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GRE Physics

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1. Atom: Bohr Formula
.5 CV²
E ~ (1/(n_f)² - 1/(n_i)²) ~ 1/?
Dv = -udm/m - v = v0 + u ln(m0/m)
1/vLC

2. Coriolis Force
M? = 2dsin(?)
F = -2*m(? x r)
?mc²
ds² = (c*dt)² - ?(x_i)²

3. Virial Theorem
L = µ N² A / l : N = number of turns - A = cross sectional area -l = length
Asin(?) = m?
0
<T> = 1/2 * <dV/dx>

4. Energy for orbits: Hyperbole - Ellipse - Parabola - Circle
E = Vmin : circle - E = 0 : parabola - E<0 : el - E>0 : h
X_C = 1/(i?C)
P(s) = (1/Z) Exp[-E(s)/(k T)] Z = S_s(Exp[-E(s)/(k T)])
L = µ N² A / l : N = number of turns - A = cross sectional area -l = length

5. Angular momentum operators L^2 and L_z
µ0 I / 2R
U = t^2 d/dt (logZ)
?? = h/mc * (1-cos(?))
L^2 |E - scl - m> = hbar^2 scl(scl+1) |E -scl -m> L_z |E - scl - m> = hbar m |E - scl - m>

6. Dulong Petit Law
Z = ?g_i*exp(-E/kT)
Interference: (m+.5)? = d sin(?) Diffraction: m? = w sin(?)
Opposing charge induced upon conductor
Cv = dE/dT = 3R

7. Clausius-Clapeyron Equation
?scl = +/-1;?m = 0 - +/-1;?S_tot = 0;(?j = ?scl + ?S_tot)
F = qv×B
Dp/dt = L / (t ?V)
B = µ0 I n

8. Lab: Precision of Measurements
E²-p²c²
Measurements close to mean
D/dt (.5*r^2 d?/dt) = 0 - r(?) = a(1-e²)/(1+ecos(?)) - T²aA³
qvb = mv²/R

9. Rocket Equation
Z_c = -i/(?C) ; Z_L = i ? L
Ct²-x²-y²-z²
1/2 CV²
Dv = -udm/m - v = v0 + u ln(m0/m)

10. Thermo: Isothermal
dU = 0 ? dS = ?dW/T
Infinitely close to equilibrium at all times
L = µ N² A / l : N = number of turns - A = cross sectional area -l = length
?s = 0 - ?l = ±1

11. Invariant spatial quantity
J/(ne) n: atom density
H = H_0 + ?H
Ct²-x²-y²-z²
Z_C + Z_L = 0. Occurs when ?=1/Sqrt[L C]

12. Thermo: 1st Law
V(r) + L²2/2mr²
?mc²
dQ = dW +dU
Cos[?] Sin[?] -Sin[?] Cos[?]

13. Solid: Resistivity of Semi-Conductor
0
1/2 CV²
Q = CVexp(-t/RC)
?~1/T

14. Biot-Savart law
4H + 2e- ? He +2? + 6?
F = mv²/r
DB = ( µ_0 I/(4Pi) ) dl(cross)rhat/r^2
P/A = s T^4

15. Rayleigh criterion
? = 1.22? / d
F = µ0 q v I / 2pr
P² ~ R³
? exp(-e/t)

16. Rocket Thrust
I ' = I cos²(?)
u dm/dt
µ0 I / 2R
D/dt (.5*r^2 d?/dt) = 0 - r(?) = a(1-e²)/(1+ecos(?)) - T²aA³

17. Resonance frequency of LC circuit
P = µ_0 q^2 a^2/(6Pi c); No radiation along the axis of acceleration
E ~ (1/(n_f)² - 1/(n_i)²) ~ 1/?
F = R/2
1/vLC

18. Parallel axis theorem
Ct²-x²-y²-z²
I = I_cm + (1/2)m d^2
µ=s^2
v(mean)

19. Anomalous Zeeman Effect

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20. Planck Radiation Law
Hbar*?³/(p²c³exp(hbar?/t)-1)
H = T + V;qdot_i = dH/dp_i - pdot_i = dH/dq_i
M? = 2dsin(?)
T = I?²/2

21. Current in resistor in RC circuit
Exp(N(µ-e)/t)
µ = m_e/2
X_C = 1/(i?C)
I = V/R exp(-t/RC)

22. Commutator identities ( [B -A C] - [A -B] )
A[B -C] = A[B -C]+[B -A]C [A -B] = -[B -A]
X_C = 1/(i?C)
Z_C + Z_L = 0. Occurs when ?=1/Sqrt[L C]
Product ( nj ^ vj ) = Product(nqj ^ vj exp (-vj F(int)/Tau))

23. Gibbs Factor
Z = ?g_i*exp(-E/kT)
Exp(N(µ-e)/t)
Series: 1/k_eq = 1/k_1 + 1/k_2; Parallel: k_eq = k_1 + k_2
ih_barL_z

24. EM: Bremsstrahlung (translation)
J = ? Fdt
Braking Radiation
N²/Z (m_elec/m_red)
Let w_i = 1/s_i^2;x_wav = S(w_i x_i) / Sw_i - s_xwav = 1/Sw_i

25. Energy in a Capacitor
Sin(?) = ?/d
P1V1 - P2V2 / (? - 1)
.5 CV²
S_mean = s/Sqrt[N]

26. Charge in Capacitor
?~1/T
Z²/n² (m_red/m_elec)
Q = CVexp(-t/RC)
?mv

27. Self Inductance
?max = 2.898 x 10 -³ / T
.5 LI²
V = -L di/dt
1s² - 2s² 2p6 - 3s² 3p6 3d¹°

28. Astro: Aperture Formula (Rayleigh Criterion)
Int ( A . dr) = Int ( del x A) dSurface
? = 1.22?/D
H = T + V;qdot_i = dH/dp_i - pdot_i = dH/dq_i
Asin(?) = m?

