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GRE Physics

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1. Energy in terms of partition function
U = t^2 d/dt (logZ)
L = L_0 Sqrt[1-v^2/c^2]
PdV +dU
I_z = I_x + I_y (think hoop symmetry)

2. SR: Spacetime Interval
E = Vmin : circle - E = 0 : parabola - E<0 : el - E>0 : h
Infinitely close to equilibrium at all times
Z = ?g_i*exp(-E/kT)
ds² = (c*dt)² - ?(x_i)²

3. Lab: Precision of Measurements
Z_C + Z_L = 0. Occurs when ?=1/Sqrt[L C]
(3/2) n R ?t
1s² - 2s² 2p6 - 3s² 3p6 3d¹°
Measurements close to mean

4. Addition of relativistic velocities

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5. Thin Film Theory: Constructive / Destructive Interference
?? = h/mc * (1-cos(?))
Const: 2t = (n +.5)? Destructive 2t = n?
T = I?²/2
F = s * T4

6. Doppler Shift in Frequency
P +1/2 ? v² + ?gh = Constant
µ0 I / 2pR
N²/Z (m_elec/m_red)
F = f* (c+v_r)/(c+v_s)

7. Heat added
I_z = I_x + I_y (think hoop symmetry)
NC?T
F = mv²/r
Cos[?] Sin[?] -Sin[?] Cos[?]

8. EM: Method of Images
C = 4pe0 ab/(a-b) = inner and outer radii
Opposing charge induced upon conductor
?? = h/mc * (1-cos(?))
Measurements close to mean

9. Quant: Eigenvalue of Hermitian Operator
Exponential - E = -ma²/2hbar² - a is strength of delta wellt
<?|O|?>
Always Real
?_max = b/T

10. Quant: Expectation Value
L = T - V dL/dq = d/dt dL/dqdot
<T> = -<V>/2
<?|O|?>
?L/A - L = length - A = cross sectional area - rho is electrical resistivity

11. EM: Reactance of Capacitor
X_C = 1/(i?C)
(3/2) n R ?t
µ = m_e/2
?scl = +/-1;?m = 0 - +/-1;?S_tot = 0;(?j = ?scl + ?S_tot)

12. RLC resonance condition
Z_C + Z_L = 0. Occurs when ?=1/Sqrt[L C]
E_n = -µ c^2 Z a^2 / (2n^2) - with µ = m_1 m_2 / (m_1 + m_2)
1/f = (n-1)(1/R1 - 1/R2) if both positive - they are convex - concave
Z_c = -i/(?C) ; Z_L = i ? L

13. Work in a capacitor
dU = 0 ? dS = ?dW/T
1/2 CV²
W' = (w-v)/(1-w v/c^2) ; observer in S sees an object moving at velocity w; another frame S' moves at v wrt S.
J = E s - s = Conductivity - E = Electric field

14. Expectation value of the energy of state |?>
When you apply a uniform electric field - it induces a dipole moment and interacts with it - and that effect depends on |mj |. So if j is an integer - splits (asymmetrically) into j+1 levels - and if j is a half integer - splits (asymmetrically) into
4H + 2e- ? He +2? + 6?
E = <?| H |?>
Asin(?) = m?

15. Thermo: Monatomic gas ?=?
? = 5/3
N²/Z (m_elec/m_red)
F = R/2
<T> = -<V>/2

16. EM: Maxwell'S equations
div(E) = ?/e_0 - curl(E) = der(B)/der(t) - div(B) = 0 - curl(B) = µ_0J + µ_0e_0*der(E)/der(t)
Measurements close to mean
Faraday/Lenz: current inducted opposes the changing field
D/dt (.5*r^2 d?/dt) = 0 - r(?) = a(1-e²)/(1+ecos(?)) - T²aA³

17. Quant: Commutator Relation [AB -C]
1s² - 2s² 2p6 - 3s² 3p6 3d¹°
PdV +dU
F = s * T4
A[B -C] + [A -C]B

18. Magnetic Field of a long solenoid
J = E s - s = Conductivity - E = Electric field
KE = 1/2 * µ (dr/dt)² L = µ r x v
Triplet: symmetric - net spin 1 Singlet: antisymmetric - net spin 0
B = µ0 I n

19. Adiabatic means
?~1/T
P = µ_0 q^2 a^2/(6Pi c); No radiation along the axis of acceleration
µ = m_e/2
Isentropic

20. Malus Law

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21. Mean electron drift speed
1/ne - where n is charge carrier density
J/(ne) n: atom density
Int ( A . dr) = Int ( del x A) dSurface
qvb = mv²/R

22. Wein'S displacement law for blackbodies (? and T)
?~T
F = -2*m(? x r)
?_max = b/T
I = V/R exp(-t/RC)

23. Bernoulli Equation
? = ?_0 Sqrt[(1+v/c)/(1-v/c)]
P +1/2 ? v² + ?gh = Constant
N²/Z (m_elec/m_red)
F = s * T4

24. EM: SHO (Hooke)
ma + kx = 0
Sin(?) = ?/d
Cos[?] Sin[?] -Sin[?] Cos[?]
? = ?0 root((1-v/c)/(1+v/c))

25. Dulong Petit Law
W' = (w-v)/(1-w v/c^2) ; observer in S sees an object moving at velocity w; another frame S' moves at v wrt S.
Infinitely close to equilibrium at all times
Cv = dE/dT = 3R
0

26. Energy for orbits: Hyperbole - Ellipse - Parabola - Circle
V = V0 + V0 a ?T
E = Vmin : circle - E = 0 : parabola - E<0 : el - E>0 : h
F = R/2
DW = P dV

