Physical Optics, Geometrical Optics, Refraction, Reflection, Snell’s Law, Critical Angle, Total Reflection, Angle Of Deviation, Dispersion, Rainbow.
.
GLOSSARY
For details on the following concepts, please consult Chapter 13.
PHYSICAL OPTICS In physical optics we treat the phenomena that arise due to the wave nature of the light, as well as interference phenomena.
GEOMETRICAL OPTICS The phenomena that arise when light can be considered to be adequately described by rays traveling in straight lines (perpendicular to the wave fronts) that change speed in moving from one medium to another. This is the case as long as the objects through which the light travels have dimensions that are much larger than the wavelength of the wave.
REFRACTION The change of direction of a ray of light in passing obliquely from one medium into another in which its wave velocity is different. We define a quantity called the “index of refraction,” n, in terms of the velocity of light in the material, v, relative to its velocity, c, in a vacuum:
n = c/v
The frequencies of the transmitted and reflected waves are the same as that of the incident waves since the rate of oscillation in the disturbance is precisely what is propagated from one location to the next. Since T = l/f is constant we can calculate the new wavelength as the distance traveled during one period, or
λ’ = vT = vλ/c = λ/n
REFLECTION The act of casting back the light, mirroring, or giving back or showing an image; the state of being reflected in this way. The angle of reflection equals the angle of incidence, i.e.
θr = θ1
SNELL’S LAW The angle of refraction is given by
n1 sin θ1 = n2 sin θ2
It is important to note, as can be seen from the geometry in the figure, that the angle of incidence, reflection and refraction also represent the angles that the wave fronts of the incident, reflected and refracted waves respectively make with the surface.
CRITICAL ANGLE When light travels from a dense (n2) to less dense (n1) medium, the angle of refraction is greater than the angle of incidence. The critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees.
sin θc = n1 / n2
TOTAL REFLECTION When light is incident from a dense (n2) to less dense (n1) medium, at an angle greater than the critical angle, no light is refracted so all the light must be reflected. We call this case one of “total reflection“. Total reflection is very useful for bending light at a surface without losing any of the energy to transmission through the surface.
ANGLE OF DEVIATION When the light leaves the prism its direction of motion is at an angle δ from the incident direction. This is the “angle of deviation“.
DISPERSION For materials there is a small dependence of the velocity of light, and therefore n, on the wavelength. This property is called “dispersion” since it can be used to disperse the various wavelengths that are included in a beam of light into different refractive paths, creating a “spectrum”. As a result of the dispersion, white light refracted in a prism will be separated into its constituent wavelengths after passing through the prism
RAINBOW A rainbow is another case in which the variation of index of refraction with wavelength leads to a spectrum.
Displacement Current, Maxwell’s Equations, Gauss’ Law, Magnetic Fields, Faraday’s Law, Ampere’s Law, Electromagnetic Waves, Electromagnetic Spectrum, Electromagnetic Wave Equation, Energy And Momentum Flux, Poynting Vector, Radiation Pressure
.
GLOSSARY
For details on the following concepts, please consult Chapter 12.
DISPLACEMENT CURRENT We know that the field within a parallel plate capacitor is uniform and is equal to E = q/ε0A, where q is the charge on the capacitor and A is the area of the capacitor. If the capacitor is being charged, then both q and E are changing, and we can write that ∆E/∆t = (∆q/∆t)/ε0A = ID/ε0A, where ID is the displacement current between the capacitor plates (compared to the conductor current in the wire).
Thus,
ID = ε0A(∆E/∆t) = ε0 (∆E A/∆t) = ε0 (∆ψ/∆t)
where ψ is the electric flux through the area.
