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Magnetic Effects Of Current And Magnetism

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Magnetic Effects of Current and Magnetism PDF Notes, Important Questions and Formulas

Magnetism


Chapter cover: Magnetic field due to a straight wire, Circular Loop, Circular Arc.

 

  1. THE MAGNETIC FIELD

    In earlier lessons we found it convenient to describe the interaction between charged objects in terms of electric fields. Recall that an electric field surrounding an electric charge. The region of space surrounding a moving charge includes a magnetic field in addition to the electric field. A magnetic field also surrounds a magnetic substance.

    In order to describe any type of field, we must define its magnitude, or strength, and its direction. Magnetic field is the region surrounding a moving charge in which its magnetic effects are perceptible on a moving charge (electric current). Magnetic field intensity is a vector quantity and also known as magnetic induction vector. It is represented by B.

    Lines of magnetic induction may be drawn in the same way as lines of electric field. The number of lines per unit area crossing a small area perpendicular to the direction of the induction being numerically equal to .

    The number of lines of B crossing a given area is referred to as the magnetic flux linked with that area. For this reason B is also called magnetic flux density. There are two methods of calculating magnetic field at some point. One is Biot-Savart law which gives the magnetic field due to an infinitesimally small current carrying wire at some point and the another is Ampere’ law, which is useful in calculating the magnetic field of a symmetric configuration carrying a steady current. The unit of magnetic field is weber/m2 and is known as tesla (T) in the SI system.


  2. BIOT-SAVART LAW

    Biot-Savart law gives the magnetic induction due to an infinitesimal current element. Let AB be a conductor of an arbitrary shape carrying a current i, and P be a point in vacuum at which the field is to be determined. Let us divide the conductor into infinitesimal current-elements. Let r be a displacement vector from the element to the point P.
    According to ‘Biot-Savart Law’, the magnetic field 
    Induction dB at P due to the current element dl is given by

    begin mathsize 12px style straight d space straight B space proportional to fraction numerator straight i left parenthesis dl cross times straight r right parenthesis over denominator straight r cubed end fraction space or space straight d space straight B equals straight k fraction numerator straight i left parenthesis dl cross times straight r right parenthesis over denominator straight r cubed end fraction end style

    When k is proportionally constant.

    Here dl vector points in the direction of current i.


    begin mathsize 12px style table attributes columnalign left end attributes row cell In    straight S. straight thin space straight I.    units comma     straight k equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction equals 10 to the power of negative 7 end exponent fraction numerator Wb over denominator amp cross times metre end fraction end cell row cell straight d space straight B equals fraction numerator straight mu subscript 0 straight i over denominator 4 straight pi end fraction fraction numerator straight i left parenthesis dl cross times sinθ over denominator straight r squared end fraction            . .... ( 1 ) end cell end table end style

    Equation (1) is the vector form of the Biot-Savart Law. The magnitude of the field induction at P is given by

    begin mathsize 12px style straight d space straight B equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction fraction numerator straight i straight space dl cross times sinθ over denominator straight r cubed end fraction end style

    Where Ѳ is the angle between dl and r.

    If the medium is other than air or vacuum, the magnetic induction is

    begin mathsize 12px style straight d straight space straight B equals fraction numerator straight mu subscript 0 straight mu subscript straight r over denominator 4 straight pi end fraction fraction numerator straight i left parenthesis dl cross times straight r right parenthesis over denominator straight r cubed end fraction               . .. ( 2 ) end style

    Where μr is relative permeability of the medium and is a dimensionless quantity.

  3. FIELD DUE TO A STRAGHT CURRENT CARRYING WIRE

    3.1 WHEN THE WIRE IS OF FINITE LENGTH

    Consider a straight wire segment carrying a current I and there is a point P at which magnetic field to b e calculated as shown in the figure. This wire segment makes angle Ѳ1 and Ѳ2 at that point with normal OP. Consider an element of length dy at a distance y from O and distance of this elements from points P is r and line joining P to Q makes an angle Ѳ with the direction of current as shown in figure. Using Biot-Savart Law magnetic field at point P due to small current element is given by

    begin mathsize 12px style straight d space straight B equals fraction numerator straight mu subscript 0 straight i over denominator 4 straight pi end fraction open parentheses dysinθ over straight r squared close parentheses end style

    As every element of the wire contributes to B in the same direction, we have



    begin mathsize 12px style straight B equals fraction numerator straight mu subscript 0 straight i over denominator 4 straight pi end fraction integral from straight A to straight B of fraction numerator dy space sinθ over denominator straight r squared end fraction end style

    From the triangle OPQ as shown in diagram, we have

    y=d tan ϕ               or         dy= d sec2 ϕ d ϕ and is same triangle,

    r=d sec ϕ and Ѳ = (900 - ϕ), where ϕ is angle between line OP and PQ

    Now e.g. (1) can be written in this form
    begin mathsize 12px style table attributes columnalign left end attributes row cell therefore     straight B equals fraction numerator straight mu subscript 0 over denominator 4 πd end fraction straight i over straight d integral from negative straight theta subscript 2 to straight theta subscript 1 of cosϕ space dϕ end cell row cell Or     straight B equals fraction numerator straight mu subscript 0 over denominator 4 straight pi end fraction straight i over straight d left square bracket sin straight space straight theta subscript 1 plus sin straight space straight theta subscript 2 right square bracket                . ... ( 3 ) end cell end table end style

    Note

    Ѳ1 and Ѳ2 must be taken with sign

    For the case shown in figure

    begin mathsize 12px style straight B    at    straight A equals fraction numerator straight mu subscript 0 straight i over denominator 4 πr end fraction left parenthesis sinθ subscript 2 minus sinθ subscript 1 right parenthesis circled times end style


    Direction of: The direction of magnetic field is determined by the cross product of the vector idl with r. Therefore, at point P, the direction of the magnetic field due to the whole conductor will be perpendicular to the plane of paper and going into the plane.

    Right-hand Thumb Rule: The direction of B at a point P due to a long, straight wire can be found by the right-hand thumb rule. The direction of magnetic field is perpendicular to the plane containing wire and perpendicular from the point. The orientation of magnetic field is given by the direction of curl fingers if we stretch thumb along the wire in the direction of current. Refer figure.



    Conventionally, the direction of the field perpendicular to the plane of the paper is represented by ⨂ if into the page and by ⦿ If out of the page. Now consider some special cases involving the application of equation (3)

     



 

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