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(a) State Biot - Savart law and ecpress it in the vector form.(b) Using Biot - Savart law, law obtain the expression for the magnetic field due to a circular coil of radius r, carrying a current I at point on its axis distant x from the centre of the coil. |
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Answer» Solution :(a) According to Biot -Savart 's law , the magnetic field dB due to a current element `Ivec(dl)`such that , `dBpropI`. . . (i) `dBpropdl` . . . (ii) `dBpropsintheta` . . . (III) `dBprop(1)/(r^(2))`. . . (iv)  From equation (i) , (ii) , (iii) and (iv) ,`dBprop(Idlsintheta)/(r^(2))` `dBprop(KIdlsintheta)/(r^(2))` VALUEOF `K=(mu_(0))/(4pi)` `dB=(mu_(0))/(4pi)(Idlsintheta)/(r^(2))` Biot -Savart ' s law in vector FORM `vec(dB)=(mu_(0))/(4pi)(I(vecdlxxvecr))/(r^(3))` (b)Consider a circular coil ofradius r , CARRYING a current I . Plane of the coil is Y- Z Plane . Magnetic field due to a current carrying element is dB and due to M'N is dB'. According to Biot - Savart 's law , `|vecdB|=|vecdB'|=(mu_(0))/(4pi)(Idlsin90^(@))/(a^(2))=(mu_(0))/(4pi)(Idl)/(a^(2))` These fields MAY be resolved into components along X axis and Y axis .Component along X axis will be dB`sinphiand dB'sinphi`. Component along Y axis will be dB `cosphiand dB'cosphi`, they will be opposite in direction and cancelled out . I lence , net magnetic field at point P due to all the elements on the loop is given by  `B=intdBsinphi` Value of `dB=(mu_(0))/(4pi)(Idl)/(a^(2))andsinphi=(r)/(a)` `B=int_(0)^(2pir)(mu_(0))/(4pi)(Idl)/(a^(2))(r)/(a)` `B=(mu_(0))/(4pi)(I_(r))/(a^(3))int_(0)^(2pir)` `B=(mu_(0))/(4pi)(Ir)/(a^(3))(2pir)` `B=(mu_(0))/(4pi)(2pir^(2))/(a^(3))` `a=(r^(2)+x^(2))^(1//2)` `B=(mu_(0))/(4pi)(2pir^(2))/((r^(2)+x^(2))^(3//2))` |
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