Magnetic Field due to a Current Element, Biot-Savart Law
A law of physics which states that the magnetic flux density (magnetic induction) near a long, straight conductor is directlyproportional to the current in the conductor and inversely proportional to the distance from the conductor.
The field near astraight conductor can be found by application of Ampère's law.
The magnetic flux density near a long, straight conductor isat every point perpendicular to the plane determined by the point and the line of the conductor.
Therefore, the lines ofinduction are circles with their centers at the conductor.
Furthermore, each line of induction is a closed line.
This observation concerning flux about a straight conductor may be generalized to include lines of induction due to a conductor ofany shape by the statement that every line of induction forms a closed path.
Biot-Savart law is a fundamental quantitative relationship between an electric current and the magnetic field it produces, based on the experiments in 1820 of the French scientists Jean-Baptiste Biot and Félix Savart.
The Biot-Savart law is applied in a specific case by adding up the contributions to the magnetic field at a given point from the whole series of short current segments that constitute a specific conductor of whatever shape.
For instance, with a very long straight wire carrying current, the value of the magnetic field at a point nearby is just directly proportional to the value of the current and inversely proportional to the perpendicular distance from the wire to the given point. Compare Ampère’s law.
$$ dB \propto \frac{Idlsine \theta}{r^2} $$ $$ \text{or } dB = k \frac{Idlsine \theta}{r^2} $$Where, K is a constant, depends upon the magnetic properties of the medium and system of the units employed. In SI system of unit-
$$ k = \frac{\mu_0 \mu _r}{4 \pi} $$Therefore final Biot Savart law derivation is,
Applications of Biot-Savart’s Law
Biot–Savart law is used to calculate magnetic responses even at the molecular/atomic level.
Biot–Savart law is used in aerodynamic theory to calculate the velocity induced by vortex lines.
Importance of Biot-Savart Law
Applicable for symmetrical current distribution.
Applicable for very small conductors also which may carry current.
Biot-Savart law is similar to the Coulomb’s law in electrostatics.
Difference between Biot Savart law and Ampere's circuital law
The Biot-Savart law is more general than the Ampère law, is valid only when the flux of electric field through the loop γ is constant in time otherwise its rate of change (displacement current) has to be added to the normal current on the right-hand side.
Similarities between Biot Savart law and Coulomb's law
Both magnetic and electric fields at a point are inversely proportional to the square of the distance between the field source and the point in question. The electric field due to a point charge as per Coulomb's law is :
E = (1/4πƐo) × (q/r2)