MAXWELL’S EQUATIONS These four equations are relationships between the electric and magnetic fields and their sources, charges and currents. The electric and magnetic fluxes are determined directly from the electric and magnetic fields and are not separate variables. Thus, these equations tell us how to calculate the electric and magnetic fields that are produced by charges, both at rest and moving. The particular form that we have used for these equations is not the most useful for actual calculations but is the easiest to understand conceptually. For purposes of calculations, these equations are expressed more formally in the language of the integral and differential calculus, which can then be solved for specific cases.
(1) GAUSS’ LAW The electric fields can be established by free charges. All electric field lines start at positive charges and end on negative charges (lines can also go to infinity, such as those of an isolated point charge, where they are presumed to land on opposite charges at that distance). By convention the number of electric field lines per unit area, the electric flux density, at a given point is chosen equal to the magnitude of the electric field at that point. It then equals the electric field at every other point as well. Gauss’ law then relates the total charge within a closed surface to the net number of electric field lines that pass through the surface.
(2) MAGNETIC FIELDS There are no magnetic monopoles that act as sources for a magnetic field. Therefore, magnetic fields do not have poles where they begin or end. All magnetic field lines must therefore close on themselves. This means that any magnetic field line that passes through a closed surface must necessarily pass through the surface again in the opposite direction, in order to close on itself. This means that the net total magnetic flux which passes through a surface is zero.
(3) FARADAY’S LAW An electric field can also be produced by a changing magnetic flux.
(4) AMPERE’S LAW The magnetic fields are created by currents, either conduction current or displacement current.
These four equations constitute Maxwell’s equations, which are the fundamental laws governing the existence of electric and magnetic fields, which are jointly called electromagnetic fields. Electric fields exert forces on any electrical charges, while magnetic fields exert forces on moving charges.
ELECTROMAGNETIC WAVES Maxwell was able to show that there were solutions to these equations that corresponded to waves propagating in free space, i.e., in regions where there are no charges or currents. These waves, which he called electromagnetic waves (EM) had special properties, which could be derived from these equations.
In the case of electromagnetic waves the time varying quantity is not the displacement but rather the electric and magnetic fields at a point in space.
These waves are transverse, and their speed, in free space is equal to
For a wave traveling in the x direction, the electric and magnetic fields associated with this wave are in the y-z plane. These fields are also perpendicular to each other, and that their magnitudes are given by:
E = cB
It is useful to consider electromagnetic waves that are sinusoidal. This means that if we take a picture of the wave at any time, the disturbance will vary sinusoidally in space along the direction of propagation. Furthermore, at any position is space, the disturbance will vary sinusoidally in time.
The disturbance associated with an electromagnetic wave is the electric and magnetic field along the wave. Light consists of electromagnetic waves in a certain frequency range to which the eye is sensitive and can “see”.
ELECTROMAGNETIC SPECTRUM Electromagnetic waves exist with wavelengths ranging from very small to very large (and corresponding frequencies from very large to very small). The various possible wavelength (and frequency) ranges constitute the electromagnetic spectrum. For small frequencies the wave is usually denoted by its frequency, and for short wavelength it is denoted by its wavelength.
All of these waves travel with a speed of c, all are transverse, and all carry perpendicular electric and magnetic fields with them.
ELECTROMAGNETIC WAVE EQUATION The equation for the disturbance of an electromagnetic sinusoidal plane wave, traveling in the + x direction, is given in terms of its disturbance (an electric field in the y direction) by
Here ω is the angular frequency of the wave, and k is the “wavenumber” of the wave and has units of m-1. E0 is the maximum value of the electric field and is thus the amplitude of the wave.
For spherical waves, since the area of a spherical surface is 4πr2, the intensity falls off as l/r2. The amplitude of the wave, A, is related to the intensity by I α A2, and therefore A falls off as l/r. The formula for the magnitude of E is given by
ENERGY AND MOMENTUM FLUX The electric and magnetic fields contain energy (substance). The energy density is uE = ε0E2/2 for electric fields and uB = B2/2μ0 for magnetic fields. The electromagnetic energy of an electromagnetic wave is just the sum of the energies of its electric and magnetic fields. The maximum energy is located at those points where the fields are at their maxima, which occurs at the crests of these waves. But these crests move with time at a speed of c, and therefore the energy is transported in the direction that the wave travels at this speed. An electromagnetic wave, therefore, carries energy and momentum with it.
