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

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

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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 [1]

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[2], 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.

History

In 1820 two Frenchman, Felix Savart and Jean-Baptiste Biot, published Note sur le magnétisme de la pile de Volta and presented it to the Academy of Sciences in what became known as the Biot-Savart Law. Savart had been trained as a medical doctor, but began focusing more on physics rather than patients. He became interested in acoustics, building his own violins and conducting research on sound, and attended a lecture in Paris given by French mathematician, Jean-Baptiste Biot.

See also

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References

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