Properties of Waves: Sound waves and the mechanics of hearing We rely on the audible frequency to communicate our taught, these are brought by the disturbance in the air and we may say that the fluctuation in a region alters the spread and concentration of point like particles in the air. The vibration may cause tension by displacing the particles and compressing them together and relaxes when the oscillating force is released. If we take a slice of a cylindrical unit volume of air we will have, Where part B is more doted than A signifying that B is more compressed that we know by the gas laws there is a change in pressure, to which our ears are sensitive. The change in volume means a change in pressure, so the volume of the cylindrical hollow of air is s∆v ∆v= Y2-Y1 or [(y(x x1+∆x1t) – y (x1t)] We differentiate thereby giving: The change in volume is related to the pressure fluctuation which is related according to the bulk modules B= - p (x1t) / dv/v Where dv/v change in volume, and p (x1t) is the pressure at points devoted by A and t, we look for the Because Then we find that, P (x,t)= B kA sin (kx- wt) So amplitude max is where the most concentration of particles and – p max is when it is most relaxed. Solve: example 16.1 (University of Physics 11th Edition by: Young and freedom page 595) In different medial we find that it was different modules thus speed varies. Graphical analysis of Wave motion: How would you draw waves on a Cartesian plane? What will be you’re x and y variables to produce a successful and logical representation of waves? Hint, Imagine and use the fact that waves are set of vibrations in a medium and its motion is being traveled over time. As time goes by vibrations are plotted against time and a periodic pattern is deduced. Write your answers here, Mechanical Waves: To understand and analyze mechanical wave, identify the types of mechanical waves, sight good and bad effects of mechanical waves. An earthquake occurred in Leyte/ Bohol which has intensity 7 according to the Richter scale and afterwards article about the devastating major quake fault in geological origin putting down the old structures of churches. “Quake is a form of disturbance in the medium which is the geological faults and energy is brought to other distant places where the old structures are built? Can you picture out how they travel inside the earth? Analogy simulation: Draw in a manila paper a structure of a city and create a wave by moving it back and forth. This will give you a lot of insights how waves (P, s) wave travels. Types of Mechanical Waves: Particles and matter react to the energy and hence a vector representing the interaction between them, the wave motion and the wave itself carries energy but the way how particles and matter reacts to it varies. We note several things that is the direction of propagation and the particles motion. A→ But the motion of individual infinistimal particles on the rope may move like B→ Thus we see that vector A and vector B (which is the representation of motion of particles) are perpendicular to each other. We call this Transverse waves. There are wave and particle motion where in the orientation of propagation of the waves we call this longitudinal waves. Can you draw your free body diagram of a longitudinal wave? Another is the combined transverse and longitudinal wave like what happen in the chaotic sea when a typhoon is coming. Periodic Waves; Most of us are certainly familiar with the crest and trough of a ripple of water in the pond that continue repeats itself over and over again and if we attempt to trace we will notice that at some length of period, we can found it is like a cycle in simple harmonic motion, cycle is defined as the complete revolution that is a closed loop of the origin and the ending position. Crest →through→crest, the question is how many times does this cycle occur in a wave, or in the ripple of the water in the pond that you are tracing a while back? The pictured concept above can be illustrated using mathematics known as mathematical modeling, we proceed as follows This is also equal to the time T we can find the velocity of the wave using its length divided by the time thus we have,λ/T=V or λf=V Denotes that the speed of propagation equals the product of and f and frequency is the function bearer due to the fact that the entire periodic motion oscillates in the same frequency. Mathematical modeling suggest that this can be a function of displacement and time given by f (x,t), a function of x which is the displacement and of t θ=ωt So this is the time dependency. We further let Ө= wt So that we can use cosine or sine functions to describe a simple sinusoidal wave. That is the x and y component. And A= r which is actually the amplitude corresponding the radius vector in the unit circle, So we, write, And when Ө= 0 we have x=A, y= rsinӨ By the relation, y/r= sinӨ, y= ∆sin wt Y=Asin2π ft, and when