Physics GRE

74 flashcards · Physics · by Alpha Centauri

1. Work-Energy

   W=ΔKEW = \Delta KEW=ΔKE Einitial+Wother=EfinalE_{initial} + W_{other} = E_{final}Einitial​+Wother​=Efinal​

2. Circular Motion force and acceleration

   F=mv2rF = \frac{mv^2}{r}F=rmv2​ a=v2ra = \frac{v^2}{r}a=rv2​

3. Kinetic energies (translational, rotational) Potential energies (gravitational, spring)

   translation = 12mv2\frac{1}{2} mv^221​mv2 , rotation = 12Iω2\frac{1}{2} I \omega^221​Iω2 gravitational = mghmghmgh , spring = 12kx2\frac{1}{2} kx^221​kx2

4. Relation between force and energy (and vice versa)

   ΔU=−∫abF⃗⋅dl⃗\Delta U = - \int_a^b {\vec{F} \cdot d\vec{l}}ΔU=−∫ab​F ⋅dl F⃗=−∇U\vec{F} = - \nabla{U}F =−∇U

5. Angular momentum

   L = r x p L=IωL = I \omegaL=Iω

6. Torque

   τ=r×F\tau = r \times Fτ=r×F τ=dLdt \tau= \frac{dL}{dt} τ=dtdL​

7. Centrifugal force Coriolis force

   Fcentrifugal=−mΩ2rF_{centrifugal} = -m \Omega^2 rFcentrifugal​=−mΩ2r Fcoriolis=−2mΩ⃗×v⃗F_{coriolis} = -2m\vec{\Omega} \times \vec{v}Fcoriolis​=−2mΩ ×v

8. Gravitational force

   F=GMmr2F = \frac{GMm}{r^2}F=r2GMm​

9. Rotational velocity

   v=rωv = r\omegav=rω

10. 3 kinematics equations

    v=v0+atv = v_0 + atv=v0​+at Δx=v0t+12at2\Delta x = v_0t + \frac{1}{2}at^2Δx=v0​t+21​at2 v2=v02+2aΔxv^2 = v_0^2 + 2a\Delta xv2=v02​+2aΔx

11. Moment of inertia

    I=mr2I = mr^2I=mr2 I=∫r2dmI = \int{r^2 dm}I=∫r2dm

12. Parallel Axis Theorem

    I=ICM+Mr2I= I_{CM} + Mr^2I=ICM​+Mr2

13. Center of mass

    rCM=∫r⃗dmM=∑ir⃗imiMr_{CM} = \frac{\int{\vec{r} dm}}{M} = \frac{\sum_i{\vec{r}_i m_i}}{M}rCM​=M∫r dm​=M∑i​r i​mi​​

14. Lagrangian

    L(q,q˙,t)=T−UL(q, \dot{q}, t) = T - UL(q,q˙​,t)=T−U

15. Euler-Lagrange Equation

    ddt∂L∂q˙=∂L∂q\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}dtd​∂q˙​∂L​=∂q∂L​

16. Momentum conjugate and implication of the Lagrangian being independent of a coordinate

    pi=∂L∂q˙p_i = \frac{\partial L}{\partial \dot{q}}pi​=∂q˙​∂L​ If L is indep. of q, then the conjugate momentum (formula above) is conserved --> ∂L∂q=0\frac{\partial L}{\partial q}=0∂q∂L​=0 and q is a cyclic coordinate

17. Hamiltonian (2 definitions, one if U doesn't explicitly depend on velocity or time)

    H(p,q)=∑ipiq˙i−LH(p,q) = \sum_i{p_i\dot{q}_i - L}H(p,q)=∑i​pi​q˙​i​−L H = T + U

18. Hamilton's equations

    p˙=−∂H∂q\dot{p} = -\frac{\partial H}{\partial q}p˙​=−∂q∂H​ q˙=∂H∂p\dot{q} = \frac{\partial H}{\partial p}q˙​=∂p∂H​

19. Lagrangian for orbits

    L=12mr˙2+12mr2ϕ˙2−U(r)L = \frac{1}{2}m\dot{r}^2 + \frac{1}{2}mr^2\dot{\phi}^2 - U(r)L=21​mr˙2+21​mr2ϕ˙​2−U(r)

20. Angular momentum for orbits

    l=mr2ϕ˙l = mr^2\dot{\phi}l=mr2ϕ˙​

21. Effective potential

    V(r)=l22mr2+U(r)V(r) = \frac{l^2}{2mr^2} + U(r)V(r)=2mr2l2​+U(r)

