AP Physics C: E&M:: Magnetic Fields

How does the MAGNETIC FORCE on a moving charge, F=q*v×B (a VECTOR CROSS PRODUCT), DIFFER FUNDAMENTALLY from the electric force, F=qE, in terms of the DIRECTION of the resulting force RELATIVE to the charge's velocity?

- **A)** The magnetic force CAN do work on a moving charge, identical to the electric force, contradicting the actual zero-work property that follows from the perpendicularity of F=qv×B to v
- **B)** The magnetic force, F=qv×B, is ALWAYS PERPENDICULAR to BOTH the charge's velocity v and the magnetic field B (a direct consequence of the cross product), meaning the magnetic force can NEVER do WORK on a moving charge (since work requires a force component ALONG the direction of motion) -- in sharp contrast to the electric force, which CAN act along (and do work along) the direction of motion
- **C)** This concept has no actual relationship between the cross-product definition of magnetic force and its direction relative to a charge's velocity
- **D)** The magnetic force is PARALLEL to the charge's velocity, identical in direction-behavior to the electric force, contradicting the actual perpendicular nature of the cross-product-based magnetic force

How is the MAGNETIC FORCE on a CURRENT-CARRYING WIRE, F=I*L×B (for a STRAIGHT wire segment of length L in a UNIFORM field), DERIVED by INTEGRATING the force on INDIVIDUAL moving charge carriers, dF=dq*v×B, using dq=I*dt and v*dt=dl (the infinitesimal DISPLACEMENT of a charge carrier)?

- **A)** The magnetic force on a current-carrying wire has no actual relationship to integrating the force on individual moving charge carriers within the wire
- **B)** This derivation requires the magnetic field to be NON-uniform along the wire, contradicting the actual UNIFORM-field assumption needed to pull B outside the integral and obtain the simple F=I*L×B formula
- **C)** Substituting dq=I*dt into dF=dq*v×B gives dF=I*dt*v×B=I*(v*dt)×B=I*dl×B (since v*dt=dl, an infinitesimal displacement along the wire); INTEGRATING this over the ENTIRE wire (from one end to the other) gives F=I*L×B for a straight wire in a UNIFORM field -- directly built from summing the magnetic force on EVERY individual charge carrier moving through the wire
- **D)** This integration produces a force INDEPENDENT of the current I, contradicting the actual direct proportionality between F and I in this formula

What is the BIOT-SAVART LAW, dB=(mu_0/(4*pi))*I*dl×r_hat/r^2 (giving the INFINITESIMAL magnetic field contribution from a SMALL current element I*dl), and how does it serve as the MAGNETIC analog of the ELECTROSTATIC 'dE=k*dq/r^2' formula used throughout the Electrostatics unit?

- **A)** This law can ONLY be applied to STRAIGHT current-carrying wires, contradicting its actual GENERAL applicability to current elements of ANY shape or orientation, integrated over the full current path
- **B)** Just as the electrostatics unit built up electric fields from continuous charge distributions by integrating dE=k*dq/r^2 over infinitesimal charge elements dq, the Biot-Savart law allows magnetic fields to be built up from current-carrying wires by integrating dB=(mu_0/4pi)*I*dl×r_hat/r^2 over infinitesimal CURRENT elements I*dl -- the SAME 'integrate infinitesimal contributions' strategy, now applied to MAGNETIC (rather than electric) sources
- **C)** The Biot-Savart law has no actual conceptual or mathematical relationship to the electrostatic dE=k*dq/r^2 formula used in the Electrostatics unit
- **D)** The Biot-Savart law's infinitesimal field contribution, dB, points in the SAME direction as the current element I*dl itself, contradicting its actual PERPENDICULAR direction (determined by the cross product dl×r_hat)

How is the Biot-Savart law used to DERIVE the magnetic field AT THE CENTER of a CIRCULAR current loop of radius R carrying current I, by INTEGRATING dB over the ENTIRE loop, exploiting the fact that EVERY current element is the SAME distance R from the center and contributes a field in the SAME direction?

- **A)** This integration produces a field INDEPENDENT of the loop's radius R, contradicting the actual inverse dependence on R shown in the final formula, B_center=mu_0*I/(2R)
- **B)** This derivation requires the current elements to contribute fields in DIFFERENT, CANCELING directions, contradicting the actual constructive addition (SAME direction) that produces a nonzero net field at the center
- **C)** Since EVERY infinitesimal current element I*dl on the loop is PERPENDICULAR to r_hat (which points radially inward to the center) and is the SAME distance R away, each contributes a field dB=(mu_0/4pi)*I*dl/R^2 in the SAME direction (perpendicular to the loop's plane); integrating over the ENTIRE loop (total length 2*pi*R) gives B_center=(mu_0*I)/(2*R) -- a clean result arising from the loop's high degree of symmetry
- **D)** The magnetic field at the center of a circular current loop has no actual relationship to integrating the Biot-Savart law over the loop's current elements

How is the Biot-Savart law used to DERIVE the magnetic field from an INFINITE STRAIGHT current-carrying wire, B=(mu_0*I)/(2*pi*r) (at perpendicular distance r from the wire), by INTEGRATING dB over the ENTIRE length of the wire?

