1.4 Ampère's circuital law

Ampère's circuital law connects magnetic fields to the electric currents that create them. It's a fundamental principle in electromagnetism, analogous to Gauss's law for electric fields but dealing with magnetic fields and currents instead.

The law has various applications, from calculating magnetic fields around wires and solenoids to understanding electromagnetic waves. It's crucial for grasping how currents shape magnetic fields in different scenarios, from simple wires to complex electromagnetic systems.

Ampère's circuital law

• Fundamental law in classical electromagnetism relates magnetic fields to electric currents that produce them
• Analogous to Gauss's law for electric fields but deals with magnetic fields and currents instead of electric fields and charges
• Integral form states line integral of magnetic field around closed loop equals product of permeability of free space and current enclosed by loop

Magnetic fields of current distributions

Magnetic field of straight wire

Top images from around the web for Magnetic field of straight wire

• 

  12.3 Magnetic Force between Two Parallel Currents – University Physics Volume 2 View original

  Is this image relevant?

• 

  22.9 Magnetic Fields Produced by Currents: Ampere’s Law – College Physics View original

  Is this image relevant?

• 

  Magnetic Force between Two Parallel Conductors | Physics View original

  Is this image relevant?

• 

  12.3 Magnetic Force between Two Parallel Currents – University Physics Volume 2 View original

  Is this image relevant?

• 

  22.9 Magnetic Fields Produced by Currents: Ampere’s Law – College Physics View original

  Is this image relevant?

1 of 3

Top images from around the web for Magnetic field of straight wire

• 

  12.3 Magnetic Force between Two Parallel Currents – University Physics Volume 2 View original

  Is this image relevant?

• 

  22.9 Magnetic Fields Produced by Currents: Ampere’s Law – College Physics View original

  Is this image relevant?

• 

  Magnetic Force between Two Parallel Conductors | Physics View original

  Is this image relevant?

• 

  12.3 Magnetic Force between Two Parallel Currents – University Physics Volume 2 View original

  Is this image relevant?

• 

  22.9 Magnetic Fields Produced by Currents: Ampere’s Law – College Physics View original

  Is this image relevant?

1 of 3

• Magnetic field lines form concentric circles around current-carrying wire
• Direction determined by right-hand rule (thumb points in current direction, fingers curl in field direction)
• Magnitude inversely proportional to distance from wire
  • Decreases as $1/r$ where $r$ is radial distance
• Example: Magnetic field around long, straight power line

Magnetic field of circular loop

• Magnetic field lines resemble dipole pattern (similar to bar magnet)
• Field strength strongest at center of loop and along axis
• Direction determined by right-hand rule (fingers curl in current direction, thumb points in field direction)
• Magnitude proportional to current and inversely proportional to loop radius
• Example: Magnetic field produced by circular current loop in Helmholtz coil

Magnetic field of solenoid

• Consists of tightly wound helical coil of wire
• Produces nearly uniform magnetic field inside solenoid (similar to bar magnet)
• Field lines parallel to solenoid axis inside and loop around outside
• Magnitude proportional to current, number of turns per unit length, and permeability of core material
  • $B = \mu_0 n I$ where $n$ is turns per unit length and $I$ is current
• Example: Magnetic field inside MRI machine solenoid

Mathematical formulation

Line integral of magnetic field

• Ampère's law relates closed line integral of magnetic field to enclosed current
• Mathematically: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$
  • $\vec{B}$ is magnetic field vector
  • $d\vec{l}$ is infinitesimal line element along path
  • $I_{enc}$ is net current passing through surface bounded by closed path
• Choosing appropriate integration path is crucial for simplifying calculations

Current enclosed by path

• Net current passing through any surface bounded by chosen closed path
• Includes all currents (both free and bound) that penetrate surface
• Current direction matters: positive if flows through surface in direction of path's normal vector
• Mathematically: $I_{enc} = \int \vec{J} \cdot d\vec{A}$
  • $\vec{J}$ is current density vector
  • $d\vec{A}$ is infinitesimal area element of surface