29. Single Slit Diffraction Maximum
J = ? Fdt
E = <?| H |?>
µ = m_e/2
Asin(?) = m?

30. Relativistic Energy
?mc²
Interference: (m+.5)? = d sin(?) Diffraction: m? = w sin(?)
v(mean)
DB = ( µ_0 I/(4Pi) ) dl(cross)rhat/r^2

31. Boltzmann / Canonical distribution
P(s) = (1/Z) Exp[-E(s)/(k T)] Z = S_s(Exp[-E(s)/(k T)])
P/A = s T^4
F = mv²/r
1/2 CV²

32. Adiabatic means
Isentropic
.5 CV²
J = ? Fdt
<?|O|?>

33. Helmholtz Free Energy
dQ = dW +dU
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
U - ts = -tlog(Z)
Dp/dt = L / (t ?V)

34. EM: Reactance of Capacitor
?s = 0 - ?l = ±1
X_C = 1/(i?C)
KE = 1/2 * µ (dr/dt)² L = µ r x v
N²/Z (m_elec/m_red)

35. Source-free RC Circuit
CdV/dt + V/R = 0 V(t) = V0 exp(-t/RC) I(t) = I(0) exp(-t/RC)
I = -(c ?t)^2 + d^2
? = 5/3
L^2 |E - scl - m> = hbar^2 scl(scl+1) |E -scl -m> L_z |E - scl - m> = hbar m |E - scl - m>

36. Bragg'S Law of Reflection
M? = 2dsin(?)
?scl = +/-1;?m = 0 - +/-1;?S_tot = 0;(?j = ?scl + ?S_tot)
F = µ0 q v I / 2pr
qvb = mv²/R

37. Perturbations
W_A < W_I
H = H_0 + ?H
Series: 1/k_eq = 1/k_1 + 1/k_2; Parallel: k_eq = k_1 + k_2
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

38. Radiation (Larmor - and another neat fact)
P1V1 - P2V2 / (? - 1)
P = µ_0 q^2 a^2/(6Pi c); No radiation along the axis of acceleration
E = Vmin : circle - E = 0 : parabola - E<0 : el - E>0 : h
1/ne - where n is charge carrier density

39. How to derive cylcotron frequency
Dp/dt = L / (t ?V)
1/f = (n-1)(1/R1 - 1/R2) if both positive - they are convex - concave
Braking Radiation
qvb = mv²/R

40. EM: Electric Field inside of Conductor
Cv = dE/dT = 3R
0
N d flux / dt
L = T - V dL/dq = d/dt dL/dqdot

41. Adiabatic processes (dS - dQ - P and V)
T = I?²/2
DS = 0 - dQ = 0 - P V^? = constant
IR + Ldi/dt = 0 - I = I0e(-tL/R) Work = 1/2 L I0^2
Exp(N(µ-e)/t)

42. EM: Series Capacitance
U - ts = -tlog(Z)
µ=s^2
C_eq = (? 1/C_i)^-1
D/dt (.5*r^2 d?/dt) = 0 - r(?) = a(1-e²)/(1+ecos(?)) - T²aA³

43. Focal point of mirrror with curvature
µ0 I / 2pR
V = V0 + V0 a ?T
C_eq = (? 1/C_i)^-1
F = R/2

44. Angular momentum - Central Force Motion
L = mr²d?/dt
KE = 1/2 * µ (dr/dt)² L = µ r x v
<T> = 1/2 * <dV/dx>
Dv = -udm/m - v = v0 + u ln(m0/m)

45. td(entropy) =
L = L_0 Sqrt[1-v^2/c^2]
I = V/R exp(-t/RC)
F = qv×B
PdV +dU

46. Quant: [L_x -L_y] = ?
ih_barL_z
P(s) = (1/Z) Exp[-E(s)/(k T)] Z = S_s(Exp[-E(s)/(k T)])
Q = CVexp(-t/RC)
? = h/p

47. Electromotive Force
DW/dq
I = -(c ?t)^2 + d^2
.5 LI²
Product ( nj ^ vj ) = Product(nqj ^ vj exp (-vj F(int)/Tau))

48. Energy in terms of partition function
? = 5/3
U = t^2 d/dt (logZ)
Ct²-x²-y²-z²
?mc²

49. Energy in Inductor
Isentropic
F = f* (c+v_r)/(c+v_s)
A[B -C] = A[B -C]+[B -A]C [A -B] = -[B -A]
.5 LI²

50. Inductance of Solenoid
I = I_cm + (1/2)m d^2
L = µ N² A / l : N = number of turns - A = cross sectional area -l = length
M? = 2dsin(?)
?scl = +/-1;?m = 0 - +/-1;?S_tot = 0;(?j = ?scl + ?S_tot)

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