Conventional Current B = In some materials current moving charges are positive: Ionic solution Holes in some materials (same charge as electron but +)

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1 Conventional Cuent In some mateials cuent moving chages ae positive: Ionic solution Holes in some mateials (same chage as electon but +) Obseving magnetic field aound coppe wie: Can we tell whethe the cuent consists of electons o positive holes? qv ˆ ev ˆ B B ( e) ( v ) ˆ The pediction of the Biot-Savat law is exactly the same in eithe case.

2 B ev ˆ ( e) v ( ) ˆ Metals: cuent consists of electons Semiconductos: n-type electons p-type positive holes Most effects ae insensitive to the sign of mobile chages: intoduce conventional cuent: I q i Conventional Cuent q nav Units: C/s A (Ampee) Andé Maie Ampèe ( )

3 Typical Mobile Electon Dift Speed Typical electon cuent in a cicuit is ~ 1 18 electons/s. What is the dift speed of an electon in a 1 mm thick coppe wie? # electons s nav n m 3 A π D ( m) m v 118 s -1 ( )( m ) m/s na 1 18 s m 3

4 The Biot-Savat Law fo Cuents Supeposition pinciple is valid ΔB i q ivi ˆ i i I q i q nav The Biot-Savat law fo a shot length of thin wie Δ B ΔB i i i qv ˆ ΔB 1 i qv ˆ ΔB naδl IΔl ˆ ΔB q ivi ˆ i i

5 Biot-Savat Law Moving chage poduces a culy magnetic field qv ˆ Single Chage: B IΔl ˆ ΔB The Biot-Savat law fo a Cuent: shot length of thin wie I q i q nav B units: T (Tesla) kg s - A -1! 1 7 T m C m/s

6 Magnetic Field of Cuent Distibutions Fou-step appoach: 1. Cut up the cuent distibution into pieces and daw ΔB. Wite an expession fo ΔB due to one piece 3. Add up the contibutions of all the pieces 4. Check the esult

7 A Long Staight Wie Step 1: Cut up the cuent distibution into pieces and daw ΔB. Oigin: cente of wie Vecto : x,,, y, x, y, Magnitude of : x + y

8 ΔB IΔl ˆ A Long Staight Wie x, y, Step : Wite an expession fo ΔB due to one piece. x, y, Unit vecto: ˆ x + y Δl : Δ l <, Δy, > ΔB field due to one piece: ΔB IΔy,1, ( x + y ) x, y, x + y

9 A Long Staight Wie ΔB IΔy,1, ( x + y ) x, y, x + y x, y,,1, Δ B x, y,,, x IΔy ( x + y ) 3,, x Δ B IxΔy ( x + y ) 3,,1 need to calculate only z component

10 A Long Staight Wie ΔB IxΔy ( x + y ) 3 Step 3: Add up the contibution of all the pieces. x, y, B Ix L / L / dy ( x + y ) 3 B Ix y x x + y L / L / B LI x x + ( L / )

11 B A Long Staight Wie LI x x + L / x, y, Special case: x<<l ( ) B B LI x ( L / ) LI ( ) x L / I B, fo x << x L What is the meaning of x?

12 B Step 4: Check esults diection B LI + ( L / ) fa away: >>L LI A Long Staight Wie IΔl 4 π B units: ( ) ( ) T T m A m A m m

13 Right-hand Rule fo Wie Conventional Cuent Diection

14 Magnetic Field of a Wie Loop Step 1: Cut up the distibution into pieces Δl Rcos( θ + dθ ), Rsin( θ + dθ ), Rcosθ, Rsinθ,,, z R cosθ, Rsinθ, Make use of symmety! Need to conside only Δ due to one dl

15 Magnetic Field of a Wie Loop Step : ΔB due to one piece Oigin: cente of loop Vecto : obs. loc souce,, z, R,, R, z Magnitude of : R + z ΔB IΔl ˆ, R, z Unit vecto: ˆ R + z Δl: Δl RΔθ,, Magnetic field due to one piece: ΔB I RΔθ,,, R, z ( R + z ) 3/

16 Magnetic Field of a Wie Loop Step : ΔB due to one piece ΔB I RΔθ,,, R, z ( R + z ) 3/ Δ B I,zRΔθ, R Δθ R + z ( ) 3/ need only z component: Δ IR Δθ ( R + z ) 3/

17 Magnetic Field of a Wie Loop Step 3: Sum the contibutions of all pieces π IR dθ ( R + z ) 3/ π dθ π π IR dθ ( R + z ) 3/ Magnetic field of a loop along its axis: π R I ( R + z ) 3/ Δ IR Δθ ( R + z ) 3/

18 Magnetic Field of a Wie Loop Step 4: Check the esults π R I ( R + z ) 3/ units: T m A ( m A) ( m ) T 3/ diection: Check seveal pieces with the ight hand ule Note: We ve not calculated o shown the est of the magnetic field

19 Magnetic Field of a Wie Loop π IR dθ ( R + z ) 3/ Δ IR Δθ ( R + z ) 3/ Magnetic field of a loop: π R I ( R + z ) 3/

20 Magnetic Field of a Wie Loop Special case: cente of the loop Δl ˆ Δl IΔl I Δl R R B ΔB Δl ˆ IΔl ˆ B I πr πi R R Using geneal fom (z) : π R I ( R + z ) 3/ π R I ( R ) 3/ π I R π I R

21 Magnetic Field of a Wie Loop Special case: fa fom the loop π R I ( R + z ) 3/ fo z>>r: ( z + R ) 3/ (z ) 3/ z 3 π R I ( z ) 3/ π R I z 3 The magnetic field of a cicula loop falls off like 1/z 3

22 A Coil of Wie What if we had a coil of wie? single loop: π R I ( R + z ) 3/ Fo N tuns: π R NI ( R + z ) 3/

23 Vectos and Pseudovectos (Axial) Compae the electic field at the cente of a unifomly chaged ing to the magnetic field in a cuentcaying ing. What do you notice about the field stength at the cente of the ing? What beaks the symmety in the cuent ing case?

24 Reflection of Cuent Ring y y x x z z B v ˆ B B Any vecto that is a esult of a coss poduct is a pseudovecto.

25 Magnetic Dipole Moment fa fom coil: magnetic dipole moment: πr I 3 z 3 z πr I AI fa fom dipole: E z p sq - vecto in the diection of B 1 p z 3 ε

26 Twisting of a Magnetic Dipole The magnetic dipole moment acts like a compass needle! In the pesence of extenal magnetic field a cuent-caying loop otates to align the magnetic dipole moment along the field B.

27 The Magnetic Field of a Ba Magnet How does the magnetic field aound a ba magnet look like? N S

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