The average energy transported per unit area and time as the wave travels with speed c in the x direction is defined as the intensity:
POYNTING VECTOR The Poynting vector S depicts the direction and rate of transfer of energy, that is power,due to electromagnetic fields in a region of space. Its magnitude is EB/μ0, and its direction is perpendicular to E and B, and obeying the right-hand rule.
Whenever energy moves in a certain direction, there is also a certain amount of momentum in that direction. This is because energy is substance.
RADIATION PRESSURE One can use sunlight in space to not only supply power but to exert a force on a spacecraft. The force that is exerted on the surface can best be characterized by the force exerted per unit area, or pressure, P = F/A.
DC CIRCUIT,R-C CIRCUIT, R-L CIRCUIT, L-C CIRCUIT, AC CIRCUIT, IN PHASE, OUT OF PHASE, RMS VALUE, CAPACITIVE REACTANCE, IMPEDANCE (R-C CIRCUIT), INDUCTIVE REACTANCE, IMPEDANCE (R-L CIRCUIT), R-L-C CIRCUIT
.
GLOSSARY
For details on the following concepts, please consult Chapter 11.
DC CIRCUIT A DC (direct current) circuit generally consists of a battery acting as the energy source, through its EMF, and causing a steady voltage to act across one or more resistors, capacitors or inductors. In the steady state, after all transient phenomena have stopped, there is no voltage across an inductor, since the current is no longer changing. There is a steady voltage across a capacitor, equal to Q/C, but there is no current flowing to or from the capacitor. The voltage across a resistor will equal V = IR.
R-C CIRCUIT A DC circuit with a resistor R and a capacitor C shows a transient response when the switch is connected or disconnected.
At the moment the switch is closed, there is current in the circuit but no charge on the capacitor, and when the capacitor is fully charged, there is no current in the circuit. The current decays exponentially, which can be expressed as,
The situation is very similar in the case of the discharge of a capacitor.
R-L CIRCUIT A DC circuit with a resistor R and an inductor L also shows a transient response when the switch is connected or disconnected.
In a similar manner, if current is decreasing in an L-R circuit, it will decay exponentially with a time constant L/R.
L-C CIRCUIT In this circuit there is no dissipation of energy because there is no resistor. The capacitor stores energy in the form of separated charges or electric fields. The inductor stores energy in the form of moving charges or magnetic fields. The total energy remains in the system.
Therefore, the separated charges and the current interchange. This would bean oscillatory situation, with a repetitive interchange of energy between the capacitor and the inductor ad infinitum.
In the above circuit wefirst close switch S1 while S2 is open, and charge the capacitor to a voltage V, and charge Qmax. We then open S1 with the capacitor charged, and close S2 to permit the capacitor to discharge through the inductor. The capacitor discharges and then charges up again. This is similar to the case of simple harmonic motion (SHM).
This frequency, f, is called the resonance frequency of the circuit.
AC CIRCUIT An AC (alternate current) circuit generally consists of asource of voltage that is varying sinusoidally.In that case we expect that the variables of the circuit will also vary sinusoidally, after there has been sufficient time for the circuit to reach a steady state. This time is usually short enough that the effect of the transients can be neglected.
Whenever we have sinusoidal variation, we can express the variables as sine or cosine functions of time. The frequency f of the sinusoidal variation can be expressed in terms of an angular frequency ω, which simplifies the equations. Of course, the frequency can also be related to the period, T, by the relationship f= 1/T.
IN PHASE In AC circuit with a resistor,both current i and voltage v vary identically with time as cos ωt. We say that the two are “in phase”. This means that they both attain their maximum value at the same time, and both go through zero at the same time. We will see that this is true only for a resistor, while for capacitors and inductors, the voltage will not be in phase with the current.
