Page 1 Magnetic Effect of Current 1 1 genius PHYSICS Oersted found that a magnetic field is established around a current carrying conductor. 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-Savartâ€™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-Savartâ€™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. The current element is taken as a vector quantity. Its direction is same as the direction of current. Current element AB = dl i 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. l d i is a current element and r is the distance of the point â€˜Pâ€™ with respect to the current element l d i . According to Biot-Savart Law, magnetic field at point â€˜Pâ€™ due to the current element l d i is given by the expression, 2 r dlsin ? i k dB ? also ? ? ? ? 2 0 sin . 4 r dl i dB B ? ? ? In C.G.S. : k = 1 ? 2 sin r idl dB ? ? Gauss In S.I. : ? ? 4 0 ? k ? 2 0 sin 4 r idl dB ? ? ? ? ? Tesla where 0 ? = Absolute permeability of air or vacuum metre Amp Wb ? ? ? ?7 10 4 ? . It's other units are metre Henry or 2 Amp N or Ampere metre Tesla ? dl i P r A B dl i Magnetic lines of forces i Page 2 Magnetic Effect of Current 1 1 genius PHYSICS Oersted found that a magnetic field is established around a current carrying conductor. 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-Savartâ€™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-Savartâ€™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. The current element is taken as a vector quantity. Its direction is same as the direction of current. Current element AB = dl i 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. l d i is a current element and r is the distance of the point â€˜Pâ€™ with respect to the current element l d i . According to Biot-Savart Law, magnetic field at point â€˜Pâ€™ due to the current element l d i is given by the expression, 2 r dlsin ? i k dB ? also ? ? ? ? 2 0 sin . 4 r dl i dB B ? ? ? In C.G.S. : k = 1 ? 2 sin r idl dB ? ? Gauss In S.I. : ? ? 4 0 ? k ? 2 0 sin 4 r idl dB ? ? ? ? ? Tesla where 0 ? = Absolute permeability of air or vacuum metre Amp Wb ? ? ? ?7 10 4 ? . It's other units are metre Henry or 2 Amp N or Ampere metre Tesla ? dl i P r A B dl i Magnetic lines of forces i 2 Magnetic Effect of Current genius PHYSICS (1) Different forms of Biot-Savarts law Vector form Biot-Savarts law in terms of current density Biot-savarts law in terms of charge and it's velocity Vectorially, 3 0 2 0 ) ( 4 ) Ë† ( 4 r r l d i r r l d i B d ? ? ? ? ? ? ? ? ? ? ? ? ? ? Direction of B d ? is perpendicular to both l d ? and r Ë† . This is given by right hand screw rule. In terms of current density dV r r J B d 3 0 4 ? ? ? ? ? ? ? where dV idl Adl idl A i j ? ? ? = current density at any point of the element, dV = volume of element In terms of charge and itâ€™s velocity, 3 0 ) ( 4 r r v q B d ? ? ? ? ? ? ? ? v q dt l d q l d dt q l id ? ? ? ? ? ? ? (2) Similarities and differences between Biot-Savart law and Coulombâ€™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. 2 0 Ë† 4 r r l id B d ? ? ? ? Biot-Savart Law r r q q F Ë† 4 1 2 2 1 0 ?? ? Coulombâ€™s Law (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 l d and the unit vector r Ë† . Direction of Magnetic Field. The direction of magnetic field is determined with the help of the following simple laws : (1) Maxwellâ€™s cork screw rule 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, then the direction of folding fingers will represent the direction of magnetic lines of force. (3) Right hand thumb rule of circular currents According to this rule if the direction of current in circular B i B dl i r Ë† B r Ë† +q E Page 3 Magnetic Effect of Current 1 1 genius PHYSICS Oersted found that a magnetic field is established around a current carrying conductor. 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-Savartâ€™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-Savartâ€™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. The current element is taken as a vector quantity. Its direction is same as the direction of current. Current element AB = dl i 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. l d i is a current element and r is the distance of the point â€˜Pâ€™ with respect to the current element l d i . According to Biot-Savart Law, magnetic field at point â€˜Pâ€™ due to the current element l d i is given by the expression, 2 r dlsin ? i k dB ? also ? ? ? ? 