Ө=0 y is not equal to A but is equal to zero, then the function f(x,t) can identify the location of the particle with regards to displacement and time. We can show that by assuming that –x/v is the earlier time and t x/v be the future time is is possible to three and predict where will the particle be and where it had been earlier dust by arranging the equations to look like f (x,t)= Acos [w(t-x/T)] Acos [ct= x/v)] or by using the relationship v= λf and f= 1/ t Here are the list of equations that we can use in dealing with problems about waves. List of equations in wave theory General fundamental quantities Main articles: Transverse wave and longitudinal wave A wave can be longitudinal where the oscillations are parallel (or antiparallel) to the propagation direction, or transverse where the oscillations are perpendicular to the propagation direction. These oscillations are characterized by a periodically time-varying displacement in the parallel or perpendicular direction, and so the instantaneous velocity and acceleration are also periodic and time varying in these directions. But the wave profile (the apparent motion of the wave due to the successive oscillations of particles or fields about their equilibrium positions) propagates at the phase and group velocities parallel or antiparallel to the propagation direction, which is common to longitudinal and transverse waves. Below oscillatory displacement, velocity and acceleration refer to the kinematics in the oscillating directions of the wave - transverse or longitudinal (mathematical description is identical), the group and phase velocities are separate. Quantity (common name/s) (Common) symbol/s SI units Dimension Number of wave cycles N dimensionless dimensionless (Oscillatory) displacement Symbol of any quantity which varies periodically, such as h, x, y (mechanical waves),x, s, η (longitudinal waves) I, V, E, B, H, D (electromagnetism), u, U (luminal waves),ψ, Ψ, Φ (quantum mechanics). Most general purposes use y, ψ, Ψ. For generality here, A is used and can be replaced by any other symbol, since others have specific, common uses. for longitudinal waves, for transverse waves. m [L] (Oscillatory) displacement amplitude Any quantity symbol typically subscripted with 0, m or max, or the capitalized letter (if displacement was in lower case). Here for generality A0 is used and can be replaced. m [L] (Oscillatory) velocity amplitude V, v0, vm. Here v0 is used. m s−1 [L][T]−1 (Oscillatory) acceleration amplitude A, a0, am. Here a0 is used. m s−2 [L][T]−2 Spatial position Position of a point in space, not necessarily a point on the wave profile or any line of propagation d, r m [L] Wave profile displacement Along propagation direction, distance travelled (path length) by one wave from the source point r0 to any point in space d (for longitudinal or transverse waves) L, d, r m [L] Phase angle δ, ε, φ rad dimensionless General derived quantities[edit] Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Wavelength λ General definition (allows for FM): For non-FM waves this reduces to: m [L] Wavenumber, k-vector, Wave vector k, σ Two definitions are in use: m−1 [L]−1 Frequency f, ν General definition (allows for FM): For non-FM waves this reduces to: In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation: Hz = s−1 [T]−1 Angular frequency/ pulsatance ω Hz = s−1 [T]−1 Oscillatory velocity v, vt, v Longitudinal waves: Transverse waves: m s−1 [L][T]−1 Oscillatory acceleration a, at Longitudinal waves: Transverse waves: m s−2 [L][T]−2 Path length difference between two waves L, ΔL, Δx, Δr m [L] Phase velocity vp General definition: In practice reduces to the useful form: m s−1 [L][T]−1 (Longitudinal) group velocity vg m s−1 [L][T]−1 Time delay, time lag/lead Δt s [T] Phase difference δ, Δε, Δϕ rad dimensionless Phase No standard symbol Physically; upper sign: wave propagation in +r direction lower sign: wave propagation in −r direction Phase angle can lag if: ϕ > 0 or lead if: ϕ < 0. rad dimensionless Relation between space, time, angle analogues used to describe the phase: Modulation indices[edit] Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension AM index: h, hAM A = carrier amplitude Am = peak amplitude of a component in the modulating signal dimensionless dimensionless FM index: hFM Δf = max. deviation of the instantaneous frequency from the carrier frequency fm = peak frequency of a component in the modulating signal dimensionless dimensionless PM index: hPM Δϕ = peak phase deviation dimensionless dimensionless Acoustics[edit] Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Acoustic impedance Z v = speed of sound, ρ = volume density of medium kg m−2 s−1 [M] [L]−2 [T]−1 Specific acoustic impedance z S = surface area kg s−1 [M] [T]−1 Sound Level β dimensionless dimensionless Equations[edit] In what follows n, m are any integers (Z = set of integers); . Standing waves Physical situation Nomenclature