The voltage at any time is just a fixed multiple of the current.
OUT OF PHASE When a generator causes a sinusoidal current to flow through the capacitor given by i = I0 cos ωt the capacitor is alternately charging each plate positively and negatively at a frequency: f = ω/2π. The voltage across the capacitor is defined to be from positive to negative plate and is always given as v = q/C. Both v and q will vary sinusoidally at the same frequency as the current. When the voltage reaches its peak, the current is zero. The current and voltage are said to be 90° or π/2 out of phase, and the current “leads” the voltage. We thus know that, for a capacitor, we can represent the voltage by v = VI0 sin ωt if the current is given by i = I0 cos ωt.
RMS VALUE In most formulas used in AC circuits, the quantity we use for the “magnitude” of currents and voltages will be the RMS value, and therefore when we write just I or V we will refer to the RMS values. The term RMS actually stands for “root-mean-square”, which refers to the method used to determine its value. To get the RMS value of a variable, we have to take the square root of the average (mean) of the square of the quantity. The current will vary as i = I0 cos ωt.I0 is the amplitude of the variation, and it represents the maximum value the current can have. We have the “RMS” value: IRMS = I0 √2.
CAPACITIVE REACTANCE We expect that if the maximum current is increased then the maximum charge on the capacitor will increase proportionally, and therefore also the maximum voltage. Consequently, we can write that V0 = χcI0, where the constant of proportionality χc is called the capacitive reactance of the capacitor. Similarly, VRMS = χcIRMS, or V = χcI. This capacitive reactance depends on the capacitance and on the frequency. Therefore, we have,
IMPEDANCE (R-C CIRCUIT) When theResistor and Capacitor are in Series, with the current given by as i = I0 cos ωt, the voltage across the entire circuit is
INDUCTIVE REACTANCE An AC generator produces a current I = I0 cos ωt in the inductor. The inductor produces a back EMF equal to (- L ∆I/∆t). This back EMF is balanced by the electrostatic voltage across the inductor, vL, as shown in the figure.
The voltage across the inductor is 90° out of phase with the current. The voltage leads the current,
IMPEDANCE (R-L CIRCUIT) When the Resistor and are in Series, with the current given by as i = I0 cos ωt, the voltage across the entire circuit and impedance is given by
We see that cos φ is again the power factor for an R-L circuit as it was for an R-C circuit. We can generally write that cos φ = X/R, where X is the reactance of the circuit, and equals XL for a circuit with inductance and -XC for a circuit with capacitance. Similarly, the impedance can then be written as Z = (R2 + X2)1/2, which will be valid for both R-C and R-L circuits. Additionally, we can write that the total voltage will vary with time as vT = V0T cos (ω t + φ), both for the case of the R-C and the R-L circuits, For the R-C circuit, φ is negative, and in the R-L circuit, φ is positive. We will find that we can extend these ideas to the last case, the R-L-C circuit also.
R-L-C CIRCUIT Here we have Resistor, Inductor and Capacitor in Series. We can write the equations giving these respective voltages as functions of time as:
From the phasor diagram we can deduce other relationships:
Self-Inductance, Mutual Inductance, Inductor, Energy in an Inductor, Energy Density in Space, Transformers
.
GLOSSARY
For details on the following concepts, please consult Chapter 10.
SELF-INDUCTANCE Self-inductance arises from the flux that a current circuit produces within its own area. It is distinguished from mutual inductance. Self-inductance depends only on the geometry of the circuit. It connects the flux with the current as follows,
The unit for inductance is Wb/A, which is given the name henry. Practical circuits have inductance much smaller than one henry, more in the range of millihenries. The main use of the concept of inductance will be in circuits where the current changes, thus causing a proportional change in flux. This changing flux induces an EMF:
Solenoid The self-inductance, L, of the solenoid is: L = μ0 n2Ad; and the inductance per unit length is: L/d = μ0 n2A.
