Ampere’s Law in Magnetism

Introduction to Ampere’s Law

• Ampere’s law is analogous to Gauss’ law in magnetism, relating the magnetic field (B) to current (I).
• Useful for certain current symmetries, making it easier than the Biot-Savart law.

Ampere's Law Statement

• Ampere's law relates the magnetic field to the total current enclosed by a curve:
  • $\oint B \cdot ds = \mu<em>0 I</em>{\text{enclosed}}$
  • The integral is over a closed path (amperian loop).
• ds represents a small displacement along the amperian loop.

Conditions for Application

• Ampere’s law is easiest to apply under specific geometries:-
  1. Long, straight wire
  2. Infinite sheet of current
  3. Infinite solenoids
  4. Toroids (loops of wire)

Example Application: Long, Straight Wire

• Magnetic field around a long straight wire:
  • Magnetic field lines are circular and centered on the wire.
  • For a circular amperian loop of radius r:
  • $\oint B \cdot ds = B(2\pi r)$
  • Since the field direction is tangent, $\theta = 0$, thus:
  • $\oint B \cdot ds = B \cdot ds$
  • Therefore,
  • $B<em>{\text{wire}}(2\pi r) = \mu</em>0 I$
  • Simplifying gives:
  • $B<em>{\text{wire}} = \frac{\mu</em>0 I}{2\pi r}$

Extra Practice

• Engage in Physlet activities for better understanding of Ampere’s law:
  1. Access Physlets from desktop.
  2. Navigate to ‘Electromagnetism’ > ‘Ch. 28: Ampere’s Law’.
  3. Complete illustrations and problem sets.

Displacement Currents and the Maxwell-Ampere Law

• Ampere's law is not limited to flat surfaces; it holds true even for non-flat surfaces.
• Consider a charging capacitor: no current flows between plates but electric flux changes.
• Displacement current defined as:
  • $I<em>d = \epsilon</em>0 \frac{d\Phi_E}{dt}$
  • Where: ( \epsilon0 ) is the permittivity of free space: $\epsilon</em>0 = \frac{1}{4\pi k} = 8.8542 \times 10^{-12} \text{C}^2/\text{(N m}^2)$
• Magnetic field dependence on the electric flux changing in time:
  • The electric field magnitude:
  • $E(t) = \frac{q(t)}{\epsilon_0 A}$
• The electric flux for a parallel plate capacitor:
  • $\Phi<em>E = \int E \cdot dA = EA = \frac{q}{\epsilon</em>0}$

Common Misunderstanding

• For a flat surface, the enclosed current equals I, giving a standard magnetic field.
• For a non-flat (mushroom cap) surface, it appears no current flows, suggesting zero magnetic field.
• However, Maxwell introduced displacement current to resolve this inconsistency.

Relationship to Maxwell-Ampere Law

• Maxwell-Ampere law corrects Ampere’s law:
  • $\oint B \cdot ds = \mu<em>0 I + \mu</em>0 \epsilon<em>0 \frac{d\Phi</em>E}{dt} = \mu<em>0 \left(I + I</em>d\right)$
• Demonstrates a changing electric field generates a magnetic field, underpinning electromagnetism.