Equations Harmonic frequencies fn = nth mode of vibration, nth harmonic, (n-1)th overtone Propagating waves[edit] Sound waves[edit] Physical situation Nomenclature Equations Average wave power P0 = Sound power due to source Sound intensity Ω = Solid angle Acoustic beat frequency f1, f2 = frequencies of two waves (nearly equal amplitudes) Doppler effect for mechanical waves V = speed of sound wave in medium f0 = Source frequency fr = Receiver frequency v0 = Source velocity vr = Receiver velocity upper signs indicate relative approach,lower signs indicate relative recession. Mach cone angle (Supersonic shockwave, sonic boom) v = speed of body v = local speed of sound θ = angle between direction of travel and conic evelope of superimposed wavefronts Acoustic pressure and displacement amplitudes p0 = pressure amplitude s0 = displacement amplitude v = speed of sound ρ = local density of medium Wave functions for sound Acoustic beats Sound displacement function Sound pressure-variation Gravitational waves Main article: Gravitational wave Sources of gravitational waves Gravitational radiation for two orbiting bodies in the low-speed limit. Physical situation Nomenclature Equations Radiated power P = Radiated power from system, t = time, r = separation between centres-of-mass m1, m2 = masses of the orbiting bodies Orbital radius decay Orbital lifetime r0 = initial distance between the orbiting bodies Superposition, interference, and diffraction Physical situation Nomenclature Equations Principle of superposition N = number of waves Resonance ωd = driving angular frequency (external agent) ωnat = natural angular frequency (oscillator) Phase and interference Δr = path length difference φ = phase difference between any two successive wave cycles Constructive interference Destructive interference Wave propagation A common misconception occurs between phase velocity and group velocity (analogous to centres of mass and gravity). They happen to be equal in non-dispersive media. In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency. The phase velocity is the rate at which the phase of the wave propagates in space. The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile. Intuitively the wave envelope is the global profile of the wave, which contains changing local profiles inside the global profile. Each propagates at generally different speeds determined by the important function called the Dispersion Relation. The use of the explicit form ω(k) is standard, since the phase velocity ω/k and the group velocity dω/dk usually have convenient representations by this function. Physical situation Nomenclature Equations Idealized non-dispersive media p = (any type of) Stress or Pressure, ρ = Volume Mass Density, F = Tension Force, μ = Linear Mass Density of medium Dispersion relation Implicit form Explicit form Amplitude modulation, AM Frequency modulation, FM General wave functions Wave equations Physical situation Nomenclature Wave equation General solution/s Non-dispersive Wave Equation in 3d A = amplitude as function of position and time Exponentially damped waveform A0 = Initial amplitude at time t = 0 b = damping parameter Korteweg–de Vries equation α = constant Sinusoidal solutions to the 3d wave equation N different sinusoidal waves Complex amplitude of wave n Resultant complex amplitude of all N waves Modulus of amplitude The transverse displacements are simply the real parts of the complex amplitudes. 1 dimensional corollaries for two sinusoidal waves The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The angle addition and sum-to-product trigonometric formulae are useful; in more advanced work complex numbers and Fourier series and transforms are used. Wave function Nomenclature Superposition Resultant Standing wave Beats Coherent interference After describing the waves in almost purely mathematically abstract manner, we can question ourselves what we are the manifestation and physical importance of and this waves, it’s time to apply and assess what we learn and how much we learn. Answer the following: A) A boy is playing with the clothesline, he unites one end, holds it taut and wriggles the end up and down with a frequency of 2:00 Hz and an amplitude of 6.075 m. the wave speed is 12 m/s at t= 0 the end has a maximum positive displacement and is instantaneously at rest. Find the amplitude angular frequency, period, wave length and wave number of the wave. b) Write a wave function describing the wave. c) Write an equation for the displacement as a function of time of the boy’s clothestine and of a point 3:00. From his end. Solution: identify everything about the problem; this is a kinematic (equation) motion problem about the clothesline. A sinusoidal wave is part (a) we look for the amplitude A, angular frequency and move. In part (b) and (c) we look for the equations. Set up, find everything, equations and clues that may help, f= 1/t, w= 2∏f, k= 2∏/ , v=f w= vk