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

Cε

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

Cε

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

Cε

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 !!!

×=