physics 225: final exam formula sheet · from physics 212 b) e= q 4!"r2 e=! 2"# 0 s e=!...
TRANSCRIPT
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