We execute, The amplitude (A) of the wave is lust the amplitude of motion So, A= 0.075 m, F= 2.0 Hz The angular frequency is given by the formula, W= 2∏f W= 2(2.0 Hz)π W= 12.6 rad/s The period is 1/f= T 1/2 = T so 0.500 seconds The wave length is v/f , where v= 12m/s We find the wave number by k= 2∏/ or by w= vk How, we are asked to find the equation, we are going to use the values of A, T, v, f that we have solve to arrive at the equation, Try to solve several problems by your own. The speed of sound at 20®C is 344 m/s. what is the wave length of a sound that has 284 Hz frequency? What will be the frequency of a sound that is 0l.0655 mm? Audible wave length provided that the sound is sufficiently great. The human ear can respond to longitudinal waves over the range of frequencies form about 20.0 Hz to 20000 Hz compute the wave length corresponding to this frequencies for waves in air v= 344 m/s and for water 1480 m/s. A certain transverse wave is d3scribed by y (x1t)= (6.50mm) cos 2∏ (x/ 28 cm- t/ 0.0360 s) Determine the wave’s amplitude, b) wavelength and frequency Speed of propagation us particle speed. Show that the equation, A cos 2∏f (x/v-t) can be written as f(x1t)= A cos [ 2∏/ x- vt] Use this to find an expression for the transverse particle velocity in the string under what circumstances is this equal to the propagation speed v? Less than v? and or greater than v? Acceleration, Energy in a wave In the current mostly we talked about speed but we recall, accelerations is the one relates mass force and motion and is given the primary definition which is the rate of change in velocity which is either by a change in speed or direction. Every wave motion has an energy associated with its energy we receive form sunlight and the destructive effect of the ocean surf and quakes bear this out to produce any kind of wave there must be a force entwined to it, so the string moves and work is done onto it/ imagine a straight clothes line, and we try to taut one end (pay attention to the highlighted part) Where Fy is the vertical force element that lifts the line and F is the horizontal vector component you’re our x which is the resultant force and motion. We try to find the slope which is equal to the rate at which it is done so the slope means the Fy max must be equal to the maximum energy because it is the force lifts the clothes line to its maximum position. We note that both Fy, T is a function of displacement and time, and because the force is conservative are use a negative describing it. Where dy/ dx is a partial derivatives that treats other as constant as one variable changes by an infinitistimal amount. So that the particle transfers its energy to its neighbor in the average rate proportional to the instantaneous velocity. And therefore power is a product of the transverse force and transverse velocity. We do the partial differentiation in the equations that we have studied earlier. And We follow the formula, Then we use w= vk and v= F/u The average power is ½1/2 (√uF)A2. We see that it depends on the angular frequency and the amplitude of the wave, as big waves are most destructive given that they wave bigger amplitude. Intensity is the power dissipation in a spherical volume/ area. Ia=P/█(4∏r^2@@) When distance such that the radial change we see that the intensity onto changes I1= I2 4∏r2I1= 4∏r2I2 It is better to visualize at the early stage how change particles interact, and does work on each other to be particular in the magnitude and direction, and that in the interaction there will be a resultant vectors. For example there are two changes with an electric around them acting on one another so there is a force that exists in them. We denote it by F. if there is a separation between the changes the change may repel or attack each other depending on the orientation of the lines of force. If we trace this and do the vector addition we will know the force is negative and it is a repulsive force. This F acts between the changes with a distance R. the alternative force tends to do work is the systems of changes. The angle in between them is equal to the slope of the straight line in this case 180®. Because we refer to the work done which is appareled to the force we use the cosine function and we will see that it is a maximum, we further use the cross products of vectors to show how coulombs arrived in his formula. When we assume that HAH is the distance between A and S and it changes so that the magnitude of any unit vector resulting from the interaction of A and B changes we find the change with respect to HAH, since the electric force represented by HAH is in the radial direction we may sight to replace by r. So that Fx=dµ(Ax B)/r F=AxB/r2 Illustrating the inverse square relationship. It is passive showing that it increases a distance approaches to zero. Here we assume that both change A and B are positive so the force is repulsive. The interaction is inverse