Template:Physeq2
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18 Electric charge and field
editScalarElectricField
edit- is Coulomb's law for the force between two charged particles separated by a distance r: ke≈8.987×109N·m²·C−2, and ε0≈8.854×10−12 F·m−1.
- is the electric force on a "test charge", q, where is the magnitude of the electric field situated a distance r from a charge, Q.
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VectorElectricField
editConsider a collection of particles of charge , located at points (called source points), the electric field at (called the field point) is:
- is the electric field at the field point, , due to point charges at the source points, , and points from source points to the field point.
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VectorElectricFieldCALCULUS
edit- is the electric field due to distributed charge, where , and denote linear, surface, and volume density (or charge density), respectively. Here, k=1/4πεo.
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19 Electric Potential and Electric Field
editDefineElectricPotential
edit- is the potential energy of a particle of charge, q, in the presence of an electric potential V.
- is a unit of energy, defined as the work associated with moving one electron through a potential difference of one volt.
- (measured in Volts) is the variation in electric potential as one moves through an electric field . The angle between the field and the displacement is θ. The electric potential, V, decreases as one moves parallel to the electric field.
- describes the electric potential if the field is not uniform.
- due to a set of charges at where .
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Capacitors
edit- is the (equal and opposite) charge on the two terminals of a capacitor of capicitance, C, that has a voltage drop, V, across the two terminals.
- is the capacitance of a parallel plate capacitor with surface area, A, and plate separation, d. This formula is valid only in the limit that d2/A vanishes. If a dielectric is between the plates, then ε>ε0≈ 8.85 × 10−12 due to shielding of the applied electric field by dielectric polarization effects.
- is the energy stored in a capacitor.
- is the energy density (energy per unit volume, or Joules per cubic meter) of an electric field.
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FieldTheoryCALCULUS
edit- in the limit that the Riemann sum becomes an integral.
- where is the del operator.
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VolumeSurfaceLineElements
editCartesian coordinates (x, y, z):
- is a volume element
- and and are area elements of the form
- is a line element.
Cylindrical coordinates (ρ, φ, z):
- is a volume element
- if azimuthal symmetry holds.
- and are surface elements in cylindrical coordinates.
- and and are line elements in cylindrical coordinates.
Spherical coordinates (r, θ, φ): Spherical symmetry holds when nothing depends on the angular variables.
- is the simplest line element in spherical coordinates.
- defines the solid angle. The solid angle of a sphere is 4π steradians.
- is the volume element of a spherical shell of radius r and thickness dr.
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GradStokesDivThms
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/SurfacesWithAndWithoutBoundary.svg/200px-SurfacesWithAndWithoutBoundary.svg.png)
To the left are closed surfaces. To the right are open surfaces, Ω, that possess closed boundaries, ∂Ω.
- is the gradient theorem.
- is the divergence theorem
Here, Ω is a (3-dimensional) volume and ∂Ω is the boundary of the volume, which is a (two-dimensional) surface. Also a surface is Σ, which, if open, has the boundary ∂Σ, which is a (one-dimensional) curve.
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GausslawSimple
edit- is Gauss's law for the surface integral of the electric field over any closed surface, and is the total charge inside that surface. The vacuum permittivity is ε0≈ 8.85 × 10−12.
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Gausslaw
edit- is Gauss's law for the surface integral of the electric field over any closed surface, and is the total charge inside that surface.
- is a useful variant if the medium is dielectric. D=εE is the electric displacement field. The permittivity, ε = (1+χ)ε0, where ε0≈ 8.85 × 10−12, and the electric susceptibility, χ, represents the degree to which the medium can be polarized by an electric field. The free charge, Qfree, represents all charges except those represented by the susceptibility, χ.
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20 Electric Current, Resistance, and Ohm's Law
editOhmsLawResistivity
edit- defines the electric current as the rate at which charge flows past a given point on a wire. The direction of the current matches the flow of positive charge (which is opposite the flow of electrons if electrons are the carriers.)
- is Ohm's Law relating current, I, and resistance, R, to the difference in voltage, V, between the terminals. The resistance, R, is positive in virtually all cases, and if R > 0, the current flows from larger to smaller voltage. Any device or substance that obeys this linear relation between I and V is called ohmic.
- relates the density (n), the charge(q), and the average drift velocity (vdrift) of the carriers. The area (A) is measured by imagining a cut across the wire oriented such that the drift velocity is perpendicular to the surface of the (imaginary) cut.
- expresses the resistance of a sample of ohmic material with a length (L) and area (A). The 'resistivity', ρ ("row"), is an intensive property of matter.
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FundamentalDefinitions
edit- Power is energy/time, measured in joules/second or J/s. Often called P (never p). It is measured in watts (W)
- Current is charge/time, measured in coulombs/second or C/s. Often called I or i. It is measured in amps or ampheres (A)
- Electric potential (or voltage) is energy/charge, measured in joules/coulomb or J/C. Often called V (sometimes E, emf, ). It is measured in volts (V)
- Resistance is voltage/current , measured in volts/amp or V/A. Often called R (sometimes r, Z) It is measured in Ohms (Ω).
- is the power dissipated as current flows through a resistor
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21 Circuits, Bioelectricity, and DC Instruments
editMiscCircuits
edit- is the effective resistance for resistors in series.
