Magnetic Effect of Current PDF

Magnetic Effect of Current 1 1 Oersted found that a magnetic field is established around a...

Magnetic Effect of Current 1 1 Oersted found that a magnetic field is established around a Magnetic current carrying conductor. i lines of forces Magnetic field exists as long as there is current in the wire. The direction of magnetic field was found to be changed when direction of current was reversed. Note :  A moving charge produces magnetic as well as electric field, unlike a stationary charge which only produces electric field. Biot Savart's Law. Biot-Savart’s law is used to determine the magnetic field at any point due to a current carrying conductors. This law is although for infinitesimally small conductors yet it can be used for long conductors. In order to understand the Biot-Savart’s law, we need to understand the term current-element. Current element It is the product of current and length of infinitesimal segment of current carrying wire. B A The current element is taken as a vector quantity. Its direction is same as the direction i of current. dl Current element AB = i dl In the figure shown below, there is a segment of current carrying wire and P is a point where magnetic field is to be calculated. i d l is a current element and r is the distance of the point ‘P’ with respect to the current element i d l. According to Biot-Savart Law, magnetic field i dlsin θ at point ‘P’ due to the current element i d l is given by the expression, dB  k also r2 P  0 i dl sin   B  dB  4  r 2. dl r idl sin  i In C.G.S. : k = 1  dB  Gauss r2 0  idl sin  In S.I. : k  dB  0  Tesla 4 4 r2 2 Magnetic Effect of Current Wb where  0 = Absolute permeability of air or vacuum  4   10 7. It's other units Amp  metre Henry are metre N Tesla  metre or 2 or Amp Ampere (1) Different forms of Biot-Savarts law Vector form Biot-Savarts law in terms of Biot-savarts law in terms of current density charge and it's velocity Vectorially, In terms of current density In terms of charge and it’s           i(d l  rˆ )  i(d l  r )   J r (v  r ) dB  0   0   dB  0 dV velocity, dB  0 q 3 4 r2 4 r3 4 r 3 4 r   Direction of is dB i idl idl  q  dl   where j    =  idl  dl  q  qv perpendicular to both d l and A Adl dV dt dt r̂. This is given by right hand current density at any point of screw rule. the element, dV = volume of element (2) Similarities and differences between Biot-Savart law and Coulomb’s Law (i) The current element produces a magnetic field, whereas a point charge produces an electric field. (ii) The magnitude of magnetic field varies as the inverse square of the distance from the current element, as does the electric field due to a point charge.  0 id l  rˆ 1 q1 q 2 dB  Biot-Savart Law F rˆ Coulomb’s Law 4 r 2 4 0 r 2 (iii) The electric field created by a point charge is radial, but the magnetic field created by a current element is perpendicular to both the length element d l and the unit vector r̂. dl E i r̂ B +q r̂ Direction of Magnetic Field. The direction of magnetic field is determined with the help of the following simple laws : (1) Maxwell’s cork screw rule  Magnetic Effect of Current 3 3 According to this rule, if we imagine a right handed screw placed along the current carrying linear conductor, be rotated such that the screw moves in the direction of flow of current, then the direction of rotation of the thumb gives the direction of magnetic lines of force. (2) Right hand thumb rule According to this rule if a current carrying conductor is held in the right hand such that the thumb of the hand represents the direction of current flow, B then the direction of folding fingers will represent the direction of magnetic lines of force. (3) Right hand thumb rule of circular currents B According to this rule if the direction of current in circular conducting coil is in the direction of folding fingers of right hand, then the direction of magnetic field will be in the direction of stretched i thumb. (4) Right hand palm rule If we stretch our right hand such that fingers point towards the point. At which magnetic field is required while thumb is in the direction of current then normal to the palm will show the direction of B magnetic field. Note :  If magnetic field is directed perpendicular and into the plane of the paper it is represented by  (cross) while if magnetic field is directed perpendicular and out of the plane of the paper it is represented by  (dot) i i i i CW ACW B B B B Out In In Out In Out In : Magnetic field is away from the observer or perpendicular inwards. Out : Magnetic field is towards the observer or perpendicular outwards. Application of Biot-Savarts Law. (1) Magnetic field due to a circular current 4 Magnetic Effect of Current If a coil of radius r, carrying current i then magnetic field on it's axis at a distance x from its centre given by μ0 2 πNir 2 B axis . 2 ; where N = number of turns in coil. 4 π ( x  r 2 ) 3/2 r P B Different cases O x i Case 1 : Magnetic field at the centre of the coil  0 2Ni  Ni (i) At centre x = 0  Bcentre . = 0  B max 4 r 2r  2i  0 i 0 (ii) For single turn coil N = 1  Bcentre  0.  (iii) In C.G.S. 1  4 r 2r 4 2i B centre  r 1 Note :  B centre  N (i, r constant), B centre  i (N, r constant), Bcentre  (N, i constant) r Case 2 : Ratio of Bcentre and Baxis The ratio of magnetic field at the centre of circular coil and on it's axis is given by 3/2 B centre  x2   1  2  B axis  r  3/2 a 5 5 a 3 (i) If x  a, Bc  2 2 Ba x   , Bc  Ba x  , Bc    Ba 2 8 2 2 Bc B (ii) If B a  then x  r (n 2 / 3  1) and if B a  c then x  r (n1 / 3  1) n n Case 3 : Magnetic field at very large/very small distance from the centre  0 2 Nir 2  0 2 NiA (i) If x >> r (very large distance)  B axis . . where A = r2 = Area of each 4 x3 4 x 3 turn of the coil. (ii) If x