square relationship increase as ∆/r → 0but the vectors on them differs when involves the sign; there is a constant of proportionality k. So that the formula has the focal form of F=kg1g2/r2 Force that acts on change: The above formula states the amount of force in between changes. It is as well correct to say that this is the amount of force acting on the nearby change because the force is radically spread out have the force that acts on over every change. F/g=E E= electric field F=kg1g2/r2 E=F/g E=kg1g2/gr2 E=kg/r2 At particular instant. The vector r is placed instead of r dealing with vectors. Electric field can be regarded as the potential difference or voltage. The Gauss Law: Imagine if we can create a thin spherical surface sheet and enclose the change it how would the change react and release the electric lines of force? Suppose that the surface volume that encloses our charge is not affected by the electric lines of force. We note that if we know the electric fields in a region we will how F and 4. Respectively, suppose that the spherical surface that enclose the volume is tinted so that we cannot see what’s inside, how would we know the contents of the box? That is of course the change has some proportionality to the strength of E. it would be easier if we regard electric field is like a flow of a fluid in our surface since that vector flow is a function of 1/r2 denoting that as r gets bigger the vector decreases and since that can actually flow out from the surface is proportional to the area of the surface we used not to know the volume. But instead the area. Whether there is a net inward flow/ outward flow, it depends on the sign of change enclosed in the surface. Change outside the surface does not give rise to a rest electric flux through the surface. The electric flux is directly proportional to the net amount of charge enclosed within but is otherwise independent of the size of the box. Gradients of a vector: As the title implies gradients are function valves that tend to increase or decrease the vector valves. Can be attributed to the divergence and curl of a function and can be answered. That is the valve of the vector changes as the ball changes its direction (directional derivatives). Let us try to visualize that there is a ball of charge travelling in an electric field and that moving the said ball of charge requires work. Let us try to designate a surface determined by an equation having the variables (x,y,z) with a function of f(x,y,z). Thus when Ө changes with respect to (x,y,z). the valve of the magnitude of work done changes. Then we may say that it changes with the direction or orientation of particle with respect to the electric field (directional derivatives). Let f be a function of variables (x,y,z) it u is any unit vector cos Ө + sin Ө, then the directional derivative of F in the direction of u, devoted by (x,y) is given by Du f(x1y)=limf(x+hcos Ө,y+sin█(Ө)-f(x1y)/@h) And can be further simplified to: Dµf (x1y)=fx (x1y) cos〖Ө+fy (x1y) sinӨ 〗 f (x1y)=12-x-4y2 u=cos〖1/6 ∏ἱ 〗+sin〖1/6∏ἱ〗 -2x – fx (x1y) – 8y= fy (x1y) -2x ∓√3- 8y 1/2 -√3x- 4y We see that: D_u F(x,y)=u∆F(x,y) From A certain point where D_u F (x,y) = 0 to certain or any u and ∆ there exist an angle in which: Duf (x,y) ≠0 Duf (x0, y0)= U x ∆f ( x0, y0) = II uII ∆f(x0, y0) II cos ∝ II ∆ (x0,y0) II cos ∝ At maximum when cos ∝ = 1 occurs when ∝=0 is in the direction of the ∆f (x,y) in a similar manner when it is -1 when ∝= ∏ where the direction is opposite that of ∆f= (x0,y0) Another concern is when A plane is use Y_o represent another vector for example the interaction of the electric and magnetic field for which each fields are represented by a plane. The question is which the magnet plane is and which is the normal plane to the surface? If there is a function representing the interaction of vectors in a certain surface like f (x,y,z)= 0 it can be found out that Fx, Fy, Fz are not all equal to zero. A proof is when we change x and y by r cos Ө or by r sin Ө. It shows that when one of the either Fx, Fy, or Fz is zero then the function is a variable of the removing two. And we can regard it as a surface suppose that there is a curve containing P0 with a parametric equation x= f (t), y= g(t), z= h(t) where the valve of t at P1= t0. A vector equation pertaining to C is R (t)= f (t) ἱ + g (t) ἱ + h (t) k. Because c is in s that has the equation F (x,y,z)= o and containing C to S. F(f (t)),g(t),h(t))=o (1) Let A function stand for 1 and assume that 1 is differentiable so that Fx, Fy, Fz are not all equal to ) at P0 and h(t), f(t), g(t) exists. There total derivative with respect to h,f,g in the function (i) is : G(t)= F2 (x0,y0,z0) (t) t. . . . . . . . . Fx (x0,y0,z0) f (ht) Which can be expressed by ∆F x DR (t0) (2) If (2) = 0 we know that Dr (t0) is in the direction of the target vector to the curve C at point F and we can conclude that the gradient vector of F at P0 is