22. Reduced mass

    μ=m1m2m1+m2\mu = \frac{m_1 m_2}{m_1 + m_2}μ=m1​+m2​m1​m2​​

23. Energy of an orbit and implications for energies of values above, below, and at 0

    E=T+V=12mr˙2+l22mr2+U(r)E = T+V = \frac{1}{2}m\dot{r}^2 + \frac{l^2}{2mr^2} + U(r)E=T+V=21​mr˙2+2mr2l2​+U(r) E > 0: hyperbolic E = 0: parabolic E < 0: elliptical

24. Kepler's Laws

    1) Planets move in elliptical orbits with the sun at a focus 2) Orbits sweep out equal areas in equal times 3) T∝a3/2T \propto a^{3/2}T∝a3/2

25. Spring force, angular frequency, and simplest equation of motion

    F=mx¨=−kxF = m\ddot{x} = -kxF=mx¨=−kx ω=km\omega = \sqrt{\frac{k}{m}}ω=mk​ ​ x(t)=Aeiωtx(t) = Ae^{i\omega t}x(t)=Aeiωt (can use as ansatz when solving normal mode problems)

26. Normal mode secular equation

    det(Ajk−ω2mjk)=0\text{det}(A_{jk} - \omega^2m_{jk}) = 0det(Ajk​−ω2mjk​)=0 The ω\omegaω 's give the normal frequencies

27. Equation of motion for damped oscillatory system and damping conditions

    mx¨+bx˙+kx=0m\ddot{x} + b\dot{x} + kx = 0mx¨+bx˙+kx=0 β=b/2m\beta = b/2mβ=b/2m Overdamped: \omega_0^2"> β2&gt;ω02\beta^2 &gt; \omega_0^2β2>ω02​ Underdamped: β2&lt;ω02\beta^2 &lt; \omega_0^2β2<ω02​ Critical: β2=ω02\beta^2 = \omega_0^2β2=ω02​

28. Resonant frequency

    ωR=ω02−2β2\omega_R = \sqrt{\omega_0^2 - 2\beta^2}ωR​=ω02​−2β2 ​

29. Driving force frequency

    D∝1∣ω02−ω2∣D \propto \frac{1}{|{\omega_0^2 - \omega^2}|}D∝∣ω02​−ω2∣1​

30. Pendulum EOM and angular frequency (both massless and physical pendulum)

    mx¨=−mgx/Lm\ddot{x} = -mgx/Lmx¨=−mgx/L ω=gL\omega = \frac{g}{L}ω=Lg​ ω=mgRI\omega = \sqrt{\frac{mgR}{I}}ω=ImgR​ ​

31. Rule for adding spring constants

    Same as adding capacitors: Series: 1/keq=1/k1+1/k2+...1/k_{eq} = 1/k_1 + 1/k_2 + ...1/keq​=1/k1​+1/k2​+... Parallel: keq=k1+k2+...k_{eq} = k_1 + k_2 + ...keq​=k1​+k2​+...

32. Bernoulli's Principle

    v22+gz+pρ=constant\frac{v^2}{2} + gz + \frac{p}{\rho} = \text{constant}2v2​+gz+ρp​=constant (think of this like conservation of energy: kinetic + grav potential energy + pressure energy). We usually equate one section of a fluid pipe with another with this equation

33. Fluid conservation

    ρ1v1A1Δt=ρ2v2A2Δt&ThickSpace;⟹&ThickSpace;v1A1=v2A2\rho_1 v_1 A_1 \Delta t = \rho_2 v_2 A_2 \Delta t \implies v_1 A_1 = v_2 A_2ρ1​v1​A1​Δt=ρ2​v2​A2​Δt⟹v1​A1​=v2​A2​ But we usually assume that the density is constant such that ρ1=ρ2\rho_1 = \rho_2ρ1​=ρ2​ (incompressible fluid)

34. Buoyant force

    F=ρVgF = \rho VgF=ρVg Note that ρV\rho VρV is just mass m

35. Density of water

    1g\cm31 g\cm^31g\cm3

36. Maxwell's Equations

    ∇⋅E⃗=ρϵ0\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}∇⋅E =ϵ0​ρ​ ∇⋅B⃗=0\nabla \cdot \vec{B} = 0∇⋅B =0 ∇×E⃗=−∂B⃗∂t\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}∇×E =−∂t∂B ​ ( = 0 if electrostatics) ∇×B⃗=μ0J⃗+μ0ϵ0∂E⃗∂t\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}∇×B =μ0​J +μ0​ϵ0​∂t∂E ​

37. Electric force

    FE=qE⃗F_E = q\vec{E}FE​=qE

38. Electric potential

    E⃗=−∇V\vec{E} = -\nabla VE =−∇V

39. Electric potential

    V(b)=−∫abE⃗⋅dl⃗V(b) = - \int_a^b{\vec{E} \cdot d\vec{l}}V(b)=−∫ab​E ⋅dl

40. Poisson's and Laplace's Equations

    ∇2V=−ρϵ0\nabla^2 V = - \frac{\rho}{\epsilon_0}∇2V=−ϵ0​ρ​ (Poisson) ∇2V=0\nabla^2 V = 0∇2V=0 (Laplace, empty space)