Toroid The self-inductance, L, of the solenoid is: L = μ0 N2A/2πr. If the toroid is filled with material of permeability μ, then: L = μ N2A/2πr.
MUTUAL INDUCTANCE Whenever one has two circuits near each other, it will be possible for a current which exists in one circuit to produce flux through the second circuit. If φ12 is the flux in circuit 2 caused by a current I1, in circuit 1, Then, the mutual inductance, M12, connects these two quantities, as follows,
The exact value of M12 is determined by the geometrical relationship between the two circuits. The change in current in circuit 1 changes the flux in circuit 2 proportionally, which produces an EMF in circuit 2, given by
There is only one mutual inductance for the two circuits meaning M12 = M21. We can measure the mutual inductance by measuring the induced EMF produced in one circuit by a known rate of change in current in the other circuit.
Coil on Solenoid
Coil on Toroid
Coil Near Long Wire
Coil at Center of Loop
INDUCTOR Any circuit element that generates an inductance when current flows through it (e.g. a coil, a solenoid, a toroid) is called an inductor. An inductor has the property that it produces a back EMF if the current is changing, but does nothing if the current is steady. While an inductor will not affect a DC circuit once a current has been established it will be of great importance during the time that the current is being turned on or off.
ENERGY IN AN INDUCTOR In order to increase the current in the inductor, an external driving voltage must be imposed on the circuit to overcome the back EMF, and this voltage will do work against the resisting EMF. The voltage will continue to do work until the current reaches its final value, at which time the current is no longer changing and no back EMF is being produced. During the time that the current is building up from zero to its final value, however, work must be done on the inductor. The work, or energy stored in the inductor is,
This result is similar to the case of storing energy in a capacitor by virtue of the charge that we have placed on the plates of the capacitor. There the energy stored = (1/2)Q2/C.
In terms of the magnetic field that have been set up in space, we have,
At any point in space, where there is a magnetic field, a certain amount of energy is stored. This energy equals the energy density times the volume of space being considered. The same general consideration holds for electric fields as well and indeed the electric field energy density is given by (1/2) ε0E2. In other words, wherever electric or magnetic fields exist in space, energy is being stored in the form of these fields
ENERGY DENSITY IN SPACE The total energy density at any point in space is the sum of the electric and the magnetic field energy densities. Since the units for energy density are the same irrespective of their source, this offers a means of comparing the relative magnitudes of electric and magnetic fields. Electric and magnetic fields with the same energy density can be considered to be comparable to each other. The electric and the magnetic fields associated with electromagnetic waves have equal energy densities. These considerations lend credence to the idea that these fields are real physical quantities that actually exist in space and are not merely mathematical contrivances that make it easier to calculate the forces exerted by the electric and magnetic interactions.
TRANSFORMERS We can induce EMFs in one circuit by changing the current in another circuit. This forms the basis of the transformer, which is used to transform voltage in one circuit into a different voltage in a second circuit. All the magnetic flux established by the first winding, called the primary coil, passes through the turns of the other winding, called the secondary coil. In order to get large fluxes, it is useful to place ferromagnetic material within the solenoid that has a large permeability, such as iron. The figure below shows a typical transformer:
Here, the primary winding, with N1 turns, is wound on one side of the rectangular ring, and the secondary winding, with N2 turns, is wound on the other side of the ring. This is a typical transformer. If one changes the voltage in the primary circuit, the current in the primary circuit will change, and therefore the flux. For a perfect transformer, the flux through one turn of the secondary is the same as the flux through one turn of the primary. Therefore, the total EMF developed in each winding will depend on the number of turns in that circuit.
