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Ampère II, Faraday, and Lenz

Ampère's Law

• Work with Ampère's law to compute magnetic fields in symmetrically structured situations.

• Formula:

  [ \oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} ]

• Surface S with boundary C; I_enc is the sum of the currents crossing S.

• The right-hand rule (thumb indicates current direction) is crucial for understanding currents.

The Integral of the Magnetic Field

• Ampère's law emphasizes that it is true generally, but specific integrals can only provide field deduction in symmetric situations.

• Field distribution from symmetries simplifies calculations, similar to Gauss's law.

Magnetic Field around a Current-Carrying Wire

• When evaluating a wire carrying a current I, consider symmetry:

  • The magnetic field is invariant under translation along the wire.

  • Current direction influences magnetic field direction:

    • Field configurations can either be circular (around the wire) or radial (ruled out).

Application of Ampère’s Law to a Wire

• Consider a wire with a circular contour:

  • Magnetic field, B, can be evaluated around this contour using symmetry arguments.

  • For a wire of zero radius:

    • Field strength determined as ( B(r) = \frac{\mu_0 I}{2\pi r} )

• For a wire of non-zero radius:

  • Current density j in the wire leads to a different magnetic field inside and outside the wire:

    • ( B(r) = \frac{\mu_0 I}{2\pi R}, \text{ for } r > R )

• Inside the wire: magnetic field increases linearly with radius r.

Solenoids

• A solenoid carrying current produces a magnetic field similar to a bar magnet.

• Ampère's law can be applied to find fields inside and outside the solenoid:

  • Inside: ( \mathbf{B}_{in} = \mu_0 n I )

  • Outside: ( \mathbf{B}_{out} = 0 )

Toroidal Solenoids

• Toroidal solenoids avoid end complications.

• Magnetic field determined using Ampère’s law:

  • Strength depends on the total number of turns adjusted per unit length.

  • Field exists only within the donut-shaped structure whenever radius a < r < b.

Faraday's Law of Induction

• Changes in magnetic fields create electric fields (E). Emf defined through the line integral of the electric field:

  • ( E = -\frac{d\Phi_B}{dt} )

• Lenz's law: induced current opposes change in magnetic flux.

Moving Conductors in Magnetic Fields

• Consider cases where a loop is dragged in a magnetic field:

  • A moving conductor cuts magnetic field lines, inducing an emf that can power a circuit (bulb).

  • Emf from the Lorentz force can be written as ( E = vB \

• Work done moving the loop translates energy to electrical energy in the bulb.

Relativity and Change in Fields

• Principles of relativity mean electrodynamics equations apply in every frame of reference, necessitating induction phenomena related to changing magnetic and electric fields.

• The interplay and anticipation of electric fields resulting from changing magnetic fields extend energy dynamics in circuits.

Key Equations

• Ampère's Law: ( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} )

• Magnetic Field due to an infinite wire: ( B = \frac{\mu_0 I}{2\pi r} )

• Inside solenoid: ( B_{in} = \mu_0 n I )

• Faraday's Law: ( E = -\frac{d\Phi_B}{dt} )