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### Chap-03 - Magnetic Effects of Current.docx

1. TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 Magnetic Effects of Current Biot - Savart law and its application to current carrying circular loop. Ampere’s C H A P T E R 3 Syllabus law and its applications to infinitely long current carrying straight wire and solenoid. Force on a moving charge in uniform magnetic and electric fields. Cyclotron. Force on a current-carrying conductor in a uniform magnetic field. Force between two parallel current-carrying conductors-definition of ampere. Torque experienced by a current loop in uniform magnetic field; Moving coil galvanometer, its current sensitivity and conversion to ammeter and voltmeter BIOT-SAVART LAW Magnetic field due to current carrying wire is given by ⮿ dl i dB  0 idl sin       CHAPTER COVERS :  Biot-Savart Law  Straight Current Carrying Conductor  Ampere Circuital Law  Circular Loop  Solenoid 4 r 2 dB  0 i (dl  r )  Motion of Charged 4 r 3 Magnetic Field Due to Straight Current Carrying Wire Magnetic field at P Particle in Magnetic Field  Force and Torque on Current Carrying Loop B  0i (cos   cos  ) 4r 1 2 i or B  0i (sin   sin) 4r  Moving Coil P Galvanometer  Conversion to Ammeter and 1. For an infinite long wire Voltmeter 1 = 2 = 0 or      , B  2 0i 2π r ‘r’ 2. For semi infinite long wire 1 = 0°, 2 = 90°, B  0i 4r dB P  r 2 r   1
2. JEE main Magnetic Effects of Current TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 r  r   2  R I x B (towards right) B  0 2I  R2 0 4 (R2  x2 )3 / 2 4 (R2  x2 )3 / 2  2M CIRCULAR LOOP 1. At the centre of a current loop dB  0idl sin90 4r 2 dl B  0i dl 4r i  0i  2r 4r 2 B  0i 2r (Outward) (Outward field) (Inward field) 2. On the axis of a loop For x >> R, B  0 2M [Current carrying loop acts as an magnetic dipole] 4 x3 where M = I × R2 is called magnetic moment M  IA units (A-m2 ) Applications outwards for anticlockwise current inwards for clockwise current I I 1. 2. P r P B  0I [sin ] 0 4r B  0I [sin   sin] P 4r i i
3. Magnetic Effects of Current JEE main TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 b a O O i    r1 O r2 r a b P 3. I 4.I B  0I P 4r [sin  1] B0   80I 4r a2  b2 ab when a = b B0  8 2 0I 4a i 5. 6. r i i O 0i Magnetic field at O B0  4r B  3 0i 0 a i1 7. B     O 0i  1  1  8. O i 0   4    i i/3 9.i 10.10. i2 B0 = 0 i r O r i At the centre of cube, B = 0 At O, B = 0 i1 i2 11.11. I 12.12. At O, B = 0, such that  I  r1  i1 i1  i2 r;r2  i2r i1  i2 B  0  0 2r 2 to  r  i i b a  r O
4. JEE main Magnetic Effects of Current TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 r O I R dl r B   I I 13.I 14. B  0I  0I B  0I 0 2r 4r 0 4r AMPERE CIRCUITAL LAW It states that the line integral of magnetic field over a closed path is equal 0 the times the algebraic sum of the current threading the closed path in free space. Ampere's circuital law has the form B.dl  0 ienc Upward Current i1 Inward Current i2 Here B.dl implies the integration of scalar dl B product B.dl around a closed loop called an Amperian loop. The current ienc is the net current encircled by the loop. Applications of Ampere's Circuital Law (1) Magnetic field due to a long thin current carrying wire  B .d i i   0 ( i1  i 2 ) or B  0i 2r dl Amperian loop B B (2) Magnetic field inside a long straight current carrying conductor Amperian loop Amperian loop  B  0ir 2R2 i.e. Bin  r r O I I i i r dl , B R
5. Magnetic Effects of Current JEE main TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 F  B v Graphical variation of magnetic field  –0i ˆ At a point in x-y plane, B  K for | x |  R 2x  –0ix ˆ x-axis B  2R2 K for | x |  R B x SOLENOID A long solenoid having number of turns/length ‘n’ carries a current I. I P The magnetic field B is given by, BP = 0nI (in between) Bend  0nI 2 (near one end) Force on a Moving Charge in Magnetic Field F  q (v  B) F = qvB sin F is perpendicular to both v & B . Note : If there are electric field E and magnetic field B both present in a region, then Lorentz force is given by F  q [E  (v  B)] .When E  (v  B) , net force because zero. In this situation, the particle can pass undeviated through the region. i y-axis – R R
