Formula Sheet
Chapter 2
a=
∆x = x2 − x1
d2 t
dv
= 2
dt
dt
v = v0 + at
vavg
∆x
x2 − x1
=
=
∆t
t2 − t1
savg =
1
x = x0 + v0 t + at2
2
total distance
∆t
v 2 − v02 = 2a(x − x0 )
∆x
dx
=
∆t→0 ∆t
dt
1
x = x0 + (v0 + v)t
2
v = lim
aavg =
∆v
∆t
1
x = x0 + vt − at2
2
Chapter 3
rx = ax + bx
~a + ~b = ~b + ~a
a=
q
a2x + a2y
& ay = a sin θ
&
tan θ =
ay
ax
~a = ax î + ay ĵ + az k̂
c = ab sin θ
~a · ~b = (ax î + ay ĵ + az k̂) · (bx î + by ĵ + bz k̂)
~a × ~b = (ax î + ay ĵ + az k̂) × (bx î + by ĵ + bz k̂)
Chapter 4
~v =
~r = xî + y ĵ + z k̂
∆~r = ~r2 − ~r1
∆~r = (x2 − x1 )î + (y2 − y1 )ĵ + (z2 − z1 )k̂
d~r
dt
~v = vx î + vy ĵ + vz k̂ =
~aavg =
dx
dy
dz
î + ĵ + k̂
dt
dt
dt
∆~v
~v2 − ~v1
=
∆t
∆t
= ∆xî + ∆y ĵ + ∆z k̂
~a =
∆~r
∆t
~a = ax î + ay ĵ + az k̂ =
~vavg =
rz = az + bz
~a · ~b = ab cos φ
(~a + ~b) + ~c = ~a + (~b + ~c)
ax = a cos θ
ry = ay + by
d~v
dt
dvy
dvx
dvz
î +
ĵ +
k̂
dt
dt
dt
Chapter 5
Fg = W = mg
F~net = m~a, (1 N = 1 kg · m/s2 )
Fnet,x = max
Fnet,y = may
F~BC = −F~CB
Fnet,z = maz
Chapter 7
Wg = mgd cos φ
1
K = mv 2
2
∆K = Kf − Ki = Wa + Wg
W = F d cos φ = F~ · d~
Pavg =
∆K = Kf − Ki = W
W
dW
−→ P =
∆t
dt
P = F v cos φ = F~ · ~v
Kf = Ki + W
Chapter 8
F (x) = −
∆U = −W
dU (x)
dx
K(x) = Emec − U (x)
Emec = K + U
W = ∆Emec = ∆K + ∆U
K2 + U2 = K1 + U1
Pavg =
∆Emec = ∆K + ∆U = 0
∆E
dE
−→ P =
∆t
dt
Chapter 16
y 0 (x, t) = y1 (x, t) + y2 (x, t)
y(x, t) = ym sin(kx − ωt)
k=
2π
,
λ
v=
ω
1
=f =
2π
T
ω
λ
= = λf
k
T
y1 (x, t) = ym sin(kx − ωt)
y2 (x, t) = ym sin(kx − ωt + φ)
−→ y 0 (x, t) = 2ym cos 21 φ sin(kx − ωt + 12 φ)
y(x, t) = h(kx ± ωt)
r
v=
y 0 (x, t) = [2ym sin kx] cos ωt
τ
µ
1
2
Pavg = µvω 2 ym
2
f=
v
v
=n ,
λ
2L
for n = 1, 2, 3, · · ·
Chapter 17
s
v=
r
v = 331
I=
B
ρ
T
m/s −→ v = 343 m/s (air, 20 ◦ C)
273 K
∆p
B=−
∆V /V
I
, where I0 = 10−12 W/m2
I0
β = (10 dB) log
f=
f=
Ps
4πr2
v
nv
=
,
λ
2L
n = 1, 2, 3, · · ·
v
v
nv
= =
,
λ
λ
4L
n = 1, 3, 5, · · ·
fbeat = f1 − f2
s = sm cos(kx − ωt)
∆p = ∆pm sin(kx − ωt)
∆pm = (vρω)sm = Bksm
I=
s1 = sm cos ω1 t & s2 = sm cos ω2 t
s(t) = 2sm cos ω 0 t cos ωt
1
1
ω 0 = (ω1 − ω2 ) & ω = (ω1 + ω2 )
2
2
1
P
= ρvω 2 s2m
A
2
f0 = f
Chapter 21
k=
i=
F =
dq
dt
1 |q1 q2 |
4π0 r2
N · m2
1
= 8.99 × 109
4π0
C2
0 = 8.85 × 10−12
E=
1 |q|
4π0 r2
E=
1 p
2π0 z 3
E=
E=
1 2λ sin θ0
4π0
r
1
Q
q
2
4π0
z z 2 + L4
σ
z
E=
1− √
20
z 2 + R2
E=
~
F~ = q E
1 p
4π0 r3
~
~τ = p~ × E
1
Qz
4π0 (z 2 + R2 )3/2
~
U = −~
p·E
E=
C2
N · m2
e = 1.602 × 10−19 C
Chapter 22
~
~ = F
E
q0
v ± vD
v ± vS
Chapter 23
~ · dA
~ −→ ΦE =
dΦE = E
I
Z
E=
~ · dA
~
E
σ
20
E=
~ · dA
~ = qenc
E
0
σ
E=
0
E=
Chapter 24
V =
λ
2π0 r
V =
U
−W∞
=
q0
q0
n
X
q
4π0 R3
Vi =
r
n
1 X qi
4π0
ri
i=1
i=1
1 p cos θ
4π0 r2

