* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.

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1 * Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17

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

3 Biot Savart s Law Fig. (a) Magnetic Field db due to current carrying element (b) Derivation of db

4 Biot Savart s Law The magnetic field db due to the current element of length dl at P is observed to be given by The product dl x r has a magnitude dl r sin θ. 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).

5 Biot Savart s Law The expression for the total magnetic field B due to the wire can be obtained by integrating the above expression as or equivalently, 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.

6 Biot Savart s Law If the current has some thickness i.e. current density is J, then the statement of the Biot Savart s law is: or equivalently, Where dl= differential current length element dv = volume element μ 0 = the Magnetic constant, r = displacement vector = the displacement unit vector,

7 Biot Savart s Law 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 μ 0

8 Biot Savart s Law The Biot Savart law is fundamental to Magnetostatics It 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.

9 Biot Savart s Law The Biot-Savart Law provides a relation between the cause (moving charge) and the effect (magnetic field) in magnetism. It is an empirical law (formulated from the experimental observations) like the Coulomb s law. Both are inverse square laws.

10 Biot Savart s Law In spite of this parallel situation, one important distinction between the Coulomb s law and the Biot Savart s law is that the magnetic field B, is in the direction of the 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 while electrostatic field E is along the displacement vector.

11 Biot Savart s Law 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 Law for Magnetism

12 Biot Savart s Law or 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).

13 Applications of Biot Savart s Law 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.

14 Applications of Biot Savart s Law For calculating the magnetic field due to an extended wire carrying current 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.

15 Applications of Biot Savart s Law

16 Applications of Biot Savart s Law Magnetic Field at the Centre of the Current Loop Consider a circular loop of radius r carrying a current I At the center of the loop, the magnitude of the magnetic field B is given by B = The direction of the magnetic field is indicated by The Right Hand Rule The magnetic field changes away from the center in both magnitude and direction

17 Applications of Biot Savart s Law

18 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. Point P is situated on the axis of loop at a distance R from the centre O of the loop. The loop carries a current I.

19 Magnetic Field due to a Circular Current Loop The magnitude of the fields db & db due to small current elements dl and dl of the circle, centered at A and A (at diagrammatically opposite points) respectively is given by Biot Savart s law as

20 Magnetic Field due to a Circular Current Loop Fig. Magnetic B-Field due to a circular current loop

21 Magnetic Field due to a Circular Current Loop 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 the axis and Perpendicular to axis i.e. along PS/ PS. Their Components db cosφ along PS and db cos φ along PS are equal and opposite and get cancelled. Components along the axis db sinφ and db sinφ both have the same direction and are added up. This applies to all such pairs of elements.

22 Magnetic Field due to a Circular Current Loop 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

23 Magnetic Field due to a Circular Current Loop 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 (i is the unit vector along OP, the x-axis) Some other examples of geometries where the Biot Savart s Law can be used to advantage in calculating the Magnetic field resulting from an Electric current distribution are as follows

24 Applications of Biot Savart Law Magnetic Field of an Infinitely Long Wire The magnetic field B at a point distance r from an infinitely long wire carrying current I has magnitude The direction of Magnetic Field is given by the Right-hand rule.

25 Applications of Biot Savart Law Magnetic Field of a 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 is independent of the length and diameter and uniform over cross-section of solenoid

26 Applications of Biot Savart Law Magnetic Field of a Solenoid Magnetic Field inside A long Solenoid

27 (A) Divergence of B-field According to the Gauss law in electrostatics, divergence of the static electric field is equal to the total density of a stationary electric charge/s at a given point. div. E =

28 However in magnetostatics a magnetic charge (i.e. monopole) is not found to exist. (The source of magnetic fields is moving electric charges, Not the static ones). Due to the absence of magnetic charges, the magnetic field is divergenceless. In Differential form (A) Divergence of B-field (where B is the Magnetic field denotes Divergence)

29 (A) Divergence of B-field This is called as the Gauss's law for magnetism (though this term is not universally adopted). It states that the magnetic field B has divergence equal to zero i.e. magnetic field is a solenoidal vector field

30 (A) Divergence of B-field It is equivalent to the statement that Magnetic Monopole (i.e. isolated North or South magnetic pole) does not exist. The basic quantity for magnetism is the Magnetic Dipole Not the magnetic charge or monopole. Hence, the law is also called as "Absence of Free Magnetic poles ".

31 (A) Divergence of B-field The statement of Gauss's law for magnetism in integral form is given as Where S is any closed surface (the boundary enclosing a three-dimensional volume); da is a vector, having magnitude equal to the infinitesimal area of the surface S and direction along the surface normal pointing outward.

32 (A) Divergence of B-field The left-hand side of the equation in integral form denotes the net flux of the magnetic field out of the surface. The law implies that the net magnetic flux into and out of a volume is zero. Thus Gauss's law for magnetism can be written in both- differential and integral- forms. These forms are equivalent due to the Divergence theorem

33 (A) Divergence of B-field The magnetic field B, like any vector field, can be represented by field lines. Gauss's law for magnetism also implies that the field lines have neither a beginning nor an end. They either form a closed loop, or extend to infinity in both directions.

34 (B) Curl of B-field Circulation is the amount of pushing, twisting or turning force along a closed boundary / path when the path is shrunk down to a single point. Circulation is the integral of a vector field along a path. A vector field is usually the source of the circulation. Curl is the circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point

35 (B) Curl of B-field The curl of a force F is calculated as follows Let the Force at position r= Direction at position r = Total pushing force = Curl =

36 (B) Curl of B-field Curl is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. The magnitude of is the limiting value of circulation per unit area

37 (B) Curl of B-field => the field is said to be an irrotational field. The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism

38 (B) Curl of B-field In magnetostatics, it can be proved that the curl of magnetic field B is given by Thus the curl of a magnetic B field at any point is equal to μ 0 times the current density J at that point. This simple statement relates the magnetic field and moving charges. The equation is mathematically equivalent to the line integral equation given by Ampere s law.

39 Divergence and Curl of B-field The equations in terms of Divergence and Curl of magnetic B-field are also called as the laws of Magnetostatics. They correspond to the curl and divergence of electric field E respectively in electrostatics as follows

40 Divergence and Curl of B-field Electrostatics Field is without curl Magnetostatics Field is without divergence Field E Source ρ relation Field B Source j relation

41 Divergence and Curl of B-field The equations for divergence and curl for vector fields are extremely powerful. Expressions for divergence and curl of a magnetic field describe uniquely any magnetic field from the current density j in the field in the same manner that the equations for the divergence and curl for the electric field describe an electric field from the electric charge density ρ in the electric field.

42 Divergence and Curl of B-field The four equations involving Curl and Divergence for Electric and Magnetic fields are the versions of Maxwell s equations for static electromagnetic fields. They describe mathematically the entire content of Electrostatics and Magnetostatics.

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