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Transcript of Review&for&Exam&III · PDF file Review&for&Exam&III...

  • Basic  Laws

    Basic  Definitions

    Important  Derivations

    Examples

    + B dl I• =∫ µ0

    U B= − •! !

    µ

    dF Idl B ! ! ! = ×

    R mv qB

    =

    Review  for  Exam  III “Leftovers” RC  circuits

  • Right-­Hand-­Rules BvqF !!!

    ×= BLIF !!!

    ×=

    Fr !!

    ×=τ B !!

    ×= µτ

    2 0 ˆ 4 r

    rsdIBd ×= !!

    π µ

  • Basic  Laws • Biot-­Savart Law:

    Direct  calculation  of  B  from  currents. dB I dl r

    r

    ! ! !

    = ×µ

    π 0

    34

    x

    Rr θ

    θ

    P

    Idx

    RHR

    We  did  the  integral  but  it  is  messy  and  takes  time  …  so

  • Basic  Laws • Biot-­Savart Law:

    direct  calculation  of  B  from  currents.

    • Ampere's  Law: calculation  of  B  in  cases  of  high  symmetry

    dB I dl r r

    ! ! !

    = ×µ

    π 0

    34

    B dl I• =∫ µ0

    ´

    If  B  “constant”  along  path then  integral  is  just  the  path

    B I R

    = µ π 0

    2

  • Basic  Laws • Faraday's  Law:

    determines  the  induced  E  field   (or  emf)  from  a  changing   magnetic  flux.

    ε = − d dt BΦ

    ! ! E dl d

    dt B•∫ = −

    Φ

    dS

    B BΦB B dS≡ •∫ ! !

    Define  Flux

    Basic  question:    Does  it  change  in  time  (ie,  increase  or  decrease?) You  will  have  to  figure  that  out

  • Lenz's  Law The  induced  current  will  appear  in  such  a  direction  that  it opposes  the  change  in  flux that  produced  it.

    v B

    S N v

    B N S

  • Change Area of loop Change magnetic fieldthrough loop Change orientation of loop relative to B

    EMF  Observed  when  dΦ/dt is  non  zero Direction  from  Lenz’s  Law

    Examples

  • BvqEqF !!!!

    ×+=

    Definitions  &  Derivations

    • Lorentz  Force:

    Force  in  an  electric  field Force  in  a  magnetic  field

    RHR

    E x    x    x    x    x    x    x    x    x x    x    x    x    x    x    x    x    x x    x    x    x    x    x    x    x    x x    x    x    x    x    x    x    x    x

    Examples  with  Magnetic  Field  only  and  a  Moving charge:

  • Magnetic  Dipole  Moment

    Bx

    .F Fθ

    θ

    µ

    Direction:  ⊥ to  plane  of  the  loop   in  the  direction  the  thumb  of  right   hand  points  if  fingers  curl  in   direction  of  current.

    • Torque  on  loop  is  then:

    • Note:  if  loop  consists  of  N  turns,  µ =  NAI

    Magnitude: µ = AI

    τ = AIB sinθ Þ ! ! ! τ µ= × B

    Remember  this:    The  torque  always  wants  to  line  µ up  with  B!

    Definitions  &  Derivations

    RHR

    I

  • Potential  Energy  of  Dipole

    Bx

    .F Fθ

    θ

    µ

    • Work  is  required  to  change  the   orientation  of  a  magnetic  dipole  in   the  presence  of  a  magnetic  field.

    • Define  potential  energy  U  (with  zero  at   position  of  max  torque) corresponding   to  this  work.

    ∫≡ θτdU Þ BU !!

    •−= µ

    Remember  this:    The  torque  always  wants  to  line  µ up  with  B … which minimizes the potential energy!

  • Examples  with  RC  circuits

    • Charging  a  capacitor:

    • Discharging  a  capacitor: R

    I I

    C ε

    a

    b + + -­ -­

    R I I

    a

    b ε = +Rdq

    dt q C

    Rdq dt

    q C

    + = 0

    Loop  rule  !

  • Charging                            Discharging RC

    t

    2RC

    0

    0 1 2 3 4 0

    0.5

    1

    t/RC

    Q f( )x

    x

    -­ε/R

    I

    t

    q

    0

    0 1 2 3 4

    0.5

    11

    0.0183156

    f( )x

    40 x

    q = C εe -t/RC

    I dq dt R

    e t RC= = − −ε /

    RC

    t

    q

    2RC

    0 0 1 2 3 4

    0

    0.5

    1

    t/RC

    Q f( )x

    x

    I

    0 t

    ε/R

    0 1 2 3 4

    0.5

    11

    0.0183156

    f( )x

    40 x

    ( )q C e t RC= − −ε 1 /

    I dq dt R

    e t RC= = −ε /

  • Examples

    • Orbit  of  charged  particle  in   uniform  magnetic  field: x    x    x    x    x    x

    x    x    x    x    x    x x    x    x    x    x    x

    v

    F

    B

    q

    x    x    x    x    x    x x    x    x    x    x    x x    x    x    x    x    x

    Fv R

    R mv qB

    =

  • Examples

    • B  for  straight  wire: × R

    (Ampere)

    • B  for  ¥ current  sheet:

    • B  for  ¥ solenoid: xxx xx •• • ••

    i i

    x x x x

    x

    x x

    x x x x

    i

    B I R

    = µ π 0

    2

    (Biot-­Savart)

    x

    Rr θ

    θ

    P

    Idx

    B = µ0 ni

    2

    B = µ0 ni

  • Examples:  B  from  long  straight  wire

    B = µ0 I 2 π

    r a 2

    • Inside  the  wire:  (r  <  a)

    • Outside  the  wire:  (r>a)

    B = µ0 I 2πr

  • Examples

    • Force  on  parallel  current-­carrying  conductors  of  length  L:

    • Currents  in  oppositedirections   Þ⇒ repulsive  force.

    • Currents  in  same  direction   Þ attractive  force.

    πd2 LIIµBLIF ba0abb =×=

    !!! L d × F

    Ib

    Ia ×

    F

    • Force  on  current  carrying  wire  in  a  magnetic  field:

    dF Idl B ! ! ! = × RHR

  • Examples • Current  induced  by  pulling  coil  through  magnetic  field:

    R LBv

    dt d

    RR I =Φ== 1εIs  the  flux  changing?

    If  so,  is  it  increasing?  Decreasing? How  can  you  express  this  quantitatively?

    The  current  through  this  bar  results  in  a  force  on  the  bar  

    RILBv R vBLFvP 2=⎟

    ⎠ ⎞

    ⎜ ⎝ ⎛==

    BLIF !!!

    ×=