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## Physics GRE (111 Cards)

Static/Kinetic Friction

where is the normal force
Force(Generally, in terms of potential, E&M, Springs, and Buoyancy)
Centripetal Force

where is the tangential velocity.
Work
Work Energy Theorem
Energy is conserved, thus
.

Use if possible whenever the problem calls for a distance and not a time.
Moment of Inertia, I

where dm is an infinitesimal mass element. Some common ones are given on the test, and most others can be found with the parallel axis theorem.
Parallel Axis Theorem
If you wish to know the moment of inertia about an axis parallel to the axis of the center of mass:

where r is the distance FROM THE CoM, not the radius.
The Lagrangian

where T is the kinetic energy and U is the potential. Notice the minus sign, very important.
Euler-Lagrange Equations (generates the equations of motion)

where is the so call conjugate momentum to q.
Hamiltonian

where L is the Lagrangian. If the potential U doesn't explictly depend upon time or velocity though, you can simply write

Classifications of Orbits
E > 0: hyperbolic orbit
E = 0: parabolic
E < 0; elliptical
E = : circular
Keplar's Laws

II. Planets sweep out equal areas in equal times. Conservation of aerial velocity.

III. If T is the period of an orbit, and a is a semimajor axis of the orbit, then:

with k the same for all planets.
Normal Modes
Find the equation of motion using Lagrangian, then write in matrix form. With the matrix written, set the determinant to 0 and solve. Should always get one solution which looks like a regular spring      and one that is greater.
Bernoulli's Principle
Maxwell's Equations in Vacuum
Scalar Potential, V

and conversely,

.
Work via Scalar Potential

where V(r) is the scalar potential
Stokes Theorem

where \vec{E} is a general vector field and \vec{S] is the surface with direction given by its normal.
Gauss Theorem

where E is a general vector field.
Boundary conditions in EM fields
Conductors
Potential is always constant throughout a conductor. Net electric field inside is always 0. Charge is always on surface.
Capacitance
C = Q/V

Capacitance is a purely geometric value like Inductance.
Parallel plate capacitance

where A is the area of the plates and d is their separation. This is derived using the approximation that .
Energy stored in a Cap
Work in EM

Magnetic fields do no work.
Energy stored in EM Fields
Cyclotron Motion
Centripetal force equals the force of the magnetic field on the particle.

where is the electromotive force (electrical potential) and is the magnetic flux through an unclosed surface ( ).
Inductance

Inductance of a solenoid

where N is the total number of terms, A is the cross sectional area of the solenoid, and is an arbitrary length scale.
Dipole Moments (E and M)
Electrical Dipole Moment

For point charges this reduces to

where is the displacement vector.

Magnetic Dipole Moment

where A is the area with direction of the normal to the surface.
Potential of a dipole
Potential and Torque of Dipole
Torque = (dipole moment) x (field)

Potential  =

I wrote it this way so it works for both magnetic and electric dipoles.

Dielectrics
Just remember

.

This increases Capacitance by a factor of .
Wave Equation

where F is any vector field. This has nice plane wave solutions for light where :

This also tells you that B is always perpendicular to E and usually smaller in amplitude (1/c is small).
Poynting vector
Tells you the flux of energy.

The intensity of radiation is just the time average of S; the sinusoids squared average to 1/2:

Energy stored in an inductor
Wavelength

where k is the wave number and lambda the wavelength.
Phase velocity vs group velocity
Phase velocity:

Group velocity:
Index of refraction

where n is the index of refraction. Essentially slows the phase velocity down by a factor of 1/n.
Malus's Law

where is the intensity of the light incident on the polarizer, and is the angle of the light's polarization with respect to the polarizer(i.e., let the light be polarized at and the polarizer be  , then )

Note this implies that for unpolarized light incident on a polarizer, you must average over all angles. This means 1/2 of the intensity of unpolarized light gets through.
Brewster's Angle
The angle at which light is completely polarized to the incident plane on a surface whose index of refraction is different from the current medium:

Maxima and Minima of a double slit

Remember to tell the difference that at must be a maxima, which means that the right hand side must be an integer for maxima (0 isn't a half integer).
Power radiated by an accelerating point charge
Called the Larmor formula, it says

where a is the acceleration. The constant of proportionality is not needed. if you must know.
Energy stored in EM fields
Gauss's and Ampere's Laws (the workhorses)
Magnetic field of infinite wire
Magnetic field of a solenoid

for an infinite solenoid along z with n turns per length.
Magnetic field of toroid

for N total turns. This is independent of the cross-section shape or area of the toroid and always holds for a toroid.
Power of an oscillating dipole

where dip mom represents either the magnetic or electric dipole moments and omega is the frequency of oscillation. The magnetic power has an extra factor of meaning that the electric dipole dominates the power term.
Voltages of common circuit elements
Rules for addition of elements in circuit
SERIES

PARALLEL

Kirchhoff's Rules
I. The sum of the currents flowing into every node must be zero.