41. General solution to Poisson's equation (hardly ever used)

    V(r⃗)=14πϵ0∫ρ(r′⃗)∣r⃗−r′⃗∣d3r′⃗V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\int{\frac{\rho(\vec{r&#x27;})}{|\vec{r} - \vec{r&#x27;}|}d^3 \vec{r&#x27;}}V(r )=4πϵ0​1​∫∣r −r′ ∣ρ(r′ )​d3r′

42. Gauss's Law and integral Maxwell

    ∯E⃗⋅da⃗=qencϵ0\oiint \vec{E} \cdot d \vec{a} = \frac{q_{enc}}{\epsilon_0}∬ ​E ⋅da =ϵ0​qenc​​ = electric flux ∮E⃗⋅ds⃗=−dΦBdt\oint \vec{E} \cdot d \vec{s} = -\frac{d\Phi_B}{dt}∮E ⋅ds =−dtdΦB​​ ( = 0 electrostatics)

43. Electric field and potential from a point charge

    E=q4πϵ0r2r^E = \frac{q}{4\pi\epsilon_0 r^2} \hat{r}E=4πϵ0​r2q​r^ V=q4πϵ0rV = \frac{q}{4\pi \epsilon_0 r}V=4πϵ0​rq​

44. Electric field above a plane (infinite sheet)

    E⃗=σ2ϵ0n^\vec{E} = \frac{\sigma}{2\epsilon_0} \hat{n}E =2ϵ0​σ​n^

45. Electric field due to line charge/cylinder

    E⃗=λ2πϵ0rr^\vec{E} = \frac{\lambda}{2 \pi \epsilon_0 r} \hat{r}E =2πϵ0​rλ​r^

46. Electric field boundary conditions

    Eout∣∣−Ein∣∣=0E_{out}^{||} - E_{in}^{||} = 0Eout∣∣​−Ein∣∣​=0 Eout⊥−Ein⊥=σϵ0E_{out}^{\perp} - E_{in}^{\perp} = \frac{\sigma}{\epsilon_0}Eout⊥​−Ein⊥​=ϵ0​σ​

47. Rules for conductors

    V is constant throughout a conductor E-field inside = 0 Net charge density inside = 0, always confined to surface E-field just outside is perpendicular to surface

48. Image charges work

    There is no energy cost to moving an image charge

49. Work to put together point charges and continuous distribution

    W=12∑i=1nqiV(ri⃗)W = \frac{1}{2}\sum_{i=1}^{n} q_i V(\vec{r_i})W=21​∑i=1n​qi​V(ri​ ​) W=12∫ρ(r⃗)V(r⃗)d3r⃗W = \frac{1}{2}\int{\rho(\vec{r}) V(\vec{r}) d^3 \vec{r}}W=21​∫ρ(r )V(r )d3r

50. Electric and magnetic field energies

    UE=ϵ02∭V∣E∣⃗2dVU_E = \frac{\epsilon_0}{2} \iiint_V \vec{|E|}^2 dVUE​=2ϵ0​​∭V​∣E∣ ​2dV UB=12μ0∭V∣B∣⃗2dVU_B = \frac{1}{2 \mu_0} \iiint_V \vec{|B|}^2 dVUB​=2μ0​1​∭V​∣B∣ ​2dV

51. Capacitance

    Q = VC hello there

52. Capacitance of parallel plate

    C=ϵ0AdC = \frac{\epsilon_0 A}{d}C=dϵ0​A​

53. Energy of a capacitor and inductor

    UC=12Q2C=12CV2U_C = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}CV^2UC​=21​CQ2​=21​CV2 UL=12LI2U_L = \frac{1}{2}LI^2UL​=21​LI2

54. Ampere's Law

    ∮B⃗⋅ds⃗=μ0Ienc\oint \vec{B} \cdot d \vec{s} = \mu_0 I_{enc} ∮B ⋅ds =μ0​Ienc​

55. Vector potential

    ∇×A⃗=B⃗\nabla \times \vec{A} = \vec{B}∇×A =B

56. Magnetic force on charge and wire

    FB=qv⃗×B⃗F_B = q\vec{v} \times \vec{B}FB​=qv ×B dFB⃗=Idl⃗×B⃗d\vec{F_B} = I d\vec{l} \times \vec{B}dFB​ ​=Idl ×B

57. Biot-Savart Law

    dB⃗=μ0I4πdl⃗×r^r2d \vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \hat{r}}{r^2}dB =4πμ0​I​r2dl ×r^​