A transformer is useful only with currents that are changing, as with AC. In that case, it is possible to use a transformer to convert a voltage applied to the primary circuit into a larger or smaller voltage in the secondary circuit. This ability to easily convert (transform) voltages in AC, which is much more difficult for DC, is the main reason why AC is the primary source of power throughout the world.
Generator, Motional EMF, Induced EMF, Magnetic Flux, Faraday’s Law, Lenz’s Law, Alternating Current, Direct Current, Induced Electric Fields
.
GLOSSARY
For details on the following concepts, please consult Chapter 9.
GENERATOR When a wire moves through a magnetic field an EMF is generated in the wire, which has the ability to move charges through the wire. This means that it is possible to build an apparatus that makes use of magnetic effects to produce EMFs that drive electrical circuits connected to the apparatus. The apparatus is called a generator, and, like a battery, it pumps positive charges within the apparatus toward the high-potential end of the apparatus, so that, in an external circuit, the charges produce a current flowing from the high- to the low-voltage terminals. As in a battery, the voltage produced by the generator on open circuit is its EMF.
MOTIONAL EMF An EMF produced in wires moving through a magnetic field is called motional EMF. For a wire of length L, with uniform electric field E, the potential difference is Vba = EL, and using the relation result E = vB, we have for our moving wire:
INDUCED EMF An EMF produced in stationary wires that are situated in a changing magnetic field is called induced EMF. Its characteristics are given by Faraday’s law.
MAGNETIC FLUX It is a concept similar to electric flux. As in the case of electric flux one can visualize the magnetic flux by drawing magnetic field lines, with the number of field lines passing through a unit area perpendicular to the lines proportional to B at that location.
In the following diagram there is a small planar area A represented by a vector A that has a magnitude equal to the area, and a direction perpendicular to the plane of the area. We use the notion of circulating current and the right and rule to determine the positive direction of A. For a magnetic field B that passes through the area in the positive direction at an angle θ to A, we define the magnetic flux as
The total magnetic flux through any area is just proportional to the total number of field lines through that area. The unit for magnetic flux is T – m2, which is given the name Weber (Wb).
FARADAY’S LAW This law says that whenever there is a change in flux within a circuit there will be an EMF induced in the circuit. This EMF depends on the time rate of change of the flux through the circuit,
where ∆φ is the change in magnetic flux through the circuit in a short time interval, ∆t. The minus sign is necessary to assure that the correct direction is given for the EMF. The requirement of the minus sign is called Lenz’s law.
LENZ’S LAW This law states that the EMF produced by a changing flux is always in a direction to produce a current whose own flux is in the opposite direction to the initial change in flux.
ALTERNATING CURRENT One can generate an EMF by rotating a coil with an angular velocity ω in a magnetic field B. Then the angle θ = ωt. The flux (φ) through a single turn of the coil is given by BAcos θ, or BAcos ωt.
By differentiating, we get the EMF to be,
The EMF varies as sin ωt. The EMF produced in this manner will change its direction and then change back again at an angular frequency ω, or at a frequency f = ω/2π and period T = 2π/ω. This is what we call an alternating voltage which produces an alternating current (AC). Thus, by rotating a coil in a magnetic field, we can easily generate an AC voltage. The magnitude of the voltage can be increased by constructing the coil out of many turns, N, of wire, in which case the voltage becomes,
DIRECT CURRENT It is also possible to construct generators to produce DC (direct current) voltage. To accomplish this, we reverse the connection to the outside wires every time the direction of the EMF in the coil reverses direction. The resultant EMF in the outside circuit will then take the form shown below.
Secondly, we use several coils (armature), in which each coil will produce a voltage which reaches its maximum at a different time, and the total voltage will vary very little with time.
INDUCED ELECTRIC FIELDS Any changing magnetic field actually produces a new type of electric field in the vicinity of the circuit that pushes the charges and creates the EMF in the stationary circuit. This new electric field is fundamentally different from the “electrostatic” field produced by point charges. . Thus, Faraday’s law has profound implications for our concept of the electric field.
