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Ampere's Law

Source lecture(s): SC134 Lec10

Intuition

If \(\vec{B}\) has enough symmetry, a clever loop turns the complicated integral into \(B\cdot\)circumference.

Formal Definition

\[\oint \vec{B}\cdot d\vec{l} = \mu_0 i_{\text{enc}}\]

Mathematical Formulation

Infinite straight wire: \(B(2\pi r) = \mu_0 i\) gives \(B = \mu_0 i/(2\pi r)\).

Derivation

Direct from experimental measurement of field around long straight conductors; for solenoids assume uniform interior field.

Worked Example

Inside a long solenoid: \(B = \mu_0 n i\) where \(n=N/L\).

Common Mistakes

  • Using Ampere with no symmetry.
  • Forgetting enclosed current direction.

Quiz

Q1. Can Ampere's law find \(B\) for an arbitrary wire shape?

Answer

Not practically—symmetry required.