2 0 sin . 4 r dl i dB B ? ? ? In C.G.S. : k = 1 ? 2 sin r idl dB ? ? Gauss In S.I. : ? ? 4 0 ? k ? 2 0 sin 4 r idl dB ? ? ? ? ? Tesla where 0 ? = Absolute permeability of air or vacuum metre Amp Wb ? ? ? ?7 10 4 ? . It's other units are metre Henry or 2 Amp N or Ampere metre Tesla ? dl i P r A B dl i Magnetic lines of forces i 2 Magnetic Effect of Current genius PHYSICS (1) Different forms of Biot-Savarts law Vector form Biot-Savarts law in terms of current density Biot-savarts law in terms of charge and it's velocity Vectorially, 3 0 2 0 ) ( 4 ) Ë† ( 4 r r l d i r r l d i B d ? ? ? ? ? ? ? ? ? ? ? ? ? ? Direction of B d ? is perpendicular to both l d ? and r Ë† . This is given by right hand screw rule. In terms of current density dV r r J B d 3 0 4 ? ? ? ? ? ? ? where dV idl Adl idl A i j ? ? ? = current density at any point of the element, dV = volume of element In terms of charge and itâ€™s velocity, 3 0 ) ( 4 r r v q B d ? ? ? ? ? ? ? ? v q dt l d q l d dt q l id ? ? ? ? ? ? ? (2) Similarities and differences between Biot-Savart law and Coulombâ€™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. 2 0 Ë† 4 r r l id B d ? ? ? ? Biot-Savart Law r r q q F Ë† 4 1 2 2 1 0 ?? ? Coulombâ€™s Law (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 l d and the unit vector r Ë† . Direction of Magnetic Field. The direction of magnetic field is determined with the help of the following simple laws : (1) Maxwellâ€™s cork screw rule 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, then the direction of folding fingers will represent the direction of magnetic lines of force. (3) Right hand thumb rule of circular currents According to this rule if the direction of current in circular B i B dl i r Ë† B r Ë† +q E Magnetic Effect of Current 3 3 genius PHYSICS 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 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 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) 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 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 3/2 2 2 2 0 ) ( 2 4 r x p N i r . µ B axis ? ? p ; where N = number of turns in coil. Different cases Case 1 : Magnetic field at the centre of the coil (i) At centre x = 0 ? r Ni B centre ? ? ? 2 . 4 0 ? = max 0 2 B r Ni ? ? (ii) For single turn coil N = 1 ? r i r i B centre 2 2 . 4 0 0 ? ? ? ? ? ? (iii) In C.G.S. 1 4 0 ? ? ? ? r i B centre ? 2 ? Note : ? N B centre ? (i, r constant), i B centre ? (N, r constant), r B centre 1 ? (N, i constant) Case 2 : Ratio of B centre and B axis The ratio of magnetic field at the centre of circular coil and on it's axis is given by 3/2 2 2 1 ? ? ? ? ? ? ? ? ? ? r x B B axis centre (i) If a c B B a x 2 2 , ? ? ? a c B B a x 8 5 5 , 2 ? ? ? a c B B a x 2 / 3 2 3 , 2 ? ? ? ? ? ? ? ? ? (ii) If n B B c a ? then ) 1 ( 3 / 2 ? ? ?? n r x and if n B B c a ? then ) 1 ( 3 / 1 ? ? ?? n r x Case 3 : Magnetic field at very large/very small distance from the centre i B B Out In i B B Out In i CW In i ACW Out B P x r i O B Page 4 Magnetic Effect of Current 1 1 genius PHYSICS Oersted found that a magnetic field is established around a current carrying conductor. 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-Savartâ€™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-Savartâ€™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. The current element is taken as a vector quantity. Its direction is same as the direction of current. Current element AB = dl i 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. l d i is a current element and r is the distance of the point â€˜Pâ€™ with respect to the current element l d i . According to Biot-Savart Law, magnetic field at point â€˜Pâ€™ due to the current element l d i is given by the expression, 2 r dlsin ? i k dB ? also ? ? ? ? 2 0 sin . 4 r dl i dB B ? ? ? In C.G.S. : k = 1 ? 