR changes in orientation angle
- **D)** Since Phi_B=∫B·dA=B*A*cos(theta) (for a uniform field), the derivative dPhi_B/dt can be NONZERO if EITHER B changes with time, OR A changes with time (e.g., a loop expanding or a rod sliding to change enclosed area), OR theta changes with time (e.g., a rotating loop) -- Faraday's law UNIFIES all three of these seemingly different induction scenarios under ONE single derivative-based equation

What is LENZ'S LAW (the NEGATIVE sign in Faraday's law, EMF=-dPhi_B/dt), and how does it GUARANTEE that an induced current ALWAYS flows in a direction that OPPOSES the CHANGE in flux that created it, consistent with CONSERVATION OF ENERGY?

- **A)** Lenz's law applies ONLY to induced EMFs caused by a CHANGING magnetic field (not by a changing area or angle), contradicting its actual GENERAL applicability to ANY source of changing flux
- **B)** An induced current REINFORCES (rather than opposes) the change in flux that created it, contradicting the actual opposing behavior guaranteed by Lenz's law
- **C)** Lenz's law has no actual relationship to conservation of energy or to the negative sign in Faraday's law
- **D)** The negative sign in Faraday's law means the induced EMF (and resulting current) always acts to OPPOSE the CHANGE in magnetic flux (not the flux itself) -- if this were NOT the case (if the induced current REINFORCED the changing flux instead), the system would experience a RUNAWAY, ENERGY-CREATING feedback loop, violating conservation of energy; Lenz's law therefore ensures energy conservation is respected in EVERY induction scenario

How is MOTIONAL EMF, EMF=B*L*v (for a conducting rod of length L sliding with speed v perpendicular to a uniform field B), DERIVED using the MAGNETIC FORCE, F=qv×B, on the INDIVIDUAL charge carriers WITHIN the moving rod, and how does this MICROSCOPIC derivation CONFIRM the MACROSCOPIC Faraday's-law result?

- **A)** As the rod moves, its charge carriers experience a magnetic force, F=qvB, that pushes them along the rod's length, creating a charge SEPARATION (and therefore an electric field) WITHIN the rod; this process continues until the induced electric force exactly BALANCES the magnetic force, at which point the rod behaves like a BATTERY with EMF=B*L*v -- and this MICROSCOPIC result EXACTLY MATCHES what Faraday's law, EMF=-dPhi_B/dt, predicts for a rod sweeping out changing enclosed area at rate L*v
- **B)** The microscopic (charge-carrier-force-based) and macroscopic (Faraday's-law-based) derivations of motional EMF produce DIFFERENT, inconsistent results, contradicting their actual full mathematical consistency
- **C)** This derivation produces an EMF that DECREASES as the rod's speed v increases, contradicting the actual DIRECT proportionality between EMF and v shown in this formula
- **D)** Motional EMF has no actual relationship to the magnetic force on individual moving charge carriers within a conducting rod

How does FARADAY'S LAW, when written in its FULL INTEGRAL FORM, ∮E·dl=-dPhi_B/dt, reveal that a CHANGING magnetic field INDUCES an electric field that is NO LONGER CONSERVATIVE (∮E·dl is NONZERO), in SHARP CONTRAST to the ELECTROSTATIC field studied throughout the Electrostatics unit?

- **A)** This concept has no actual relationship between Faraday's law and the conservative-versus-non-conservative nature of different types of electric fields
- **B)** This non-conservative induced field applies ONLY within the wire of a circuit loop, contradicting its actual GENERAL existence throughout ALL of space wherever a changing magnetic flux is present, wire or no wire
- **C)** The induced electric field from a changing magnetic field is ALSO conservative (∮E·dl=0), identical to the electrostatic field, contradicting the actual NON-conservative nature revealed by Faraday's law in integral form
- **D)** While the ELECTROSTATIC field (produced by STATIC charges) satisfies ∮E·dl=0 (a CONSERVATIVE field, as established in the Electrostatics unit), the INDUCED electric field produced by a CHANGING magnetic field satisfies ∮E·dl=-dPhi_B/dt (GENERALLY NONZERO) -- this induced field is FUNDAMENTALLY NON-CONSERVATIVE, meaning moving a charge around a CLOSED loop in this field CAN do NET work, a genuinely NEW type of electric field not encountered anywhere in the Electrostatics unit

What is SELF-INDUCTANCE, and how is the SELF-INDUCED EMF in a coil, EMF=-L*(dI/dt) (proportional to the RATE OF CHANGE of the coil's OWN current, rather than an external flux), DERIVED by applying Faraday's law to the coil's OWN magnetic flux, which is PROPORTIONAL to its OWN current?