II. The sum of the voltages across any closed loop must be zero.
Joule's Heating Law
Energy is dissipated according to :

Power in a circuit
Optical path length

where n is the index of refraction. This is what introduces a phase difference between two waves, one in a vacuum and another in a medium with index of refraction n, when they recombine. The vacuum wave travels d, and the medium wave travels nd.
Phase shift when light reflects off boundary between two regions of different index of refraction
The Raylaigh Criterion for circular aperture

where D is the diameter of the hole. This basically just tells you that the angular separation between two objects seen through a hole must be greater than for them to appear as separate objects through the hole.
Single slit diffraction minima
Minima occur when
Constructive interference vs destructive
Constructive occurs when there is an overall phase shift of for some m, while destructive occurs when the phase shift is for some m.
Bragg diffraction
Maxima occur at

where theta is the angle of the incident X-rays wrt the plane of the crystal. Similar to double slit fomula, but the X-rays must traverse the distance between the two layers twice, hence the extra factor of 2.
Snell's Law

where both angles are taken from the normal to the surface of refraction.
Focal length equation in Geometric Optics

where f is the focal length of the lens, s is the position of the object, and s' the position of the image.
Magnification

where s' is the position of the image and s is the position of the object wrt to the lens.
Rayleigh Scattering

where I is the intensity of scattered light, is the intensity of the light we see and is the particle size. This effect is responsible for the blue sky and the red sunset.
Doppler Effect (sound)

where v is the velocity of the wave, is the velocity of the receiver, and is the velocity of the source. The signs of is positive if the motion is toward the source, and is positive if the source is moving toward the receiver.
Speed of sound
The Partition Function

where
Average energy in an ensemble
Probability of state i in ensemble
Entropy

where is the number of possible microstates and k is Boltzmann's constant.

where T is temperature.

If its reversible and all you care about is the change in entropy:

Equipartition Theorem
Every degree of freedom contributes 1/2 kT  to the internal energy and thus 1/2 k to the heat capacity.

Generally at low energies most of the degrees of freedom freeze out leaving 3/2 kT. At very high energies for a diatomic molecule, it caps at 7/2 kT.
Stirling's Approximation
The Three Laws of Thermodynamics
1. Energy cannot be created or destroyed:

where U is the internal energy, Q is heat, W work.

2. There is no process who's sole effect is to transfer heat from a hot body to a cold body. Another way to put it is that no engine can convert energy from a heat reservoir entirely into mechanical energy.
Mathematically:

3. Entropy is zero at absolute zero.
Reversible Process
A slow process that creates no extra entropy (the total change between the system and the outside is 0). In a reversible process the following is true:

Quasistatic
A process so slow its considered constantly in equilibrium. A reversible process is always quasistatic.

Work done goes as if P constant, or .
Isothermal
.

If taken to be a slow process, work can be calculated with quasistatic work.
No heat is exchanged,

which can be used to solve the work using quasistatic work. where f is the degrees of freedom.
Isentropic
No change in entropy at all. This is both adiabatic and reversible.
Thermodynamic identity
Temperature and Pressure

both of which come straight from the thermodynamic identity. If that's easier to memorize than skip this.
Heat Capacity
There is two, defined for either constant volume or pressure:

.

An important side note, for ideal gases always.
RMS Velocity
Fermi-Dirac Distribution
The partition function for fermions looks like

which gives an occupancy (average number of particles) of

Bose-Einstein Distribution
For bosons the partition function looks like

which gives an occupancy
Energy of a free particle

which should be easy to remember as and the normal energy formula for a free particle

where is the reduced mass. ( for positronium.)
Ground State Energy for Hyrdrogen

13.6 is super important, but the form of the equation is important for Hydrogen-like atoms.
Energy of excited states of hydrogen

where the second one is substituting in the definition of the Bohr radius and the last number is only valid for Hydrogen.
Spin Eigenvectors
First Order Correction to a perturbed Hamiltonian

where is the unperturbed nth Energy, similarly for the wavefunctions while lambda is the perturbation parameter..
Useful trig identities

Infinite Square well energy

where a is the width of the well.
Stefan Boltzman Law

where P is power radiated, and A is area.
Wien's displacement Law

where is the peak wavelength emitted by a blackbody.
Lorentz Transformations

For the reverse transformations, change the sign of v and switch the primed with the unprimed.

where w is the speed an object in frame S, v is the speed of S', and u is the speed of the object in the S' frame.
Relativistic momentum
Relativistic Kinetic Energy
The rest energy is and the total energy is given by which implies that the kinetic energy is

100
0
Relativistic dot product

This is ignoring the metric tensor and some other jazz.
101
0
Invariant interval
If we define

There are 3 classifications for its value:

102
0
Energy-Momentum Invariance
is invariant (the same in all inertial reference frames) and conserved.
103
0
Invariant Dot Product

where p is the momentum 4-vector. SUPER USEFUL for calculations.
104
0
Relativistic Doppler Shift (Light)

where
105
0
The 4-vectors
106
0
Boost Matrix along x-axis

which reduces to the regular Lorentz transformations as long as you remember . NOTE, to get the inverse transformation you just need to change the signs of the off diagonal elements and switch the primes.
107
0
Poison Distribution

where is the average amount of counts in a given interval.

Remember its times a term of the power series for

It has error for large N.

108
0
Common and useful constants
13.6 eV - Binding Energy of Hydrogen

.5 MeV/ - Mass of Electron (Proton is ~2000 times this, 1 GeV

1.22 - Rayleigh criterion coefficient for a circle aperture.

- Wien displacement law constant

2.7 K - T of cosmic microwave background

- hc in the given units.

109
0
Impedance of L, C, and R

where is the driving frequency of the AC circuit.
110
0
Max energy of an electron emitted from photoabsorption (Photoelectric effect)

where is the energy of the incident photon which is absorbed and is the work function of the material.
111
0
Wavelength shift of light under Compton scattering

where is the angle the photon is scattered at.
Flashcard set info:
Author: CoboCards-User
Main topic: Physics
Topic: GRE
School / Univ.: UC Santa Cruz
City: Santa Cruz, CA
Published: 20.10.2015

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