58. Magnetic field of a wire

    B⃗=μ0I2πrϕ^\vec{B} = \frac{\mu_0 I}{2 \pi r} \hat{\phi} B =2πrμ0​I​ϕ^​

59. Magnetic field of a solenoid and toroid

    Bsol=μ0nIB_{sol} = \mu_0 nIBsol​=μ0​nI (note n = turns/unit length, and enclosed current is InL) Btor=μ0NI2πrB_{tor} = \frac{\mu_0 NI}{2 \pi r}Btor​=2πrμ0​NI​

60. Magnetic Field Boundary Conditions

    Bout∣∣−Bin∣∣=μ0K⃗×n^B_{out}^{||} - B_{in}^{||} = \mu_0 \vec{K} \times \hat{n}Bout∣∣​−Bin∣∣​=μ0​K ×n^ Bout⊥−Bin⊥=0B_{out}^{\perp} - B_{in}^{\perp} = 0Bout⊥​−Bin⊥​=0

61. Cyclotron radius and frequency

    R=mvqBR = \frac{mv}{qB}R=qBmv​ ω=qBm\omega = \frac{qB}{m}ω=mqB​

62. Faraday's Law

    ε=∫CE⃗⋅ds⃗=−ddt∫SB⃗⋅da⃗=−dΦBdt\varepsilon = \int_C \vec{E}\cdot d\vec{s} = -\frac{d}{dt} \int_S \vec{B} \cdot d\vec{a} = -\frac{d\Phi_B}{dt}ε=∫C​E ⋅ds =−dtd​∫S​B ⋅da =−dtdΦB​​

63. Lenz's Law

    Induced currents always oppose changes in magnetic flux

64. Mutual and Self Inductance

    Φ21=M21I1\Phi_{21} = M_{21} I_1Φ21​=M21​I1​ (note that M12=M21M_{12} = M_{21}M12​=M21​ so Φ12=M21I2=M12I2 \Phi_{12} = M{_21} I_2 = M_{12}I_2 Φ12​=M2​1I2​=M12​I2​ ΦB=LI\Phi_B = LIΦB​=LI

65. Flux through inductance

    ε=−LdIdt\varepsilon = -L \frac{dI}{dt}ε=−LdtdI​

66. Inductance of a solenoid and toroid

    Lsol=μ0N2AlL_{sol} = \frac{\mu_0 N^2 A}{l}Lsol​=lμ0​N2A​ Ltor=μ0a2N22RL_{tor} = \frac{\mu_0 a^2 N^2}{2R}Ltor​=2Rμ0​a2N2​

67. Electric dipole moments (discrete and continuous)

    p⃗=qr1⃗−qr2⃗=qd⃗\vec{p} = q\vec{r_1} - q\vec{r_2} = q\vec{d}p ​=qr1​ ​−qr2​ ​=qd (or sum of qd for multiple charges) p⃗=∫r⃗ρ(r⃗)d3r⃗\vec{p} = \int{\vec{r} \rho(\vec{r}) d^3 \vec{r}}p ​=∫r ρ(r )d3r

68. Dipole Potential

    V(r)=14πϵ0p⃗⋅r^⃗r2 V(r) = \frac{1}{4\pi \epsilon_0} \frac{\vec{p} \cdot \vec{\hat{r}}}{r^2} V(r)=4πϵ0​1​r2p ​⋅r^ ​

69. Dipole torques (electric and magnetic)

    N⃗=p⃗×E⃗\vec{N} = \vec{p} \times \vec{E}N =p ​×E N=m⃗×B⃗N = \vec{m} \times \vec{B}N=m ×B

70. Dipole potential energies (electric and magnetic)

    U=−p⃗⋅E⃗U = -\vec{p} \cdot \vec{E}U=−p ​⋅E U=−m⃗⋅B⃗U = -\vec{m} \cdot \vec{B}U=−m ⋅B

71. Magnetic dipole moment

    m⃗=IA⃗\vec{m} = I\vec{A}m =IA (A is the area vector)

72. Polariziation/bound charge and volume densities

    σb=P⃗⋅n^\sigma_b = \vec{P} \cdot \hat{n}σb​=P ⋅n^ ρb=−∇⋅P⃗\rho_b = -\nabla \cdot \vec{P}ρb​=−∇⋅P

73. Dielectric constant substitution and capacitance change

    \epsilon = \kappa \epsilon_0"> ϵ0−&gt;ϵ=κϵ0\epsilon_0 -&gt; \epsilon = \kappa \epsilon_0ϵ0​−>ϵ=κϵ0​ C=ϵAd=κϵ0AdC = \frac{\epsilon A}{d} = \kappa \frac{\epsilon_0 A}{d}C=dϵA​=κdϵ0​A​

74. Speed of light in terms of constants

    c=1/ϵ0μ0c = 1/\sqrt{\epsilon_0 \mu_0}c=1/ϵ0​μ0​ ​