Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Electromagnetism → Magnetism ↓


Ampere's Law


Ampere's law is a fundamental law of electromagnetism that relates magnetic fields to the electric currents that produce them. It is one of Maxwell's equations, which form the theoretical basis of classical electromagnetism, classical optics, and electrical circuits. Understanding Ampere's law is essential to understanding how electricity and magnetism are intrinsically linked.

Understanding magnetism and Ampere's law

Magnetism is a force that acts at a distance and is caused by magnetic fields. Magnetic objects have the ability to exert force on other magnetic materials without making contact. Magnetism and electricity are two aspects of the same electromagnetic force.

Ampere's law mathematically describes the magnetic field produced by an electric current. It states that the integrated magnetic field around a closed loop is proportional to the electric current passing through the loop. Specifically, this law is given as:

        ∮ B · dl = μ₀Iₑₙc

Where:

  • ∮ denotes closed line integral.
  • B is the magnetic field.
  • dl is a differential length vector along the closed path.
  • μ₀ is the permittivity of free space, which is a constant.
  • Iₑₙc is the current enclosed by the loop.

Ampere's law in simple words

Imagine an electric current flowing through a wire. The current flowing through the wire creates a magnetic field around it. Ampere's law helps you calculate the strength of that magnetic field. This law can be visualized by walking a path around the wire and integrating the magnetic field.

Visual example

I B

In this diagram, the gray circle shows the cross-section of a wire with a current I flowing out of the page. The blue circle shows the magnetic field lines B Notice how the field wraps around the wire. Ampere's law lets you calculate the value of the field on this circle.

Applying Ampere's law

To apply Ampere's law, follow these steps:

  1. Select a path: Select an imaginary loop (often a circle) around the stream.
  2. Integrate the magnetic field: Calculate the sum of the magnetic fields over this path.
  3. Calculate current: Determine the current passing through the loop.
  4. Use Ampere's law: Substitute the values into the equation to solve for the unknown.

Example: Long straight wire

A typical example is a long straight wire carrying a steady current I The magnetic field at a distance r from the wire can be found using Ampere's law.

Let's use Ampere's law to find the magnetic field:

  1. Choose a circular path of radius r centered on the wire.
  2. Because of symmetry, the magnetic field B remains constant along this path and is directed tangentially.
  3. The line integral becomes: ∮ B · dl = B ∮ dl = B(2πr) since ∮ dl is the circumference of the circle.
  4. Attached is Section I
  5. Substitute into Ampère's law: B(2πr) = μ₀I .
  6. Solve for B: B = μ₀I / (2πr) .

This tells us that the magnetic field decreases with distance, and its direction follows the right-hand rule. Bend the fingers in the direction of the current; the thumb points in the direction of the magnetic field.

Limitations and considerations

Ampere's law is powerful but has limitations. It is mainly used in cases with high symmetry, such as infinitely long wires or solenoids. In asymmetric cases, it becomes challenging to apply Ampere's law directly without additional techniques such as the Biot-Savart law or numerical methods.

Visualization of electric current and magnetic field

B I

This example shows a cross-section of a wire with current flowing horizontally. The blue line is the magnetic field line wrapped around the wire. The dashed line shows how the magnetic field lines emerge, encircle the wire, and align with Ampere's law.

Practical uses of Ampere's law

Ampere's law is used in engineering and physics applications such as:

  • Design of electromagnets: evaluation of magnetic fields in solenoids.
  • Electrical engineering: Ensuring proper current distribution in a circuit.
  • Magnetic field sensors: Calculating the strength of the field applied by multiple currents.

Example: Solenoid

Consider a solenoid, which is a coil of wire designed to produce a magnetic field when carrying a current. Using Ampere's law:

  1. Choose an Ampere loop, which is a rectangular shape inside the solenoid that is parallel to the length of the solenoid.
  2. Due to cancellation and symmetry the magnetic field inside is uniform and the magnetic field outside is zero.
  3. The integration is simple: Bℓ = μ₀NI where N is the number of turns and ℓ is the length of the solenoid.
  4. Solve for B: B = μ₀NI / ℓ .

This equation shows why solenoids are used in applications that require strong, uniform magnetic fields, such as MRI machines where a high level of precision is necessary.

Ampere-Maxwell law

Ampere's law was later generalized by James Clerk Maxwell to include time-varying electric fields. The revised equation, known as the Ampere-Maxwell law, adds a term for the displacement current generated by the changing electric fields:

        ∮ B · dl = μ₀(Iₑₙc + ε₀(dΦₑ/dt))

This incorporated time-varying fields into the theory, making it possible to describe dynamic electrical and magnetic phenomena.

Summary

Ampere's law is the basis of electromagnetism, showing the relationship between electricity and magnetism. By integrating the magnetic field around the path enclosing an electric current, it gives information about how currents generate magnetic fields.

Despite being mainly applicable to symmetrical situations, Ampere's law remains important in designing electrical devices such as solenoids and in understanding electromagnetic theory. Its development into the Ampere–Maxwell laws forms the basis of modern physics, which covers the dynamic interaction of electric and magnetic fields.


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Ampere's Law

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