Q free on inner surface - Q free on inner surface Interior points electric field must be zero - q bound + q bound Symmetry – fields must be uniform – field lines perpendicular to plates

Q free on inner surface - Q free on inner surface plate separation d area of plates A

conductordielectric Gauss’s Law

frequency dielectric Constant (polar molecules)

dy F F me

Electric displacement Electric field Polarization

A B C only some of the windings are shown Integration paths

L dA 3 BzBz dA 1 BzBz dA 2 BrBr

Z Y X

Bz1Bz1 s Bz2Bz2 x I B r = 0 A I enclosed = 0

s Bz2Bz2 x I B r = 0 C I enclosed = n s I B z1 = 0 I enclosed = 0 x I Bz2Bz2 B z1 = 0 x x xx

B single turn of wire with current I around integration loop B dr = 0 and B r = 0 outside loop B z = 0

B Fe H Fe B gap H gap B air H air i coil windings gap region iron core

XXXXXXXXXXXX Circulation loop: square of length L Cross-section through electromagnet Current i out of page Current i into page

width L thickness t area A q = - e electrons are the charge carriers in copper

+ -

dy F +q+q -q-q

x L-x V rr C = C A + C B C CACA CBCB

Induced dipole moment – helium atom -e +2e Zero electric field – helium atom symmetric  zero dipole moment -e +2e -e A B effectively charge +2e at A and -2e at B dipole moment p = 2 e d

Induced dipole moment – sulfur atom -8e +16e Zero electric field – helium atom symmetric  zero dipole moment -8e +16e -8e A B effectively charge +16e at A and -16e at B dipole moment p = 16 e d

-q-q +q+q r 1  r – (d/2)cos  r 2  r + (d/2)cos  r  P ErEr EE (d/2)cos 

f+f -f-f     dA -b-b +b+b

+q+q -q-q

+f+f -b-b +b+b -  f O r S

+ dd  r Pcos  surface S Area of the shaded ring between  and  + d   Width of ring r d  Radius of ring r sin 

+  element of charge dq e electric field at O due to charge dq e E0E0 E 0 cos 

a +Ze a d d << a

F F F d +Q+Q - Q 

0 π/2 π 0 + p E - p E  U

+ - U = - p E Lowest energy state + - U = U = + p E highest energy state  = 0  = 180 o  = 90 o

1/T  r - 1

T PoPo

p E / k Tp E / k T slope = 1/3

non-conducting liquid air conducting sphere q a Gaussian surface S r Symmetry  field lines must be radial

non-conducting liquid air conducting sphere q Symmetry  E airt = E liquidt  E air = E liquid = E E airt E liquidt

field lines of E field lines of D +

field lines of E field lines of D greater concentration of charge on surface bounded by liquid

+ - induced dipoles due to shift in electron cloud rotation orientation of polar molecules - + shift in atoms due to ionic nature of bond

NS H Fe H air Circulation loop: square side L 5 6

B-field lines – form continuous loops Gauss’s Law for magnetism Cylindrical Gaussian surface

Bound surface currents i m (right hand screw rule)  N pole imim

un-magnetized piece of iron N Bar magnet bought near un-magnetized piece of iron N N  Bar magnet will attract the iron that was initially un-magnetized north pole attracts south pole

Fe ramp Cu ramp plastic ramp N N N

Circulation loop for circulation integration used in applying Ampere’s Law N N H iron H air

d I I B H (0,0) B d

B, H gap M gap = 0 B = B gap = B iron H iron M iron B B, H gap M gap = 0 B = B gap = B iron H iron M iron B PERMANENT MAGNET ELECTROMAGNET

X Y Z thickness t width w area A = w t magnetic field in Z direction current in X direction Schematic diagram of a Hall Probe

I X Y. Z direction out of page charge carriers electrons (-) eg wire, N-type semiconductor charge carriers positive (+) eg holes in P-type semiconductor + _ VHVH VHVH width w

I area A length L + _ V resistance R resistivity  conductivity  number density n _ v electron

X Y Z object image electron beam A

+Y +X +Z BzBz ByBy vyvy FxFx Electron at A moving parallel to +Y-axis Electron acted upon by the radial component of the magnetic field  force on electron in +X direction  +X- component to the velocity axis for the motion of the electron beam radial component of magnetic field due to B z

+Y +X +Z BzBz ByBy vxvx FyFy Electron at B has a velocity component in the +X direction Electron acted upon by the axial component of the magnetic field B y  force on electron in -Z direction i.e. towards to axis  focusing action axis for the motion of the electron beam radial component of magnetic field FzFz due to B y due to B z

i free

external magnetic field

Electrostatic capacitor Electrolytic capacitor

Electrostatic capacitor Electrolytic capacitor

Electrochemical double layer capacitor conductive electrode conductive electrode separator activated carbon d

Electric Field

Zero applied stress Compressive stress Induces a voltage Applied voltage produces An expansion + -

Ferroelectric material Antiferroelectric material

Q on inner surface - Q on inner surface Interior points electric field must be zero Symmetry – electric field must be uniform – electric field lines perpendicular to conductive plates Q on outer surface Interior points electric field must be zero

Q on inner surface - Q on outer surface Interior points electric field must be zero Symmetry – fields must be uniform – field lines perpendicular to plates Interior points electric field must be zero Q on inner surface + Q on outer surface

+V+V +q+q +q+q +q+q -q-q -q-q -q-q Electric field between Adjacent plates

... Series branch V Capacitors in series (charge on each plate) Capacitors in parallel (voltage across each capacitor is the same)

V Capacitors in parallel V  +Q 1 +Q 2 -Q 2 -Q 1 Q =Q 1 +Q 2 Capacitors in series C1C1 C2C2 C eq = C 1 +C 2 V C1C1 C2C2 +Q -Q V Q 1/C eq = 1/C 1 +1/C 2

fuel air w h l fuel

- + +Q+Q r Induced dipole

5a5a Slab 1 Slab 2 a a a a a

S1S1 S2S2 S3S3 S4S4

+ Q - Q + Q b1 - Q b1 + Q b2 - Q b2 C1C1 C2C2 + Q - Q Capacitors in series

E = 0

V1V1 +Q/2 - Q/2 C 1 = Q / 2V 1 Q = 2 C 1 V 1 V2V2 +q A +q B - q B - q A C1C1 C1C1 C1C1 q A = C 2 V 2 =  r C 1 V 2 q B = C 1 V 2 Q = q A + q B = C 1 V 2 (  r + 1) = 2 C 1 V 1 V 2 = 2 V 1 / (  r + 1) q A = 2 C 1 V 1  r / (  r + 1) q B = 2 C 1 V 1 / (  r + 1) C2C2

+Qf+Qf -Qf-Qf

+q - q rr d t

Dielectric is neutral Homogenous dielectric – uniformly polarized The electrical field is reduced in the dielectric material

Flat plate L = 1 Max polarization Thin long rod L = 0 Zero polarization Sphere L = 1/3 Concentration of charges At surface given by

- Q + Q

+Q -Q +Q b1 -Q b1 +Q b2 -Q b2 E1E1 E2E2

R1R1 R2R2