Ampere’s Circuital Law

• Magnetic fields are produced by moving charges (currents).
• Calculating B using the Biot–Savart Law is mathematically lengthy, especially for extended conductors.
• Ampere's Circuital Law provides a simpler alternative method for finding B when the current distribution has symmetry (straight wire, solenoid, toroid).
• It is the magnetic analogue of Gauss's Law in electrostatics.

• Proposed by André-Marie Ampère (1775–1836), a French physicist and mathematician.
• Known as the "Newton of Electricity," he laid the foundation of electrodynamics.
• The SI unit of current, the ampere (A), is named in his honour.
• Later, James Clerk Maxwell corrected the law for time-varying fields by adding the displacement current term.

Statement

The line integral \[\oint\vec{B}\cdot d\vec{l}\] taken around any closed loop equals μ₀ times the net steady current passing through the loop.

Proof (for a long straight wire)

• Consider an infinitely long straight wire carrying current I.

• By Biot–Savart law, field at distance r:
  B = \[\frac{\mu_0I}{2\pi r}\]

• Choose a circular Amperian loop of radius r, concentric with the wire.

• By symmetry, B is constant in magnitude and tangential (parallel to \[d\vec l\]) everywhere:
  \[\oint\vec{B}\cdot d\vec{l}=B\oint dl\] = B(2πr)

• Substituting B:
  \[\oint\vec{B}\cdot d\vec{l}=\frac{\mu_0I}{2\pi r}(2\pi r)\] = μ₀​I

Conclusion

\[\oint\vec{B}\cdot d\vec{l}=\mu_0I\]
The result is independent of the loop's radius, confirming the law's validity.

• Curl the fingers of the right hand along the direction of integration (\[d\vec l\]).
• The thumb points in the direction of positive current.
• Current in the thumb's direction → positive (+); opposite direction → negative (–).
• I_enc​ = algebraic sum of all enclosed currents with proper signs.

Problem: A long straight wire of radius a carries a steady current I, uniformly distributed across its cross-section. Find B for (i) r < a and (ii) r > a.

Solution:

Case (i): Inside the wire (r < a)

• Current enclosed (uniform distribution):
  \[I_{enc}=I\frac{\pi r^2}{\pi a^2}=I\frac{r^2}{a^2}\]

• Applying the law: B(2πr) = μ₀I\[\frac {r^2}{a^2}\]
  B = \[\frac{\mu_0Ir}{2\pi a^2}\] ​​(B ∝ r)

Case (ii): Outside the wire (r > a)

• Ienc = I, so B(2πr) = μ₀I
  B = \[\frac {μ_0I}{2πr}\] (B ∝ \[\frac {1}{r}\])

• Electromagnets in cranes, doorbells, and relays use solenoid fields.
• Toroidal coils in transformers and inductors confine the magnetic field within the core (no leakage).
• MRI machines rely on strong, uniform solenoid-generated fields.
• Power transmission cables — the field around them follows the straight-wire formula.