t ' = ! (t " #x / c)

x ' = ! (x " #ct)

y ' = y

z ' = z

t = ! (t '+ "x '/ c)

x = ! (x '+ "ct ')

y = y '

z = z '

! "v

c! "

1

1# $ 2 I = (c!t)2" (!x)

2" (!y)

2" (!z)

2= (c!# )

2

!F =

d!p

dt

!p = m

!v

W =

!F !i!d

!l = !E"

E = mc2= (pc)

2+ (m

0c2)2

KE = E ! m0c2

m = !m0

!p = !m

0

!v ! =

pc

E! =

E

m0c2

!µ

" #

$ %$& 0 0

%$& $ 0 0

0 0 1 0

0 0 0 1

'

(

))))

*

+

,,,,

xµ! (ct, x, y, z)

!µ"dx

µ

d#= $ u c,ux ,uy ,uz( )

pµ ! m

0"µ

=E

c, p

x, p

y, p

z

#$%

&'(

d! =dt

"u

!u=

1

1" (u / c)2

f '

f=

1! "

1+ "E! = hf

!ux=

ux" v

1" uxv / c

2

!uy,z =uy,z/ "

1# uxv / c

2

!I (t) = I

maxei(! t+" )

!VR(t) = !I (t)R

!VL(t) = !I (t) !X

L

!VC(t) = !I (t) !X

C

!E (t) = !I (t)! !Z

!XL= iX

L= i!L

!XC= !iX

C= !i

1

"C

!E(!r ) =

1

4!"0

dq

!r #!rq

!r #!rq

3$

dq = !dVq !!or!!" dAq !!or!!# dlq

!E = !

!"V

V (!r ) = !

!E "d!l

!r0

!r

#

V (!r ) =

1

4!"0

dq

|!r #!rq |

$

!B(!r ) =

µ0

4!

!I dlq "

!r #!rq

!r #!rq

3$

!! "!B = µ

0

!J + µ

0#0

$!E

$t

!! "!B = 0

!! "!E =

#

$0

!! "!E = #

$!B

$t

!F = q(

!E +!v !!B)From Physics 212

E =Q

4!"0r2

E =!

2"#0s

E =!

2"0

charged sphere(exterior)

charged ∞ line

charged ∞ sheet

Physics 225: Final Exam Formula Sheet

ei!= cos! + i sin!

B =µ0I

2! s

B = µ0nI

B = µ0NI

!J (!r ) !

d!I

dA"

I =!J !d!A

Surface

"where

∞ wire

∞ solenoid(n turns/length)

toroid of N turns

!I dlq

or!J dVq

!

"##

$##

Maxwellʼs Equations

!!f "d

!l

a

b

# = f (!rb ) $ f (

!ra )

!! "!E( ) #d

!A

Area

$ =!E #d!l

%Area"$

GGS

!! "!E !dV

Vol

# =!E "d!A

$Vol"#

f (x0) =

f(n)(x

0)

n!(x ! x

0)n

n=0

"

#

•!(1+ x)n! 1+ nx

•!sin x ! x

•!cos x ! 1"x2

2

•!ex! 1+ x

Limit x/a → ∞: • (xn+a) → xn • (x–n+a) → a for positive n …“forever and a day”

•!ln(1+ x) ! x

•!tan x ! x

•!sin!1x " x

•!tan!1x " x

•!cos!1x "

#

2! x

Integral Table

sin2! d!

0

2"

# = cos2! d!

0

2"

# = "

ei!= cos! + i sin!

imag

xφ

!z = x + iy = re

i!

r

real

y

!z* ! x " iy = re

" i#

| !z | ! !z * !z = r

cosn! !sin! !d! = "

cosn+1!

n +1#

dx

a2+ x

2=1

atan

!1 x

a

"#$

%&'(

dx

a2+ x

2

= ln x + a2+ x

2( )!dx

(a2± x

2)3/2

=x

a2a2± x

2!

x!dx

a2± x

2( )3/2

= !1

a2± x

2!x!dx

a2± x

2

= ± a2± x

2

!x!dx

a2± x

2= ±

1

2ln a

2± x

2( )!

(x ! acos") sin" !d"

(x2+ a

2 ! 2ax cos")3/2=1

x2

a ! x cos"

x2+ a

2 ! 2ax cos"#

dx

a2 ! x2

= sin!1 x

a

"#$

%&'(

a2 ! x2 dx =

x

2a2 ! x2 +

a2

2tan

!1 x

a2 ! x2

"

#$%

&'(

x2± a

2dx =

x

2x2± a

2±a2

2ln x + x

2± a

2

!

!v = (

!v ! r̂

i)!r̂

i

i=1

3

"

df (x1,..., xn ) =

!f

!xi!dxi

i=1

n

"

d!lpath =

d!l

du!du

cos!

2

"#$

%&'=

1+ cos!

2

sin!

2

"#$

%&'=

1( cos!

2

x2

a2 ! x2

dx = !x

2a2 ! x2 +

a2

2tan

!1 x

a2 ! x2

"

#$%

&'(

d!A =

!!l

!u"!!l

!v#

$%&

'(dudv

dV =!!l

!u"!!l

!v#

$%&

'()!!l

!wdudvdw

x!dx

(a ± x)2=

a

a ± x+1

2ln a ± x( )!

dx

(a ± x)2!=!!

1

a ± x!

Taylor

cosa!cosb =1

2cos(a + b) + cos(a ! b)[ ]

0

1

90°

10tan

1cos

0sin

60°45°30°0°

1

2

3

2

1

2

1

2

1

2

3

2

3

1

3

!

sin(a + b) = sina cosb + cosa sinb

cos(a + b) = cosa cosb ! sina sinb

Complex Numbers

sina!cosb =1

2sin(a + b) + sin(a ! b)[ ]

(circuit formulaeon previous page)

1st order approx for : x !1

Conceptualversion:

d!lu!"!l

"udu

d!lpath

= d!lu

d!A = d

!lu! d!lv

dV = (d

!lu! d!lv) "d!lw

!

!v !

!v "!v