- **A)** Since a coil's OWN magnetic flux is PROPORTIONAL to the current flowing through it, Phi_B=L*I (defining L, the SELF-INDUCTANCE, as this proportionality constant), applying Faraday's law, EMF=-dPhi_B/dt=-d(L*I)/dt=-L*(dI/dt) (since L is a CONSTANT, geometry-dependent property), shows that a CHANGING current in a coil induces an EMF that OPPOSES the CHANGE in its OWN current -- a phenomenon called self-inductance
- **B)** This self-induced EMF REINFORCES (rather than opposes) changes in the coil's own current, contradicting the actual opposing behavior required by Lenz's law
- **C)** This derivation requires the self-inductance L to DEPEND on the current I flowing through the coil, contradicting the actual CONSTANT, geometry-only dependence of L (analogous to capacitance's geometry-only dependence)
- **D)** Self-inductance has no actual relationship to applying Faraday's law to a coil's own, current-proportional magnetic flux

How is the SELF-INDUCTANCE of an IDEAL SOLENOID, L=mu_0*n^2*A*l (n turns per unit length, cross-sectional area A, length l), DERIVED by COMBINING the solenoid's magnetic field formula (from Ampere's law, in the Magnetic Fields unit) with the DEFINITION of self-inductance, L=N*Phi_B/I?

- **A)** Using B=mu_0*n*I (from the earlier Magnetic Fields unit) and computing the flux through ALL N=n*l turns of the solenoid, N*Phi_B=(n*l)*(B*A)=(n*l)*(mu_0*n*I)*A=mu_0*n^2*I*A*l; dividing by I (per the definition, L=N*Phi_B/I) gives L=mu_0*n^2*A*l -- directly built by COMBINING this unit's self-inductance definition with the solenoid field formula derived earlier in the Magnetic Fields unit
- **B)** The self-inductance of a solenoid has no actual relationship to the solenoid's magnetic field formula derived earlier in the Magnetic Fields unit
- **C)** This derivation produces a self-inductance that is INDEPENDENT of the solenoid's cross-sectional area A, contradicting the actual DIRECT proportionality between L and A shown in this formula
- **D)** This derivation shows self-inductance L DECREASING as the number of turns per length, n, INCREASES, contradicting the actual proportionality to n-SQUARED (a stronger-than-linear increase) shown in this formula

How is the ENERGY STORED in an INDUCTOR carrying current I, U=(1/2)*L*I^2, DERIVED using an INTEGRAL, W=∫[0 to I] L*I'*dI'=∫ P dt, representing the CUMULATIVE work needed to ESTABLISH the current against the coil's OWN self-induced opposing EMF?

- **A)** This derivation produces U=L*I^2 (missing the factor of 1/2), contradicting the correct energy formula, U=(1/2)*L*I^2
- **B)** The energy stored in an inductor has no actual relationship to integrating the power required to establish current against the coil's self-induced EMF
- **C)** The instantaneous power required to overcome the self-induced EMF is P=I*EMF_applied=I*(L*(dI/dt)) (working against the coil's own opposition); integrating this power over the time needed to bring the current from 0 to its final value I gives W=∫[0 to I] L*I'*dI'=(1/2)*L*I^2 -- a derivation that directly PARALLELS the earlier capacitor-charging-integral derivation of U=(1/2)*C*V^2 from the Conductors, Capacitors, and Dielectrics unit
- **D)** This integration requires the current to remain CONSTANT throughout the process, contradicting the actual CHANGING current (from 0 to its final value) that necessitates the integral in the first place

How is the DIFFERENTIAL EQUATION governing an RL CIRCUIT (a resistor R and inductor L in series with a battery of EMF, epsilon), L*(dI/dt)+I*R=epsilon, DERIVED by applying KIRCHHOFF'S VOLTAGE LAW around the single loop, and how does this EQUATION DIRECTLY PARALLEL the RC-circuit differential equation from the Electric Circuits unit?

- **A)** Applying Kirchhoff's voltage law around the RL loop -- EMF minus the self-induced EMF across the inductor (L*dI/dt) minus the voltage drop across the resistor (I*R) equals zero -- gives L*(dI/dt)+I*R=epsilon, which is MATHEMATICALLY ANALOGOUS to the RC-circuit differential equation, R*(dQ/dt)+Q/C=epsilon, with CURRENT I playing the role that CHARGE Q played for the RC circuit
- **B)** The RL-circuit differential equation has no actual relationship to the analogous RC-circuit differential equation derived in the earlier Electric Circuits unit
- **C)** This derivation requires treating the inductor's self-induced EMF as CONSTANT (rather than proportional to dI/dt), contradicting the actual TIME-VARYING self-induced EMF that depends on the RATE OF CHANGE of current
- **D)** This differential equation involves the SECOND derivative of current, d^2I/dt^2 (rather than the first derivative, dI/dt), contradicting the actual FIRST-order differential equation that correctly describes RL circuit behavior