R
1

Z

4π0 R
1
dq
1
V =
−→ 4π

4π0
r
 10 R
V =
U = qV
∆U = q∆V = q(Vf − Vi )
4π0
∆K = −q∆V
1/2 #
L + L2 + d2
λ
ln
V =
4π0
d
σ p 2
V =
z + R2 − z
20
"
∆K = −q∆V + Wapp
Wapp = q∆V
E=−
Z
f
∆V = −E∆x −→ Vf − Vi = −
~ · d~s
E
i
V =
λds
r
σda
r
ρdv
r
1 q
4π0 r
Ex = −
∆V
∂V
−→ Es = −
∆s
∂s
∂V
∂V
∂V
, Ey = −
, Ez = −
∂x
∂y
∂z
U =W =
1 q1 q2
4π0 r
Chapter 25
n
X 1
1
1
1
1
=
+
··· +
=
Ceq
C1 C2
Cn
Ci
q = CV
i=1
C=
0 A
d
C = 2π0
U=
L
ln
b
a
1
u = 0 E 2
2
ab
C = 4π0
b−a
C = κCair
C = 4π0 R
Ceq = C1 + C2 + · · · + Cn =
q2
1
1
= CV 2 = qV
2C
2
2
n
X
i=1
Ci
E=
Eair
κ
Chapter 26
i=
∆q
dq
−→ i =
dt
∆t
ρ=
1
E
=
σ
J
~ = ρJ~
E
iin = iout
R=ρ
Z
i=
~ −→ J = i
J~ · dA
A
J~ = ne~vd
R=
V
i
L
A
ρ − ρ0 = ρ0 α(T − T0 )
P = iV
P = i2 R =
V2
R
Chapter 27
E=
dW
dq
Req = R1 + R2 + · · · Rn =
n
X
Ri
i=1
n
Pemf = iE
X 1
1
1
1
1
=
+
··· +
=
Req
R1 R2
Rn
Ri
i=1
Chapter 28
~
F~B = q ~v × B
~ ×B
~
F~B = iL
FB = iLB sin φ
FB = |q|vB sin φ
E
v=
B
n=
iB
eV l
~ ×B
~
dF~B = idL
τ = N iAB sin θ
µ
~ = N iA~n
V = vBd
µ = N iA
|q|vB =
~
~τ = µ
~ ×B
mv
|q|B
~
U (θ) = −~
µ·B
ω
1
|q|B
= =
2π
T
2πm
Wa = ∆U = Uf − Ui
r=
f=
mv 2
r
Chapter 29
I
~ = µ0 id~s × r̂
dB
4π r2
~ · d~s = µ0 ienc
B
B=
µ0 = 4π × 10−7 T · m/A ≈ 1.26 × 10−6 T · m/A
Fba
B=
µ0 i
2πR
B=
µ0 iφ
4πR
µ0 Lia ib
= ib LBa sin 90 =
2πd
◦
ΦB =
B=
B (z) =
2 (R2 + z 2 )3/2
E=
dΦB
dt
~ · d~s = − dΦB
E
dt
≈
µ0 iR2
2z 3
N ΦB
i
L
= µ0 n 2 A
l
EL = −L
di
dt
1
UB = Li2
2
B 2 L2 v
B 2 L2 v 2
=⇒ P =
R
R
I
µ0 iN 1
2π r
µ0 iR2
~ · dA
~ −→ ΦB = BA
B
ΦB = BLx =⇒ E = BLv
F =
r
~
~ (z) = µ0 µ
B
2π z 3
L=
E = −N
B = µ0 in
Chapter 30
Z
µ0 i
2πR2
uB =
E2 = −M
B2
2µ0
di2
di1
& E1 = −M
dt
dt
Chapter 31 & 32
I
1
ω=√
LC
I
~ · dA
~ = qenc
E
0
I
~ · dA
~=0
B
I
~ · d~s = − dΦB
E
dt
~ · d~s = µ0 0 dΦE + µ0 ienc
B
dt
id = 0
dΦE
dt
Chapter 33
∆p =
E = Em sin (kx − ωt)
B = Bm sin (kx − ωt)
c=
1
E
≈ 3.00 × 108 m/s
=√
B
0 µ0
~ ×B
~
~= 1E
S
µ0
I=
2
1 Em
1 2
E
=
cµ0 2
cµ0 rms
I=
∆p =
Ps
4πr2
∆U
I
=⇒ pr =
c
c
2I
2∆U
=⇒ pr =
c
c
1
I = I0
2
I = I0 cos2 θ
θ1 = θ10
n1 sin θ1 = n2 sin θ2
sin θc =
n2
n1
tan θB =
n2
n1
Chapter 34
m=−
i = −p
1 1
1
2
+ = =
p
i
f
r
n1 n2
n2 − n1
+
=
p
i
r
1 1
1
1
1
+ = = (n − 1)
−
p
i
f
r1 r2
|m| =
h0
h
i
p
M = m1 m2
mθ =
25 cm
f
M = mmθ = −
mθ = −
s 25 cm
fob fey
fob
fey