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what are the applicatios of biot savarts law?


Biot Savart's Law and its applications

Consider a straight infinitely long wire carrying a steady current I. Point P is at a perpendicular distance (AP=) R from the wire. Consider a small element dl of the wire at the point O on the wire. The line joining points O to P (OP=vector r) makes an angle q with the direction of the current element dl.

Magnetic Field dB due to current carrying element

Derivation of dB

The magnetic field dB due to the current length element dl at P is observed to be given by 

                          dB = \frac {\mu_0 I}{4 \pi} \frac {dl \times r}{r^3}

                          dB = \frac {\mu_0}{4 \pi} \frac {I dl sin \theta}{r^2} k

The product dl*r has a magnitude dlrsin? . It is directed perpendicular to both dl and r. i.e. it is perpendicular to the plane of the paper and going into it, according to the right handed corkscrew rule (direction in which a right handed corkscrew advances when turning from dl to r).

The expression for the total magnetic field B due to the wire can be obtained by integrating the above expression as

B = \int \frac {\mu_0}{4 \pi} \frac {I dl \times r}{r^3}      B = \frac {\mu_0 I}{4 \pi} \oint \frac {dl \times \hat r}{r^2}

It is called as the Biot–Savart law which gives the magnetic field B generated by a steady electric current I when the current can be approximated as running through an infinitely-narrow wire.

If the current has some thickness i.e. current density is J, then the statement of the law is:

B = \int \frac {\mu_0}{4 \pi} \frac {(J dV) \times \hat r}{r^2}, or (equivalently), B = \int \frac {\mu_0}{4 \pi} \frac {(J dV) \times r}{r^3}

Where dl =differential current length element and dV = differential volume element.

?0= the magnetic constant, \hat r= the displacement unit vector, r= displacement vector.

The magnetic field B at a point P due to an infinite (very long) straight wire carrying a current I is proportional to I, and is inversely proportional to the perpendicular distance R of the point from the wire.

The vector field B depends on the magnitude, direction, length, and proximity of the electric current, and also on a fundamental constant called the magnetic constant ?0.

\mu_0 = 4 \pi \times 10^{-7} N A^{-2}.

The Biot–Savart law is fundamental to magnetostatics & plays a role similar to Coulomb's law in electrostatics. The Biot-Savart Law relates magnetic fields to the electric currents which are their sources just as Coulomb's law relates electric fields to the point charges which are their sources. It provides a relation between the cause (moving charge) and effect (magnetic field) in magnetism. It is an empirical law (formulated from the experimental observations) like Coulomb’s law. Both are inverse square laws.

In spite of this parallel situation, one important distinction is that the magnetic field B, is in direction of vector cross product dl x r i.e. along the perpendicular direction of the plane constituted by the current length element dl and displacement vector r (i.e. it is not along displacement vector as in electrostatic field).This necessitates representation of B-Field by vector notation and 3-D space for its visualization. The magnetic field B as computed using the Biot-Savart law always satisfies Ampere's circuital law and Gauss's law for magnetism (explained later).

Though the above statement of Biot-Savart law is for a macroscopic current element, it can be applied in the calculation of magnetic field even at the atomic/molecular level (in which case quantum mechanical calculation or theory is used for obtaining the current density).

Biot Savart’s Law’s Applications

Biot-Savart’s law is stated for a small current element (Idl) of wire – not for the extended wire carrying current. However, magnetic field due to extended wire carrying current can be found by using the superposition principle i.e. the magnetic field is a vector sum of the fields created by each infinitesimal section of the wire individually. The point in space at which the magnetic field is to be computed is chosen, it is held fixed and integration is carried out over the path of the current(s) by applying the equation of Biot Savart’s Law.

Magnetic Field due to a Circular Current Loop


Magnetic field at any point on the axis of a circular loop can be obtained as follows Consider a circular loop of radius a having its centre at O carrying a current I. Point P is situated on the axis of loop at a distance R from the centre O of the loop. The magnitude of the fields dB & dB’ due to small current elements dl of the circle, centered at A and A’ (at diagrammatically opposite points) respectively is given by Biot Savart’s law 
                         dB = \frac {\mu_0}{4 \pi} \frac {I dl}{|AP|^2} = \frac {\mu_0}{4 \pi} \frac {I dl}{(R^2 + a^2)}

The direction of the field dB is normal to a plane containing dl and AP i.e. along PQ and that of dB’ is along PQ’. The fields can be resolved into two components in mutually perpendicular directions along axis and PS/ PS’. Their Components dB cos? along PS and dB’cos ? along PS’ are equal and opposite and get cancelled. Components along the axis dB sin? and dB’ sin? both have the same direction and are added up. This applies to all such pairs of elements. Thus the resultant field due to the loop is directed along the axis of the loop and its magnitude is obtained by integrating the expression 

                          dB_{along OP} = \frac {\mu_0}{4 \pi} \frac {I dl}{r^2} {sin \phi}
                                       = \frac {\mu_0}{4 \pi} \frac {I dl a}{(R^2+a^2)(R^2+a^2)^{1/2}}
                                       = \frac {\mu_0}{4 \pi} \frac {I a}{(R^2+a^2)^{3/2}} dl

The magnetic field B due to the circular current loop of radius a at a point on its axis and a distance R away is given by integrating the above expression as

B = \frac {\mu_0 I a^2}{2(R^2+a^2)^{3/2}} i (i is the unit vector along OP, the x-axis)

Some other examples of geometries where the Biot-Savart law can be used to advantage in calculating the magnetic field resulting from an electric current distribution are as follows

Infinitely Long Wire: The magnetic field B at a point distance r from an infinitely long wire carrying current I has magnitude 

                              B = \frac {\mu_0 I}{2 \pi r}

and its direction is given by the right-hand rule.

Long Solenoid: The magnetic field B inside the long solenoid of length L with N turns of wire wrapped evenly along its length is uniform throughout the volume of the solenoid (except near the ends where the magnetic field becomes weak) and is given by 

                              B = \mu_0 \frac {N}{L} I = \mu_0 n I, (where n = N/L.)

B is independent of the length and diameter and uniform over cross-section of solenoid.




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