# Biot-Savart Law

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## What is Biot Savart Law?

Biot-Savart law is a law that relates magnetic fields to the electric current. This law was discovered by French physicists Jean-Baptiste Biot and Felix Savart in 1820. The two scientists noticed the deflection of a magnetic compass needle and concluded that any current element projects a magnetic field into the space around it. The Biot-Savart law gives the magnetic field produced due to a current-carrying segment. Biot-Savart law states that the magnetic field is directly proportional to the length of the conductor and the current flowing in the conductor. The Biot-Savart law is a fundamental law in magnetostatics and plays a similar role to that of Coulomb’s law in electrostatics.

## Biot-Savart Law Equation:

Jean-Baptiste Biot and Felix Savart derived a mathematical expression through many observations and calculations. The expression states that the magnetic flux density, dB is directly proportional to the length of the element dl, the current I, the sine of the angle and θ between the direction of the current and the vector joining a given point of the magnetic field and the current element is inversely proportional to the square of the distance of the given point from the current element, r.

Following the above given details, we can mathematically express,

Here, k is a constant and can be expressed as,

Hence, equation 1 becomes,

Equation 2 is the expression for Biot-Savart law.

## Derivation of Biot-Savart Law using Point Charges:

The expression of Biot-Savart law for point charges was derived in 1888 by Oliver Heaviside, an English physicist. Maxwell’s equation is used for expressing the electric field and magnetic field.

H= v x D

Here, q is the charged particle and v is the constant velocity.

## Derivation of Biot-Savart Law using Electric Field:

Using electric fields generated by stationary charges, we have,

Here, Φ is the electric scalar potential.

The equation obtained from electric fields that are generated by stationary charges are given below,

The Biot-Savart law equation before the substitution of current density, j is given below,

We have,

Here, I is the vector current and dl is the element length.

Now, substituting for j, we have the final equation,

## Applications of Biot-Savart Law:

Some of the important applications of Biot-Savart law are listed below.

• Biot-Savart law can be used to calculate magnetic responses even at the atomic or molecular level.
• Biot-Savart law is used in aerodynamic theory to calculate the velocity induced by vortex lines.
• Biot-Savart law can be used to calculate the force between two parallel and lengthy current-carrying conductors.
• Biot-Savart law can be used to calculate the magnetic field along the axis of a circular current-carrying coil.
• Biot-Savart law can be used to calculate the magnetic field in the centre of a current-carrying arc.
• Biot-Savart law can be used to calculate the magnetic field at a point on the axis of a solenoid.