Differential form of Ampère's law

• Obtained by shrinking integration path to infinitesimal size and applying Stokes' theorem
• Relates curl of magnetic field to current density: $\nabla \times \vec{B} = \mu_0 \vec{J}$
  • $\nabla \times$ is curl operator that measures field's rotation
• Useful for calculating magnetic fields in problems with high symmetry
• Example: Deriving magnetic field inside infinite solenoid using differential form

Applications of Ampère's law

Calculating magnetic fields

• Ampère's law provides straightforward method for calculating magnetic fields in situations with high symmetry
• Exploit symmetry to choose integration path along which magnetic field is either constant or zero
• Simplifies line integral and allows solving for unknown field strength
• Example: Calculating magnetic field inside toroidal solenoid

Symmetry considerations

• Ampère's law most useful when problem exhibits certain symmetries
  • Infinite straight wires, infinite sheets of current, infinite solenoids, toroids
• Symmetry allows choosing integration path that simplifies calculations
  • Circular loops for infinite wires, rectangular loops for infinite sheets, etc.
• Without sufficient symmetry, Biot-Savart law may be more appropriate

Limitations and assumptions

• Ampère's law assumes steady-state currents (no time-varying electric fields)
  • Fails for rapidly changing fields like in electromagnetic waves
• Assumes all currents are enclosed by integration path
  • Fails if path misses or bisects currents
• Requires integration path to be closed loop
  • Open paths or non-loop paths invalid
• Example: Attempting to use Ampère's law with an open integration path

Displacement current

Ampère-Maxwell law

• Generalization of Ampère's law to include time-varying electric fields
• Adds displacement current term to account for changing electric flux: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \varepsilon_0 \frac{d}{dt} \int \vec{E} \cdot d\vec{A}$
  • $\vec{E}$ is electric field vector
  • $\varepsilon_0$ is permittivity of free space
• Necessary for explaining electromagnetic wave propagation

Consistency with Faraday's law

• Ampère-Maxwell law resolves inconsistency between Ampère's law and Faraday's law
• Faraday's law implies changing magnetic fields produce electric fields
• But Ampère's law (without displacement current) fails to predict magnetic fields from changing electric fields
• Displacement current term restores symmetry between electric and magnetic fields

Electromagnetic waves

• Ampère-Maxwell law and Faraday's law together lead to wave equations for electric and magnetic fields
• Predict existence and propagation of coupled electric and magnetic field oscillations (electromagnetic waves)
• Waves travel at speed of light $c = 1/\sqrt{\mu_0 \varepsilon_0}$
• Example: Light, radio waves, X-rays are all electromagnetic waves

Comparison with Biot-Savart law

Similarities and differences

• Both Ampère's law and Biot-Savart law relate magnetic fields to electric currents
• Biot-Savart law is more general
  • Applies to any current distribution (not just symmetric ones)
  • Gives magnetic field at any point in space (not just on closed paths)
• Ampère's law is more specialized
  • Restricted to highly symmetric situations
  • Gives average magnetic field on closed paths

Advantages of Ampère's law

• Often provides quicker, simpler way to calculate magnetic fields than Biot-Savart law
• Exploits symmetry to reduce complicated integrals to algebraic expressions
• Gives direct physical insight into relationship between currents and fields
• Example: Deriving magnetic field of infinite solenoid using Ampère's law vs. Biot-Savart law

Limitations of Ampère's law

• Not applicable to problems lacking sufficient symmetry
  • Biot-Savart law must be used instead
• Fails for time-varying fields
  • Ampère-Maxwell law must be used instead
• Only gives average magnetic field on closed paths
  • Biot-Savart law needed for field at specific points
• Example: Calculating magnetic field of finite solenoid requires Biot-Savart law