- is the effective resistanc vor resistors in parallel.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Voltage_divider.svg/120px-Voltage_divider.svg.png)
- and are Kirchoff's Laws[1]
- for the voltage divider shown.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Capacitor_resistor_series.svg/180px-Capacitor_resistor_series.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Capacitor_Square_wave_charge-discharge.svg/360px-Capacitor_Square_wave_charge-discharge.svg.png)
- Simple RC circuit[2] The figure to the right depicts a capacitor being charged by an ideal voltage source. If, at t=0, the switch is thrown to the other side, the capacitor will discharge, with the voltage, V , undergoing exponential decay:
where V0 is the capacitor voltage at time t = 0 (when the switch was closed). The time required for the voltage to fall to is called the RC time constant and is given by
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22 Magnetism
editVectorMagneticForce
edit- is the force on a particle with charge q moving at velocity v with in the presence of a magnetic field B. The angle between velocity and magnetic field is θ and the force is perpeduclar to both velocity and magnetic field by the right hand rule.
- expresses this result as a cross product.
- is the force a straight wire segment of length carrying a current, I.
- expresses thus sum over many segments to model a wire.
- CALCULUS: In the limit that we have the integral, .
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DefineMagneticFieldVector
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Magnetic_field_element_%28Biot-Savart_Law%29.svg/180px-Magnetic_field_element_%28Biot-Savart_Law%29.svg.png)
- is the contribution to the field due to a short segment of length carrying a current I, where the displacement vector r points from the source point to the field point.
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DefineMagneticFieldVectorCALCULUS
edit- and the volume integral , where is current density.
- is Ampere's law relating a closed integral involving magnetic field to the total current enclosed by that path.
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DefineMagneticFieldScalar
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Manoderecha.svg/140px-Manoderecha.svg.png)
- is the magnetic field at a distance r from an infinitely long wire carrying a current, where μ0 = 4π × 10−7 N A. This field points azimuthally around the wire in a direction defined by the right hand rule. Application of the force law on a current element, we have
- is the force between two long wires of length separated by a short distance . The currents are I1 and I2, with the force being attractive if the currents are flowing in the same direction.
Call with {{Physeq2|transcludesection=DefineMagneticFieldScalar}}These equations are exact in that they serve to define both μ0 as well a the ampere.
CyclotronMotion
editFor a particle moving perpendicular to B, we have cyclotron motion. Recall that for uniform circular motion, the acceleration is a=v2/r, where r is the radius. Since sin θ =1, Newton's second law of motion (F=ma) yields,
Since, sin θ =0, for motion parallel to a magnetic field, particles in a uniform magnetic field move in spirals at a radius which is determined by the perpendicular component of the velocity:
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HallEffect
editThe Hall effect occurs when the magnetic field, velocity, and electric field are mutually perpendicular. In this case, the electric and magnetic forces are aligned, and can cancel if qE=qvB (since sinθ = 1). Since both terms are porportional to charge, q, the appropriate ratio of electric to magnetic field for null net force depends only on velocity:
- ,
where we have used the fact that voltage (i.e. emf or potential) is related to the electric field and a displacement parallel to that field: ΔV = -E Δs cosθ
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23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
editFaradayLaw
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Moving_Conductor_In_Magnetic_Field.jpg/200px-Moving_Conductor_In_Magnetic_Field.jpg)
- is a consequence of the magnetic force law as seen in the reference frame of a moving charged object, where E is the electric field perceived by an observer moving at velocity v in the presence of a magnetic vield, B. Also written as, E = vBsinθ, this can be used to derive Faraday's law of induction. (Here, θ is the angle between the velocity and the magnetic field.)
- is the magnetic flux, where θ is the angle between the magnetic field and the normal to a surface of area, A.
- is Faraday's law where t is time and N is the number of turns. The minus sign reminds us that the emf, or electromotive force, acts as a "voltage" that opposes the change in the magnetic field or flux.
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24 Electromagnetic Waves
editDisplacementCurrent
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Displacement_current_in_capacitor.svg/240px-Displacement_current_in_capacitor.svg.png)
- is called the displacement current because it replaces the current density when using Ampère's circuital law to calculate the line integral of the magnetic field around a closed loop.
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25 Geometric Optics
editThinLensEquation
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Lens3.svg/500px-Lens3.svg.png)
relates the focal length f of the lens, the image distance S1, and the object distance S2. The figure depicts the situation for which (S1, S2, f) are all positive: (1)The lens is converging (convex); (2) The real image is to the right of the lens; and (3) the object is to the left of the lens. If the lens is diverging (concave), then f < 0. If the image is to the left of the lens (virtual image), then S2 < 0 .
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26 Vision and Optical Instruments
edit27 Wave Optics
editInterferenceDiffraction
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/bc/TwoSlitYoung%27s.png/120px-TwoSlitYoung%27s.png)
- where describes the constructive interference associated with two slits in the Fraunhoffer (far field) approximation.
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Beats
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Beats52.png/240px-Beats52.png)
- where is the high frequency carrier and is the slowly varying envelope. Here,
- and . Consequently, the beat frequency heard when two tones of frequency and is .
- models the addition of two waves of equal amplitude but different path length, .
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28 Special Relativity
edit29 Introduction to Quantum Physics
edit30 Atomic Physics
edit31 Radioactivity and Nuclear Physics
edit32 Medical Applications of Nuclear Physics
edit33 Particle Physics
edit34 Frontiers of Physics
editLight: Refraction, ray optics, diffraction, and polarization
editField theories
editDefine g, E, B (static case)
edit- ↑ https //en.wikipedia.org/w/index.php?title=Kirchhoff%27s_circuit_laws&oldid=579357795
- ↑ From https://en.wikipedia.org/w/index.php?title=RC_circuit&oldid=598786790