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 ↓


Biot-Savart law


The Biot-Savart law is an essential principle in physics that helps us understand how magnetic fields are produced by electric currents. It is named after Jean-Baptiste Biot and Félix Savart who developed this law in the early 19th century. This law is very important in electromagnetism and is widely used to calculate the magnetic fields produced by various electric distributions.

The Biot-Savart law provides a mathematical equation that relates the magnetic field produced to the current and the geometry of the current-carrying conductor. This law is particularly useful because it can be applied to conductors with arbitrary shapes, allowing us to determine the magnetic field at any point in space.

Understanding the basics

To understand the Biot-Savart law, let's start by reviewing the basic concept of a magnetic field. A magnetic field is a vector field that shows the magnetic effect on moving charges, magnetic materials, and other magnetic objects. The direction of the magnetic field is represented by the field lines, and the strength of the field is represented by how close these lines are to each other.

Electric currents, which are flows of electric charge, are a common source of magnetic fields. When electric current passes through a conductor, it produces a magnetic field around the conductor. The Biot-Savart law describes exactly how this magnetic field is produced.

Mathematical formulation

The Biot-Savart law is mathematically formulated as follows. Consider a small portion of a conductor carrying current ( I ). The vector quantity ( mathbf{dL} ) represents an infinitesimal length of the conductor. The magnetic field ( mathbf{dB} ) produced at a point in space by this small portion is given by the Biot-Savart law:

[ mathbf{dB} = frac{mu_0}{4pi} frac{I , mathbf{dL} times mathbf{hat{r}}}{r^2} ]

Where:

  • ( mu_0 ) is the permittivity of free space and is a constant, approximately equal to ( 4pi times 10^{-7} , text{T}cdottext{m/A} ).
  • ( I ) is the electric current flowing through the conductor.
  • ( mathbf{dL} ) is a vector representing a small segment of a current carrying wire.
  • ( mathbf{hat{r}} ) is a unit vector pointing from the element ( mathbf{dL} ) to the point where the magnetic field ( mathbf{dB} ) is being calculated.
  • ( r ) is the distance from the current element to the point where the field is calculated.
  • The cross product ( times ) indicates that the direction of ( mathbf{dB} ) is perpendicular to both ( mathbf{dL} ) and ( mathbf{hat{r}} ).

Visual example: straight wire

One of the simplest applications of the Biot-Savart law is to calculate the magnetic field due to a long, straight, current-carrying wire. For this example, consider an infinitely long straight wire carrying a constant current ( I ). We want to find the magnetic field at a perpendicular distance ( r ) from the wire.

Using the Biot-Savart law, the magnetic field ( mathbf{B} ) at a distance ( r ) from the wire can be determined by integrating the contributions of all infinitesimal elements ( mathbf{dL} ) along the wire:

[ B = frac{mu_0 I}{2pi r} ]
Current I. Distance R

The magnetic field lines form concentric circles around the wire, and the direction of the field lines follows the right-hand rule. If the thumb of your right hand points in the direction of the current, your fingers will bend in the direction of the magnetic field.

Circular loop

Let's consider another example where the Biot-Savart law applies: a circular loop of wire. If current ( I ) flows around the loop of radius ( R ), we can determine the magnetic field at the center of the loop using the Biot-Savart law.

For a circular loop, symmetry allows the direct integration of the contributions from all segments of the loop:

[ B = frac{mu_0 I}{2R} ]
Current I. Center

The magnetic field at the centre of the loop is perpendicular to the plane of the loop, and its direction can again be determined using the right-hand rule.

Helical coil

The Biot-Savart rule can also be used to calculate the magnetic field from helical or solenoidal coils, which are commonly used in electromagnets and inductors. In these coils, the wire forms a spiral helix.

Detailed computations require integrating the contributions from each loop of the helix, which is often performed using computational tools due to the complexity of the geometry.

Helical Coil

Applications of Biot-Savart law

The Biot-Savart law is used in various fields of physics and engineering. Its ability to predict magnetic fields from different power distributions makes it invaluable in the design and evaluation of electromagnetic systems. Some of the major applications include:

  • Electric motors: Calculating magnetic fields in motors to improve efficiency and performance.
  • Magnetic sensors: Designing magnetic sensors that can measure the strength and direction of magnetic fields.
  • Biomedical equipment: Designing MRI machines and other devices that rely on magnetic fields.
  • Power transmission: Analyzing magnetic effects in power lines and transmission systems.

Limitations and considerations

The Biot-Savart law is powerful, but there are some limitations and conditions for its use. It assumes that the medium through which the magnetic field propagates is uniform and isotropic, which means having the same properties in all directions.

In addition, this law is mainly valid for constant or steady currents. For changing currents, we must use the more general Maxwell's equations, which describe how electric and magnetic fields interact more generally.

Conclusion

The Biot-Savart law is a cornerstone in understanding magnetism and electromagnetism, giving insight into how current distributions generate magnetic fields. Its explicit mathematical formulation allows engineers and physicists to design and analyze systems with precision.

Whether it is in the analysis of a simple electrical circuit or in the design of complex systems such as medical imaging equipment, the Biot-Savart law plays an essential role. As you continue your studies in electromagnetism, a fundamental understanding of this law will serve as a vital tool in exploring the depths of this fascinating field.


Undergraduate → 2.3.4


U
username
0%
completed in Undergraduate

← Prev (2.3.3)
Magnetic dipole moment
Next (2.3.5) →
Magnetic Materials and Hysteresis

Comments 



Biot-Savart law

https://www.physicsflow.com/ug/2.3.4?pfal=en