27. Mech: Centripetal Force
S = (hbar/2) s ;with S = S_x xhat + S_y yhat + S_z zhat -s = s_x xhat + s_y yhat + s_z zhat
F = mv²/r
P = µ_0 q^2 a^2/(6Pi c); No radiation along the axis of acceleration
When you apply a uniform electric field - it induces a dipole moment and interacts with it - and that effect depends on |mj |. So if j is an integer - splits (asymmetrically) into j+1 levels - and if j is a half integer - splits (asymmetrically) into

28. Mech: Rotational Energy
IR + Ldi/dt = 0 - I = I0e(-tL/R) Work = 1/2 L I0^2
DB = ( µ_0 I/(4Pi) ) dl(cross)rhat/r^2
Int ( A . dr) = Int ( del x A) dSurface
T = I?²/2

29. A reversible process stays..
E = s/e_0
NC?T
C_eq = ?C_i
Infinitely close to equilibrium at all times

30. Thermo: Partition Function
H = H_0 + ?H
Z = ?g_i*exp(-E/kT)
?~1/T
? = 1.22?/D

31. Perturbations
I = I_cm + (1/2)m d^2
Isentropic
? = h/mv
H = H_0 + ?H

32. Kepler'S Three Laws
? (t-vx/c²)
<T> = -<V>/2
P +1/2 ? v² + ?gh = Constant
D/dt (.5*r^2 d?/dt) = 0 - r(?) = a(1-e²)/(1+ecos(?)) - T²aA³

33. Delta Function Potential - type of WF
S = k ln[O] ; dS = dQ/T
(° of Freedom)kT/2
Exponential - E = -ma²/2hbar² - a is strength of delta wellt
DW = P dV

34. Biot-Savart law
Cos[?] Sin[?] -Sin[?] Cos[?]
?mc²
U = t^2 d/dt (logZ)
DB = ( µ_0 I/(4Pi) ) dl(cross)rhat/r^2

35. Relativistic Energy
L = L_0 Sqrt[1-v^2/c^2]
X_C = 1/(i?C)
Product ( nj ^ vj ) = Product(nqj ^ vj exp (-vj F(int)/Tau))
?mc²

36. Electromotive Force
I = Im (sinc²(a)) ; a = pai sin(?) / ?
DW/dq
Z_C + Z_L = 0. Occurs when ?=1/Sqrt[L C]
div(E) = ?/e_0 - curl(E) = der(B)/der(t) - div(B) = 0 - curl(B) = µ_0J + µ_0e_0*der(E)/der(t)

37. Mech: Force of Friction
I_z = I_x + I_y (think hoop symmetry)
F_f = µ*F_N
dQ = dW +dU
P1V1 - P2V2 / (? - 1)

38. Relativistic length contraction
P/A = s T^4
Z_c = -i/(?C) ; Z_L = i ? L
div(E) = ?/e_0 - curl(E) = der(B)/der(t) - div(B) = 0 - curl(B) = µ_0J + µ_0e_0*der(E)/der(t)
L = L_0 Sqrt[1-v^2/c^2]

39. Relativistic Momentum
?mv
u dm/dt
M? = 2dsin(?)
Ct²-x²-y²-z²

40. 3 Laws of Thermo
Z_C + Z_L = 0. Occurs when ?=1/Sqrt[L C]
A[B -C] = A[B -C]+[B -A]C [A -B] = -[B -A]
1. Heat is energy 2. Entropy never decreases 3. Entropy approaches a constant value as t -> 0...
F = mv²/r

41. EM: Electromagnetic inertia
U - ts = -tlog(Z)
?~1/T
DS = 0 - dQ = 0 - P V^? = constant
Faraday/Lenz: current inducted opposes the changing field

42. Relativistic interval (which must remain constant for two events)
I = -(c ?t)^2 + d^2
Asin(?) = m?
SR: ?=? - ß=? E = ?mc² = v(p²c² + m²c4)
Isentropic

43. Mech: Parallel Axis Theorem (Moment of Inertia)
I_z = I_x + I_y (think hoop symmetry)
I = I_cm + md²
?s = 0 - ?l = ±1
? = h/mv

44. EM: AC Resonance
C = 4pe0 ab/(a-b) = inner and outer radii
X_L = X_C or X_total = 0
J = ? Fdt
<T> = 1/2 * <dV/dx>

45. Thermo: Isothermal
Interference: (m+.5)? = d sin(?) Diffraction: m? = w sin(?)
dU = 0 ? dS = ?dW/T
u dm/dt
X_L = X_C or X_total = 0

46. Rocket Equation
(° of Freedom)kT/2
Infinitely close to equilibrium at all times
C_eq = (? 1/C_i)^-1
Dv = -udm/m - v = v0 + u ln(m0/m)

47. Coriolis Force
Interference: (m+.5)? = d sin(?) Diffraction: m? = w sin(?)
? (t-vx/c²)
F = -2*m(? x r)
? = 1.22? / d

48. Atom: Positronium Reduced Mass
F = mv²/r
S_mean = s/Sqrt[N]
Hbar*?³/(p²c³exp(hbar?/t)-1)
µ = m_e/2

49. Source-free RC Circuit
U - ts = -tlog(Z)
P = µ_0 q^2 a^2/(6Pi c); No radiation along the axis of acceleration
CdV/dt + V/R = 0 V(t) = V0 exp(-t/RC) I(t) = I(0) exp(-t/RC)
C = 4pe0 ab/(a-b) = inner and outer radii

50. Energy in a Capacitor
CdV/dt + V/R = 0 V(t) = V0 exp(-t/RC) I(t) = I(0) exp(-t/RC)
? (t-vx/c²)
S = k ln[O] ; dS = dQ/T
.5 CV²

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