Magnetic Field of a Wire Fundamental Laws for Calculating B-field - PowerPoint PPT Presentation

Magnetic Field of a Wire • Fundamental Laws for Calculating B-field – Biot-Savart Law (long method, but works always) – Ampere’s Law (high symmetry) • B-Field of a Straight Wire – For a thin straight conductor carrying current the magnetic field is: We expect B from wire to be proportional to I / R . Springr2004 OSU Sources of Magnetic Field 1 Chapter 28

Springr2004 OSU Sources of Magnetic Field 2 Chapter 32

Magnetic Field of a Wire • B-Field of a Straight Wire  I  0 B dl   r 2 r I • Where  T m A . 7     4 10 0 • is the Permeability of free space • Use the right hand rule to determine the direction. (Place thumb in direction of current and B Field is in direction of fingers grabbing the wire.) Springr2004 OSU Sources of Magnetic Field 3 Chapter 32

Ampere’s Law I d l r Springr2004 OSU Sources of Magnetic Field 4 Chapter 32

Today Example B-field Calculations (Application of Ampere’s Law) – Inside a Long Straight Wire – Infinite Current Sheet – Solenoid – Toroid – Biot-Savart Law – Circular Loop Springr2004 OSU Sources of Magnetic Field 5 Chapter 32

Ampere’s Law A) Evaluate the integral  dl around each loop 1) 0 3) 2  R R 2)  R 2 4) not enough information dl 1) 0 3) ab a 2) 2( a + b) 4) Not enough info b dl Springr2004 OSU Sources of Magnetic Field 6 Chapter 32

Ampere’s Law - Examples Two identical loops are placed in proximity to two identical current carrying wires. A B   3) For which loop is  B • dl the greatest? 1) A 2) B 3) Same Springr2004 OSU Sources of Magnetic Field 7 Chapter 32

C B Two identical loops are placed in proximity to two identical current carrying wires.   D) Now compare loops B and C. For which loop is  B • dl the greatest? 1) B 2) C 3) Same Springr2004 OSU Sources of Magnetic Field 8 Chapter 32

B Field Inside a Long Wire Suppose a total current I flows through the wire of radius a R into the screen as shown. r r To calculate B field as a function of r, from center of the wire: Take an amperian loop of radius r outside the wire,     using Ampere’s Law: B dl I 0 enc       2 B rd rB I 0 enc  I  0 The enclosed current is all of current through wire: B  2 r The B-field diminishes as 1/ r outside the wire Springr2004 OSU Sources of Magnetic Field 9 Chapter 28

B Field Inside a Long Wire Now B-field inside the wire: We choose an amperean loop R of radius r inside the wire: r r     B dl I 0 enc       B rd 2 rB I 0 enc But the enclosed current is a fraction of total; since current is uniform:   Ir 2 r    0 therefo e r B I I   enc 2 2 R 2 R Inside the wire the B-field is linear with r. Springr2004 OSU Sources of Magnetic Field 10 Chapter 28

B Field Inside a Long Wire Inside the wire: ( r < R ) B 1 μ I r B  0 π 2 2 R • Outside the wire:( r > R )  0 I r R B = r 2  Springr2004 OSU Sources of Magnetic Field 11 Chapter 32

B Field of  Current Sheet Consider an  sheet of current described y by n wires/length each carrying current I I x into the screen as shown. Calculate the B field. x x x What is the direction of the field? x From the Symmetry  +/  y direction w x x Calculate using Ampere's law for a x x square of side w x x          x • B d l Bw 0 Bw 0 2 Bw x I  • constant constant nwi   μ  0 ni    B  therefore, μ B d l I 0 2 Springr2004 OSU Sources of Magnetic Field 12 Chapter 32

B-Field of A Solenoid A uniform magnetic field can be produced by a solenoid A solenoid is defined by a current I flowing through a wire that is l wrapped n turns per unit length d l on a cylinder of radius a and length L .      B dl Bl NI 0  NI    0 B nI 0 l Springr2004 OSU Sources of Magnetic Field 13 Chapter 32

B-Field of A Toroid • Toroid defined by a solenoid of N total turns with current I connected at both ends. It becomes a donut shape with a coil wrapped around it. outside Toroid B = 0 d l integrating B on circle outside toroid, enclosed current = I +(- I) = 0 r       Inside the B dl B 2 r NI 0  Toroid NI  0 B  2 r Get a concentrated field inside the toroid. Springr2004 OSU Sources of Magnetic Field 14 Chapter 32

Examples Two cylindrical conductors, one solid and the other hallow in middle, each carry current I into the screen as shown. The conductor has radius R =4 a . The conductor on the right has a hole in the middle and carries current only between R = a and R =4 a . I I a 4 a 4 a At R = 5 a which conductor produces stronger B-field? 1) Left conductor 3) Both are the same 2) Right Conductor 4) both are zero Springr2004 OSU Sources of Magnetic Field 15 Chapter 32

Examples Two cylindrical conductors, one solid and the other hallow in middle, each carry current I into the screen as shown. The conductor has radius R =4 a . The conductor on the right has a hole in the middle and carries current only between R = a and R =4 a . I I a 4 a 4 a At R = 2 a which conductor produces stronger B-field? 1) Left conductor 3) Both are the same 2) Right Conductor 4) both are zero Springr2004 OSU Sources of Magnetic Field 16 Chapter 32

Examples Use Ampere’s Law in both cases by drawing a loop in the plane of the screen at R = 5 a and R = 2a . Both fields have cylindrical symmetry, so the fields are tangent to the loop at all points, thus the field at R =5 a only depends on current enclosed I I enc = I in both cases I a 4 a Field only depends on enclosed current 4 a Springr2004 OSU Sources of Magnetic Field 17 Chapter 32

Examples A current carrying wire is wrapped around an iron core, forming an electro-magnet. Which direction does the magnetic field point inside the iron core? 1) left 4) right 2) up 5) down 3) out of the screen 6) into the screen Which side of the solenoid should be labeled as the magnetic north pole? 1) left 3) right 2) up 4) down Springr2004 OSU Sources of Magnetic Field 18 Chapter 32

B Field Inside a Long Wire Inside the wire: ( r < R ) R 1 μ I r B  0 π 2 2 R • Outside the wire:( r > R )  r 0 I B = r 2  Springr2004 OSU Sources of Magnetic Field 19 Chapter 32

B Field of  Current Sheet Consider an  sheet of current described y by n wires/length each carrying current I I x into the screen as shown. Calculate the B field. x x x What is the direction of the field? x From the Symmetry  +/  y direction w x x Calculate using Ampere's law for a x x square of side w x x          x • B d l Bw 0 Bw 0 2 Bw x I  • constant constant nwi   μ  0 ni    B  therefore, μ B d l I 0 2 Springr2004 OSU Sources of Magnetic Field 20 Chapter 28