You want to find the magnetic field inside an infinite solenoid using Ampère's law. Which Amperian loop shape is most appropriate, and why?
AA circle concentric with the solenoid cross-section, because B is radial
BA rectangular loop with one side inside and one side outside the solenoid, because B is uniform inside and zero outside
CA circular loop outside the solenoid, because all the current is enclosed
DA sphere enclosing the solenoid, to capture the full flux
For a solenoid, B inside is uniform and parallel to the axis; B outside is approximately zero. A rectangular Amperian loop with one long side inside (where B · dl = BL) and one long side outside (where B · dl = 0) lets you write ∮ B · dl = BL = μ₀ n L I, giving B = μ₀nI. A circular loop does not exploit the geometry correctly because B is axial, not azimuthal.
Question 2 True / False
Ampère's law (∮ B · dl = μ₀I_enc) is only valid when the current distribution has a high degree of symmetry.
TTrue
FFalse
Answer: False
Ampère's law is always valid for steady currents — it is an exact law of magnetostatics, true for any current distribution and any closed loop. The symmetry requirement applies to practical calculation: the integral ∮ B · dl simplifies to BL or B(2πr) only when symmetry guarantees B is constant in magnitude and parallel to dl along the chosen loop. Without symmetry, the integral is too complex to evaluate analytically, so Biot-Savart is used instead.
Question 3 Short Answer
Explain the analogy between Ampère's law and Gauss's law, and what condition both laws require for practical use.
Think about your answer, then reveal below.
Model answer: Both laws relate a field integrated over a closed surface or loop to the source enclosed: Gauss's law relates the electric flux through a closed surface to the enclosed charge (∮ E · dA = Q_enc/ε₀), while Ampère's law relates the line integral of B around a closed loop to the enclosed current (∮ B · dl = μ₀I_enc). Both are always mathematically valid, but both simplify to usable algebraic equations only when symmetry makes the field constant and aligned with the integration path or surface.
The structural parallel is deep: Gauss's law uses a closed surface (Gaussian surface) and a volume integral of charge; Ampère's law uses a closed loop (Amperian loop) and a line integral of current. In both cases, the law itself does not depend on symmetry — it is an identity. Symmetry is required to pull the field outside the integral sign, reducing ∮ B · dl to B × (path length) or ∮ E · dA to E × (area).