Biot-Savart Law

The Biot-Savart Law is an equation that describes the quantitative relationship between an electrical current and the magnetic field it generates. This law is seen as a magnetic equivalent of Coulomb's Law, and states that the magnetic field decreases with the square of a distance from a point of current.Page in progress by Andrea Boyd. [contd]

The Main Idea

An electron current flowing through a conductor, such as a wire, or a moving electric charge produces a detectable magnetic field. The Biot-Savart law describes this phenomenon by relating the magnetic field to the magnitude, direction, length, and proximity of the electric current.

A Mathematical Model

The general formula for a single charge is [math]\displaystyle{ \vec B=\frac{\mu_0}{4\pi}\frac{q\vec v\times\hat r}{r^2} }[/math] where [math]\displaystyle{ \mu_0 }[/math] is the constant [math]\displaystyle{ 1e-7 \frac{tesla * m^2}{coloumb * \frac{m}{s}} }[/math] exactly. [math]\displaystyle{ \vec q }[/math] is the velocity of the point charge [math]\displaystyle{ q }[/math] and [math]\displaystyle{ \hat r }[/math] is the unit vector pointing from the source towards the observation location (remember [math]\displaystyle{ r= r(obs)-r(source) }[/math])

Additionally, in problems where the angle is given, another form of the Biot-Savart law may be used: [math]\displaystyle{ B=\frac{\mu_0}{4\pi}\frac{qv sin(\theta)}{r^2} }[/math]

A Computational Model

For a computational model of a wire carrying current, see the wiki page on the Biot-Savart Law for Currents [[1]].

Examples

A point charge, [math]\displaystyle{ \vec q }[/math] is moving in the +z direction with a velocity [math]\displaystyle{ \vec v }[/math] in the -y direction . At a point in time, the point charge's location is [math]\displaystyle{ \lt x, y, 0\gt }[/math]. If the source location is [math]\displaystyle{ \lt 0, -y, z\gt }[/math], what is the magnetic field generated at that point?

Applications

Magnetic Response

The Biot-Savart law has applications in nuclear magnetic resonance (NMR) spectroscopy, used to measure the chemical signals given off by compounds. The law can be used to calculate the magnetic responses at the atomic or molecular level, provided that the current density can be obtained mathematically. For more about NMR spectroscopy, see the wiki page [2]

Aerodynamics

In aerodynamics, the Biot-Savart law may be used to calculate the velocity induced by vortex lines, which are lines that are everywhere tangent to the vorticity vector. A vorticity vector is a pseudovector field that describes the tendency of something to rotate, in other words, the curl of the velocity field of a fluid.

In this example,[3], the yellow arms represent whirring space from a black hole and the red lines represent vortex lines. Astrophysicists may use the Biot-Savart Law to calculate the velocity.

Medical Technology

Aside from applications in aerospace engineering and chemistry, the Biot-Savart law also plays an important role in MRI imaging techniques. It's important to understand how a looped wire can create a current so these technologies can be as accurate as possible without harming patients who receive them.

History

Felix Savart was born on June 30, 1791 in Mezieres, France to a family with a strong association with military engineering schools. While completing his formal education in 1808 at the university in Metz, Savart decided to pursue medicine and become a physician. After serving a short stint in Napoleon's army in the the first engineering battalion, he resumed his medical training and graduated from Strasbourg in 1816. During his medical studies, Savart became interested in first century Roman writer Aulus Cornelius Celsus and his famous medical book De medicinia. Savart began working on a translation and set up a medical practice in Metz in 1817, but gradually became more interested in physics than patients, particularly in sound and acoustics. He began building violins as a way to explore the form of the instrument through mathematical principles.

In 1819, Savart officially closed the doors of his medical practice and went to Paris to find a publisher for the translation of De medicina. While there, he attended a lecture on acoustics at by Jean-Baptiste Biot at the Faculty of Sciences. The two met there and began collaboration, when in 1820, Hans Christian Oersted published that a compass needle placed near a wire carrying current pointed at right angles to the wire. Biot and Savart began looking more closely into the field produced by a wire, and by using the oscillation of a magnetic dipole to determine the strength of the field close to a wire carrying current, they discovered what is now called the Biot-Savart Law [[4]].

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