- **A)** The magnetic field from an infinite straight wire has no actual relationship to integrating the Biot-Savart law over the wire's length
- **B)** This derivation requires the wire to carry a CONSTANTLY CHANGING current, contradicting the actual STEADY (DC) current assumption used in this standard Biot-Savart integration
- **C)** This integration produces a field that falls off as 1/r^2 (identical to a point charge's electric field), contradicting the actual 1/r falloff characteristic of an infinite current-carrying wire
- **D)** Integrating the Biot-Savart law over an infinitely long straight wire (accounting for the CHANGING angle between dl and r_hat at different points along the wire) produces B=(mu_0*I)/(2*pi*r) -- a field that FALLS OFF as 1/r, directly PARALLELING the analogous 1/r falloff derived earlier (via Gauss's law) for the ELECTRIC field of an infinite line of charge

What is AMPERE'S LAW, ∮B·dl=mu_0*I_enclosed (the line integral of B around ANY CLOSED LOOP equals mu_0 times the ENCLOSED current), and how does it provide a POWERFUL SHORTCUT for finding magnetic fields in HIGHLY SYMMETRIC situations, DIRECTLY PARALLELING the role of GAUSS'S LAW for electric fields?

- **A)** Ampere's law has no actual relationship to symmetry or to simplifying the calculation of magnetic fields for symmetric current distributions
- **B)** For current distributions with SUFFICIENT symmetry (e.g., a straight wire, a solenoid), a well-chosen 'Amperian loop' allows B to be pulled OUTSIDE the line integral (since B is CONSTANT in magnitude and PARALLEL to dl everywhere on the loop), reducing ∮B·dl to simply B times the loop's circumference -- turning Ampere's law into an ALGEBRAIC equation for B, EXACTLY PARALLELING how Gauss's law simplifies electric field calculations for symmetric charge distributions
- **C)** Ampere's law requires performing a MORE DIFFICULT integration than the direct Biot-Savart approach, contradicting its actual role as a SIMPLIFYING shortcut for symmetric current distributions
- **D)** Ampere's law can ONLY be used to find the TOTAL enclosed current, NEVER to find the magnetic field itself, contradicting its actual widespread use as a field-finding shortcut for symmetric situations

How is Ampere's law used to RE-DERIVE the magnetic field from an INFINITE STRAIGHT wire, B=(mu_0*I)/(2*pi*r), using a CIRCULAR Amperian loop of radius r CENTERED on the wire, and how does this CONFIRM the earlier Biot-Savart-based result?

- **A)** This derivation requires the Amperian loop to be a SQUARE (rather than a circle), contradicting the actual circular loop choice that exploits the wire's cylindrical symmetry
- **B)** Ampere's law produces a DIFFERENT result from the Biot-Savart-based derivation for an infinite straight wire, contradicting the actual CONSISTENT result obtained by both methods
- **C)** This application of Ampere's law requires KNOWING the magnetic field's value IN ADVANCE, contradicting its actual use as a method for FINDING the unknown field from the enclosed current alone
- **D)** By symmetry, B is CONSTANT in magnitude and TANGENT to (parallel to dl along) the circular Amperian loop everywhere, so ∮B·dl=B*(2*pi*r)=mu_0*I_enclosed=mu_0*I, giving B=(mu_0*I)/(2*pi*r) -- EXACTLY matching the result derived earlier using the more laborious Biot-Savart integration, confirming Ampere's law as a correct, much SIMPLER alternative derivation method for this symmetric situation

How is Ampere's law used to DERIVE the magnetic field INSIDE an IDEAL SOLENOID (a tightly wound coil with n turns per unit length carrying current I), B=mu_0*n*I, using a RECTANGULAR Amperian loop with one side INSIDE the solenoid and one side FAR OUTSIDE it?

- **A)** Choosing a rectangular Amperian loop with one side of length L INSIDE the solenoid (where B is assumed uniform) and one side FAR OUTSIDE (where B≈0 for an IDEAL solenoid), and noting that the loop's other two sides contribute ZERO to the line integral (since B is PERPENDICULAR to them), gives ∮B·dl=B*L=mu_0*(n*L)*I (since n*L turns are enclosed, each carrying current I), so B=mu_0*n*I -- a field that is UNIFORM and INDEPENDENT of position within an ideal solenoid's interior
- **B)** This derivation requires the field OUTSIDE the solenoid to be EQUAL in magnitude to the field INSIDE, contradicting the actual near-zero external field assumed for an IDEAL solenoid
- **C)** This integration produces a field that DECREASES with distance from the solenoid's central axis, contradicting the actual UNIFORM field characteristic of an ideal solenoid's interior
- **D)** The magnetic field inside a solenoid has no actual relationship to applying Ampere's law with a rectangular Amperian loop