2 sin r idl dB ? ? Gauss In S.I. : ? ? 4 0 ? k ? 2 0 sin 4 r idl dB ? ? ? ? ? Tesla where 0 ? = Absolute permeability of air or vacuum metre Amp Wb ? ? ? ?7 10 4 ? . It's other units are metre Henry or 2 Amp N or Ampere metre Tesla ? dl i P r A B dl i Magnetic lines of forces i 2 Magnetic Effect of Current genius PHYSICS (1) Different forms of Biot-Savarts law Vector form Biot-Savarts law in terms of current density Biot-savarts law in terms of charge and it's velocity Vectorially, 3 0 2 0 ) ( 4 ) Ë† ( 4 r r l d i r r l d i B d ? ? ? ? ? ? ? ? ? ? ? ? ? ? Direction of B d ? is perpendicular to both l d ? and r Ë† . This is given by right hand screw rule. In terms of current density dV r r J B d 3 0 4 ? ? ? ? ? ? ? where dV idl Adl idl A i j ? ? ? = current density at any point of the element, dV = volume of element In terms of charge and itâ€™s velocity, 3 0 ) ( 4 r r v q B d ? ? ? ? ? ? ? ? v q dt l d q l d dt q l id ? ? ? ? ? ? ? (2) Similarities and differences between Biot-Savart law and Coulombâ€™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. 2 0 Ë† 4 r r l id B d ? ? ? ? Biot-Savart Law r r q q F Ë† 4 1 2 2 1 0 ?? ? Coulombâ€™s Law (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 l d and the unit vector r Ë† . Direction of Magnetic Field. The direction of magnetic field is determined with the help of the following simple laws : (1) Maxwellâ€™s cork screw rule 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, then the direction of folding fingers will represent the direction of magnetic lines of force. (3) Right hand thumb rule of circular currents According to this rule if the direction of current in circular B i B dl i r Ë† B r Ë† +q E Magnetic Effect of Current 3 3 genius PHYSICS 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 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 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) 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 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 3/2 2 2 2 0 ) ( 2 4 r x p N i r . µ B axis ? ? p ; where N = number of turns in coil. Different cases Case 1 : Magnetic field at the centre of the coil (i) At centre x = 0 ? r Ni B centre ? ? ? 2 . 4 0 ? = max 0 2 B r Ni ? ? (ii) For single turn coil N = 1 ? r i r i B centre 2 2 . 4 0 0 ? ? ? ? ? ? (iii) In C.G.S. 1 4 0 ? ? ? ? r i B centre ? 2 ? Note : ? N B centre ? (i, r constant), i B centre ? (N, r constant), r B centre 1 ? (N, i constant) Case 2 : Ratio of B centre and B axis The ratio of magnetic field at the centre of circular coil and on it's axis is given by 3/2 2 2 1 ? ? ? ? ? ? ? ? ? ? r x B B axis centre (i) If a c B B a x 2 2 , ? ? ? a c B B a x 8 5 5 , 2 ? ? ? a c B B a x 2 / 3 2 3 , 2 ? ? ? ? ? ? ? ? ? (ii) If n B B c a ? then ) 1 ( 3 / 2 ? ? ?? n r x and if n B B c a ? then ) 1 ( 3 / 1 ? ? ?? n r x Case 3 : Magnetic field at very large/very small distance from the centre i B B Out In i B B Out In i CW In i ACW Out B P x r i O B 4 Magnetic Effect of Current genius PHYSICS (i) If x >> r (very large distance) ? 3 0 3 2 0 2 . 4 2 . 4 x NiA x Nir B axis ? ? ? ? ? ? ? where A = ?r 2 = Area of each turn of the coil. (ii) If x << r (very small distance) ? centre axis B B ? , but by using binomial theorem and neglecting higher power of ; 2 2 r x ? ? ? ? ? ? ? ? ? ? 2 2 2 3 1 r x B B centre axis Case 4 : B-x curve The variation of magnetic field due to a circular coil as the distance x varies as shown in the figure. B varies non-linearly with distance x as shown in figure and is maximum when 0 min 2 ? ? x , i.e., the point is at the centre of the coil and it is zero at x = ? ?. Point of inflection (A and A ?) : Also known as points of curvature change or pints of zero curvature. (i) At these points B varies linearly with x ? ? dx dB constant ? 