Unit 12: Magnetic Fields and Electromagnetism — AP Physics C E&M Practice

Magnetic Fields in AP Physics C E&M

Unit 12 introduces the second major field in electromagnetism: the magnetic field. Unlike the electric field, magnetic forces act only on moving charges and never do work on them. AP Physics C E&M treats magnetic field calculations using two powerful calculus-based laws — the Biot-Savart law and Ampere's law — analogous to Coulomb's law and Gauss's law for electric fields.

Magnetic Force on Charges and Current-Carrying Conductors

Lorentz Force

The magnetic force on a moving charge is F = qv × B. The cross product demands careful attention to direction using the right-hand rule. Because F is always perpendicular to v, the magnetic force does no work and changes only the direction of motion, producing circular or helical trajectories. AP FRQs test radius of curvature: r = mv/(|q|B).

Force on a Current-Carrying Conductor

For a current element, the force is dF = I dl × B. The total force on a straight conductor of length L in a uniform field is F = IL × B. AP-style problems extend this to curved conductors, requiring integration: F = I∫dl × B over the conductor's path.

Biot-Savart Law

The Biot-Savart law gives the differential magnetic field contribution from a current element: dB = (μ₀/4π) · (I dl × r̂)/r². To find the total field, this expression must be integrated over the entire current-carrying conductor. Key AP setups include:

• Magnetic field at the centre of a circular loop: By symmetry, all dB contributions point along the axis; the integral gives B = μ₀I/(2R).
• Magnetic field along the axis of a circular loop: Transverse components cancel; axial components are summed via integration, giving B = μ₀IR²/[2(R²+x²)^(3/2)].
• Magnetic field of a finite straight wire: Setting up dB, expressing dl and r in terms of an angle or position variable, and integrating between limits is a standard AP FRQ task.

Ampere's Law

Ampere's law states that the line integral of the magnetic field around any closed loop equals μ₀ times the enclosed current: ∮B·dl = μ₀I_enc. Like Gauss's law, its power is in simplifying calculations for symmetric current distributions.

Magnetic Field of an Infinite Straight Wire

Using a circular Amperian loop of radius r centred on the wire: B(2πr) = μ₀I, so B = μ₀I/(2πr). This derivation must be reproducible in full for AP FRQs.

Magnetic Field Inside a Solenoid

Using a rectangular Amperian loop enclosing n turns per unit length: B·L = μ₀nLI, giving B = μ₀nI inside (and zero outside for an ideal solenoid). The symmetry argument justifying why the field is uniform and axial inside must be stated explicitly.

Magnetic Field of a Toroid

For a toroid with N total turns and Amperian loop radius r: B(2πr) = μ₀NI, giving B = μ₀NI/(2πr) inside the toroid and zero outside. This 1/r dependence distinguishes the toroid from the uniform-field solenoid.

Key FRQ Skills for Unit 12

1. Biot-Savart integration: Set up dB, identify the symmetry argument to eliminate transverse components, and evaluate the integral with correct limits.
2. Amperian loop selection: Choose the loop shape and size consistent with the current distribution's symmetry; justify B being constant and parallel to dl on each segment.
3. Computing I_enc: For non-uniform current distributions (e.g., a wire with current density J(r)), integrate J over the enclosed cross-sectional area to find I_enc.
4. Right-hand rule application: Determine the direction of B from Biot-Savart or Ampere's law before computing the magnitude.

Common Errors in Unit 12

• Choosing an Amperian loop that does not share the symmetry of the current distribution
• Forgetting to account for current direction when applying the right-hand rule to determine B direction
• Confusing the Biot-Savart 1/r² dependence (point source) with the Ampere's law 1/r result (infinite wire)
• Using Ampere's law in situations lacking sufficient symmetry to justify constant B on the loop