6. JEE main Magnetic Effects of Current TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 × × × r × F × F O × ×F × MOTION OF CHARGED PARTICLE IN A UNIFORM MAGNETIC FIELD B. Case 1 : v  B × v × v × × × × F  qvB  Results mv2   r r  mv  qB × 2mE qB v × (E is kinetic energy) 1. Speed is constant 2. Kinetic energy is constant 3. Work done by the magnetic force is zero 4. Velocity and momentum change continuously in direction, not in magnitude. 5. T  2r v  2m qB (Independent of speed and radius) 6. Frequency f  qB 2m (Cyclotron frequency) Case 2 : v is not perpendicular to B . Let be the ‘’ is angle between v and B . v The particle moves in a helical path such that r  mv sin qB ,T    2m qB , Pitch = v cos  × T v sin   v cos   r B pitch Case 3 : If charge particle is moving parallel or antiparallel to field B then force is zero and it moves in a straight line Applications mv 1. r  qB 2r O Time spent in magnetic field v t  m qB 2. r  mv qB Time spent in magnetic field O t  2m qB v r q, v  q × × × × × × ×  × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×
7. Magnetic Effects of Current JEE main TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 3. r  mv ,t  (2  2) m qB qB 4. To prevent the charge from hitting the screen, d > r where ‘r’ is radius of circular path Screen i.e., d  mv qB or B  mv qd 5. sin  d  dqB × × × r mv Deflection y = r – r cos  = r (1 – cos ) O × × × v  r × × × × × × y q  Cyclotron v d × × × The cyclotron is a machine used to accelerate charge particle (like proton, deutron, -particle). It uses both electric & magnetic field. Speed of particle is increased by electric field. Cyclotron frequency  qB 2m q2 B2 R2 Kmax  max 2m To accelerate electron Betatron and Synchrom are used. Force on a current carrying conductor in uniform field B Bsin  F  IBl sin   F  I ( l  B) q, v  v r 2 O O × × × r × × × × r × × × × × q v × d × × I 
8. JEE main Magnetic Effects of Current TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 × × I × × O R BI(2R) × × I T × O × R × × × × × F F Attraction r ×   Some Important Cases × × × × L × × × F = BIl F = BI(2R) × (1) i × × × (2) × × × l I × (3) (4) × × L × × × × i × × × × × × × × (F      2iBL ) × × × × × BI(2R) × Net F = 0 T = BIR (tension) Torque on a Current Carrying Loop B Case 1 :M  IA Here M  B   M  B  MB (maximum) Potential energy U   M . B  0 Case 2 : Here M || B (a)   M  B  0 (minimum) Potential energy = –MB (minimum) (a) If  = 0° (Stable equilibrium) (b) If  = 180° Potential energy = MB (maximum)  Unstable equilibrium (c) Here angle between M and B is   = MBsin, U = –MBcos Work done in rotating from 1 to 2 is W = MB (cos1 – cos2) FORCE BETWEEN TWO CURRENT CARRYING WIRES F (force/unit length) is given by B M (c) F  0I1I2 . 2r The force on a segment of length ‘l’ is  0I1I2  l 2r to    I1 I2 to       Repulsion (b) M I  B I M I I I1 I2
9. Magnetic Effects of Current JEE main TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 i ig G i – ig S   MOVING COIL GALVANOMETER Principle When a current carrying coil is placed in a magnetic field, torque acts on it. In moving coil galvanometer radial field is used which is obtained from magnet having concave shape poles. In this type of field plane of the coil is always parallel to the magnetic field so maximum torque acts on it.  = NIAB [ = 90° due to radial field] = C where  is angle formed by the pointer and C is restoring torque/twist in the suspension wire. I  C ,   NBA current sensitivity NBA I C current sensitivity   NBA I C voltage sensitivity    V IR  NBA CR Conversion to Ammeter A small resistance called shunt S is connected in parallel with the galvanometer. ig  Full scale deflection current ig × G = (i – ig) S i   S  i   S  G    S  G  i   Ammeter  1  ig  N S Cylinder Scale Pointer Coil Soft iron core g
10. JEE main Magnetic Effects of Current TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 g Conversion to Voltmeter A high resistance R is connected in series with the galvanometer. i R Voltmeter V = ig (G + R) G R  V  G ig V Some Important Points : 1. An ideal ammeter has zero resistance. 2. An ideal voltmeter has infinite resistance. 3. An ammeter is connected in series in an electric circuit 4. A voltmeter is connected in parallel across the device whose potential difference is to be measured.           
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