0 2 2 ? dx B d . (ii) They locates at 2 r x ? ? from the centre of the coil. (iii) Separation between point of inflextion is equal to radius of coil (r) (iv) Application of points of inflextion is "Hamholtz coils" arrangement. Note : ? The magnetic field at 2 r x ? is r Ni B 5 5 4 0 ? ? (2) Helmholtz coils (i) This is the set-up of two coaxial coils of same radius such that distance between their centres is equal to their radius. (ii) These coils are used to obtain uniform magnetic field of short range which is obtained between the coils. (iii) At axial mid point O, magnetic field is given by B R Ni R Ni B 432 . 1 716 . 0 5 5 8 0 0 ? ? ? ? ? , where R Ni B 2 0 ? ? (iv) Current direction is same in both coils otherwise this arrangement is not called Helmholtz's coil arrangement. (v) Number of points of inflextion ? Three (A, A ?, A ? ?) Note : ? The device whose working principle based on this arrangement and in which uniform magnetic field is used called as "Halmholtz galvanometer". (3) Magnetic field due to current carrying circular arc : Magnetic field at centre O A ? A B0 x = â€“ r/2 x = 0 x = r/2 a O2 O + â€“ a O1 + â€“ O x O2 O1 2 a x ? ? 2 a x ? Resultant field (Uniform) A ? A ? ? A Page 5 Magnetic Effect of Current 1 1 genius PHYSICS Oersted found that a magnetic field is established around a current carrying conductor. 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-Savartâ€™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-Savartâ€™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. The current element is taken as a vector quantity. Its direction is same as the direction of current. Current element AB = dl i 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. l d i is a current element and r is the distance of the point â€˜Pâ€™ with respect to the current element l d i . According to Biot-Savart Law, magnetic field at point â€˜Pâ€™ due to the current element l d i is given by the expression, 2 r dlsin ? i k dB ? also ? ? ? ? 2 0 sin . 4 r dl i dB B ? ? ? In C.G.S. : k = 1 ? 2 sin r idl dB ? ? Gauss In S.I. : ? ? 4 0 ? k ? 2 0 sin 4 r idl dB ? ? ? ? ? Tesla where 0 ? = Absolute permeability of air or vacuum metre Amp Wb ? ? ? ?7 10 4 ? . It's other units are metre Henry or 2 Amp N or Ampere metre Tesla ? dl i P r A B dl i Magnetic lines of forces i 2 Magnetic Effect of Current genius PHYSICS (1) Different forms of Biot-Savarts law Vector form Biot-Savarts law in terms of current density Biot-savarts law in terms of charge and it's velocity Vectorially, 3 0 2 0 ) ( 4 ) Ë† ( 4 r r l d i r r l d i B d ? ? ? ? ? ? ? ? ? ? ? ? ? ? Direction of B d ? is perpendicular to both l d ? and r Ë† . This is given by right hand screw rule. In terms of current density dV r r J B d 3 0 4 ? ? ? ? ? ? ? where dV idl Adl idl A i j ? ? ? = current density at any point of the element, dV = volume of element In terms of charge and itâ€™s velocity, 3 0 ) ( 4 r r v q B d ? ? ? ? ? ? ? ? v q dt l d q l d dt q l id ? ? ? ? ? ? ? (2) Similarities and differences between Biot-Savart law and Coulombâ€™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. 2 0 Ë† 4 r r l id B d ? ? ? ? Biot-Savart Law r r q q F Ë† 4 1 2 2 1 0 ?? ? Coulombâ€™s Law (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 l d and the unit vector r Ë† . Direction of Magnetic Field. The direction of magnetic field is determined with the help of the following simple laws : (1) Maxwellâ€™s cork screw rule 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, then the direction of folding fingers will represent the direction of magnetic lines of force. (3) Right hand thumb rule of circular currents According to this rule if the direction of current in circular B i B dl i r Ë† B r Ë† +q E Magnetic Effect of Current 3 3 genius PHYSICS 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 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 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) 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 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 3/2 2 2 2 0 ) ( 2 4 r x p N i r . µ B axis ? ? p ; where N = number of turns in coil. Different cases Case 1 : Magnetic field at the centre of the coil (i) At centre x = 0 ? r Ni B centre ? ? ? 2 . 4 0 ? = max 0 2 B r Ni ? ? (ii) For single turn coil N = 1 ? r i r i B centre 2 2 . 4 0 0 ? ? ? ? ? ? (iii) In C.G.S. 1 4 0 ? ? ? ? r i B centre ? 