OpenStax College Physics/Equations
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Oscillatory Motion and Waves
- , is Hooke's law, where F is the restoring force, k is the force constant, and x is the displacement from equilibrium.
- is the potential energy stored in the deformation of a spring or elastic system.
- where f is the frequency and T is period. It is convenient to define omega as, , for a spring-mass system. For a simple pendulum, .
- describe the position, velocity, and acceleration of simple harmonic motion, respectively.
- is the wave velocity, where λ is wavelength.
- The node is a point of zero motion and the antinode is a point of maximum motion of a standing wave.
- is the beat frequency when two waves of slightly different frequency are superimposed.
- is intensity of sound, where P is the power passing through an area A, Δp is the pressure amplitude, and ρ is the density of the medium.
- is the intensity level in decibels, where I0 = 10-12W/m2 is the threshold level of hearing.
- is the observed Doppler shifted frequency for a stationary observer when the source of frequency fs is moving at speed . The (+/-) sign refers to motion (away/towards) the observer.
- is the frequency perceived by an observer moving at speed with respect to a stationary source of frequencyfs. The (+/-) sign refers to motion (towards/away) from the source.
- is the resonant frequency of the n-th mode of a standing sound wave in a tube that is closed at one end. The mode numbers are .
- with are the resonant modes for a tube open at both ends.
- is the intensity reflection coefficient, which is the ratio of the intensity of the reflected wave to that of the incident wave at a boundary between two media. The acoustical impedance is .
Electric Charge and Electric Field
The charge of the electron is qe = −e, where the fundamental charge is e≈1.6 x 10−19 C. (C≡Coulombs is the SI unit of charge).
- is Coulomb's law for the force between two point charges q1 and q2 separated by a distance r. Coulomb's constant is k ≈ 8.99x109 N m2 C−2.
- if the force on a test charge qt due to the electric field at the location of the test charge. The vector sum is over source charges, and the magnitude of each contribution to the field is where qn is the source charge and rnt is the distance from the source point to the test charge.
Electric Potential and Electric Field
- is the component of the electric field parallel to a small displacement (the electric field points from high to low voltage, and ΔV=Ed if one moves a distance d along a uniform electric field E.)
- is the electric potential at a distance r from a point charge Q.
- is the charge stored by a capacitor of capacitance C with a voltage drop V across its terminals. For a parallel plate capacitor, for plates of area A separated by a (small) distance d. The permittivity of free space is ε0 = 8.85×10−12F/m. If a dielectric fills the gap, replace ε0 by κε0
- is the total capacitance in series.
- is the total capacitance in parallel.
- is the energy stored in a capacitor.
Electric Current, Resistance, and Ohm's Law
- relates the current through the wire to the density of carriers n, charge of each carrier q, cross-sectional area A, and carrier drift velocity vd.
- is w:Ohm's law, where 1Ω=1V/A. where ρ is resistivity, L is length, and A is area. For changes in temperature T, both R and ρ typically increase linearly with ΔT, as , where α is the temperature coefficient of resistivity.
- is electrical power (energy divided by time). For alternating current, the average power is , where rms denotes root mean square). If , where X0 is the peak value and f is frequency.
Circuits and DC Instruments
If resistors are in series, the equivalent resistance is . If they are in parallel,
- relates the terminal voltage to current I, internal resistance r and emf is the current, and r is the internal resistance.
- Kirchoff's node rule is . The loop rules is . The voltage drop if the path crosses through a voltage source from the negative to the positive terminals, or if it travels opposite the current when passing through a resistor.
- is the decay constant for a resistor and capacitor.
- While charging, , so that if the voltage is , it will rise by in the next RC time.
- While discharging, the voltage will drop from to in one RC time.
MagnetismSI unit for magnetic field is the tesla: 1T=1·N ·C−1(m/s)−1 = 1N·A−1m−1,The magnetic force on a moving charge q in the presence of a magnetic field B is where v is speed and θ is the angle between the velocity and magnetic field. The direction of the force is given by the cross product version of the right-hand-rule (RHR-1).
- is the orbital radius of a charged particle in a magnetic field (m,v,q are mass, speed, and charge, respectively.)
- is the Hall emf across a distance ℓ, for moving charged particles (v B, ℓ are mutually perpendicular vectors.)
- is the force on a wire of length ℓ, current I, in the presence of uniform magnetic field. The direction follows RHR-1.
- is the torque on N turns of wire around an area A, where θ is the angle between the uniform magnetic field and the perpendicular to the loop.
- is the magnetic field at a distance r from a long straight wire, where μ0 = 4π × 10−7 T·m/A is the permeability of free space.
- is the magnetic field at the center of a loop of radius R.
- is the magnetic field inside a long, thin solenoid, where n=N/ℓ is the number of turns per unit length.
- Parallel wires attract (repel) if the currents are parallel (antiparallel), with .
- The net force on any charged particle is zero if and the velocity, magnetic, and electric fields are mutually perpendicular (see Wein filter.)
Electromagnetic Induction, AC Circuits, and Electrical Technologies
- If a coil has N turns the induced emf is
- The motional if a wire of length , its velocity and the magnetic field are mutually perpendicular.
- If a coil is rotating at angular speed ω in a magnetic field, , with the peak
- In an ideal transformer, the primary and secondary voltages and currents are related by and .
- Mutual induction
- Self inductance where N is the number of turns and is the current in one turn.
- Self inductance of a long thin solenoid , and where is the number of turns, is area and is length, where μ0 = 4π×10−7
H/m ≈ 1.2566...×10−6
N/A2 or T⋅m/A or Wb/(A⋅m) or V⋅s/(A⋅m).
- is the energy stored in an inductor.
Vision and Optical Instruments
Introduction to Quantum Physics
Radioactivity and Nuclear Physics
Medical Applications of Nuclear Physics
Frontiers of Physics
- frequency is revolutions per second and period is seconds per revolution: (rev/sec)(sec/rev)=1
- Note that vmax=ωX and amax=ω2X.
- occurs when two wave are out of phase; constructive interference occurs the waves are in phase and add to maximum amplitude.
- Audible frequencies typically range between 20 and 20,000 Hz. Infrasound (ultrasound) is below (above) that range.
- a=Γ2 where &Gamma is defined using Wikipedia:reflection coefficient.
- For more information about this sum, see w:Special:Permalink/791000226#Superposition_principle.
- Here the area is assumed to be normal to the flow of current. Also, I=JA where J is current density.
- the latter follows from energy conservation