2 ? Note : ? N B centre ? (i, r constant), i B centre ? (N, r constant), r B centre 1 ? (N, i constant) Case 2 : Ratio of B centre and B axis The ratio of magnetic field at the centre of circular coil and on it's axis is given by 3/2 2 2 1 ? ? ? ? ? ? ? ? ? ? r x B B axis centre (i) If a c B B a x 2 2 , ? ? ? a c B B a x 8 5 5 , 2 ? ? ? a c B B a x 2 / 3 2 3 , 2 ? ? ? ? ? ? ? ? ? (ii) If n B B c a ? then ) 1 ( 3 / 2 ? ? ?? n r x and if n B B c a ? then ) 1 ( 3 / 1 ? ? ?? n r x Case 3 : Magnetic field at very large/very small distance from the centre i B B Out In i B B Out In i CW In i ACW Out B P x r i O B 4 Magnetic Effect of Current genius PHYSICS (i) If x >> r (very large distance) ? 3 0 3 2 0 2 . 4 2 . 4 x NiA x Nir B axis ? ? ? ? ? ? ? where A = ?r 2 = Area of each turn of the coil. (ii) If x << r (very small distance) ? centre axis B B ? , but by using binomial theorem and neglecting higher power of ; 2 2 r x ? ? ? ? ? ? ? ? ? ? 2 2 2 3 1 r x B B centre axis Case 4 : B-x curve The variation of magnetic field due to a circular coil as the distance x varies as shown in the figure. B varies non-linearly with distance x as shown in figure and is maximum when 0 min 2 ? ? x , i.e., the point is at the centre of the coil and it is zero at x = ? ?. Point of inflection (A and A ?) : Also known as points of curvature change or pints of zero curvature. (i) At these points B varies linearly with x ? ? dx dB constant ? 0 2 2 ? dx B d . (ii) They locates at 2 r x ? ? from the centre of the coil. (iii) Separation between point of inflextion is equal to radius of coil (r) (iv) Application of points of inflextion is "Hamholtz coils" arrangement. Note : ? The magnetic field at 2 r x ? is r Ni B 5 5 4 0 ? ? (2) Helmholtz coils (i) This is the set-up of two coaxial coils of same radius such that distance between their centres is equal to their radius. (ii) These coils are used to obtain uniform magnetic field of short range which is obtained between the coils. (iii) At axial mid point O, magnetic field is given by B R Ni R Ni B 432 . 1 716 . 0 5 5 8 0 0 ? ? ? ? ? , where R Ni B 2 0 ? ? (iv) Current direction is same in both coils otherwise this arrangement is not called Helmholtz's coil arrangement. (v) Number of points of inflextion ? Three (A, A ?, A ? ?) Note : ? The device whose working principle based on this arrangement and in which uniform magnetic field is used called as "Halmholtz galvanometer". (3) Magnetic field due to current carrying circular arc : Magnetic field at centre O A ? A B0 x = â€“ r/2 x = 0 x = r/2 a O2 O + â€“ a O1 + â€“ O x O2 O1 2 a x ? ? 2 a x ? Resultant field (Uniform) A ? A ? ? A Magnetic Effect of Current 5 5 genius PHYSICS r i r i B 4 . 4 0 0 ? ? ? ? ? ? r i B ? ? ? . 4 0 ? r i B ) 2 ( . 4 0 ? ? ? ? ? ? Special results Angle at centre Magnetic field at centre in term of B 0 360 o (2 ?) B 0 180 o ( ?) B 0 / 2 120 o (2 ?/3) B 0 / 3 90 o ( ?/2) B 0 / 4 60 o ( ?/3) B 0 / 6 30 o ( ?/6) B 0 / 12 (4) Concentric circular loops (N = 1) (i) Coplanar and concentric : It means both coils are in same plane with common centre (a) Current in same direction (b) Current in opposite direction ? ? ? ? ? ? ? ? ? ? 2 1 0 1 1 1 2 4 r r i B ? ? ? ? ? ? ? ? ? ? ? 2 1 0 2 1 1 2 4 r r i B ? ? ? Note : ? ? ? ? ? ? ? ? ? ? ? ? 1 2 1 2 2 1 r r r r B B (ii) Non-coplanar and concentric : Plane of both coils are perpendicular to each other Magnetic field at common centre 2 2 2 1 0 2 2 2 1 2 i i r B B B ? ? ? ? ? (5) Magnetic field due to a straight current carrying wire Magnetic field due to a current carrying wire at a point P which lies at a perpendicular distance r from the wire as shown is given as ) sin (sin . 4 2 1 0 ? ? ? ? ? ? r i B From figure ) 90 ( 1 ? ? ? ? o and ) 90 ( 2 ? ? ? ? o Hence ) cos (cos . 4 ? ? ? ? ? ? r i B o O r i O i r ? i r ? O i i r1 r2 i i r1 r2 i2 i1 B2 B1 ? ? If magnetic field at the centre of circular coil is denoted by B 0 ? ? ? ? ? ? ? r i ? ? ? 2 . 4 0 Magnetic field at the centre of arc which is making an angle ? at the centre is ? ? . 2 0 ? ? ? ? ? ? ? B B arc P ?2 ?1 i r Y X ? ?Read More

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