perpendicular to the target vector of every curve using the foregoing mathematical theorems we will try to prove the formula about the gauss law and the theorems about electric flux. Important implications of these mathematical concept is a surface with an electrically static portion satisfy the condition F2= 0 and hence the unit target vector is to zero so that electrons are static. What remains is the option Fy is not equal to zero Fy is the unit or through vector. This one perpendicular to the surface. The tendency of the electron is to stay on the surface of a conductor and if there is a curve path, the unit normal electric field adds up together producing the steepest accent. We suppose that this vector is flowing in the surface that ∆= 1 to avoid complexity and let the electric field flow to be like fluid flow in the area. We have, E= v, where v is fluid flow and E= electric field of lines A= area then to account for all E in the area we multiply E ∙ A= ɸ Where ɸ= electric flux in the area. If A changes such that of a tuning plate of a capacitor then the rate of change of A can be a variable of Ө or time we may express it by dA/dt or dA/dt or simply dA we use integration to account for all electric field of lines inside dA. ⨜c ∙dA= ɸ c Where (∙) is an operation telling you to use the dot product for vectors. F/g=E kg/r2=E E∙A kg/r2 (4∏r2) k^1= Eo 9/E= EA This implies that the electric flux does not depends on the area or to the value of R. but only to the amount of q and when ɸ is an scalar and Ө is a mode indicating direction the conclusion is still the same. Work done on the change by the electric field. If the charge discussed previously is regarded a free moving charge and a test charge of is placed in space, the two would react in accordance to the law of charges. The motion of the electron from one place to another is paid at the expense of the electric force exerted by the test charge that obeys the inverse square relationship. Thus at the location R2 it felt small force thus requiring more time to do work same is true is we tend to move the charge from Ri to Rn where in Ri it is acted upon by a greater force. However, it does not depend on the path of travel due to the fact that the force that acts on it do not change with regards to the length of the flux is like a fluid flow in a given surface. dv/dt=vA,dv/dt=vA cos〖ɸ @ 90°=v1 A〗 (Sequel, electric potential) → work → current → resistance, power It is easy to give a formula but doing it poses a huge risk in pit falls which will lead to big mistakes. The following formula are used to describe the flux; a flux is a Latin word which means flow. Flow of what? Electric force denoted by the electric lines of force in a certain surface. We analyze it by postulating, using gauss divergence theorem device from green’s theorem in the figure. Let R be an electric lines of force. And its magnitude is denoted by its length (III) what if there exists another vector denoted by S and was able to move A particle to the same point where R ended. Can we say that S does more work on the particle. State in another if R, S is emitting from a closed surface but met on the same end, can we say that s is a greater vector than R. there are cases where in the valve of a vector depends on the initial and final point vectors that are independent such is evaluated using the line integral. Preparations for vectors, Electricity and Magnetism It is fundamental for a student of physics to know the vectors in the study of vector fields. Examples of vector fields are the gravitational field and electric field/ magnetic field. But to the potential functions that depends on R and the total distance travel is not the one used but the displacement rather in calculus this is referred to its line integrals. It is easy to say that a force is conservative like when we say that gravity is a conservative vector. How can we convince ourselves without particularly believing in that spoon feed facts in the article (7-7) page 167 under the heading conservative and dissipative forces (College Physics by: Zemansky) it was stated that a force like gravitational forces acts on a body, the work they can do depends only on the end points of the motion such that the only forces that acts on the body. Still incomplete doing work must be mutual and by the law of interaction the counter part of the force must be equal and of motion or work persist hence we say that there is a net work done. The potential and kinetic relationship acts in a way to explain the total work done in a system. Say for example you are carving an amount of charge and it has to be moved around the electric field of last charge of. From point A to B. We know that the distance covered is the length of the arc AB. Recall that F11 is the one responsible for work and the displacement in that direction multiplied by the force by vertically adding the vector arrows we equally perform (Rb-Rn) which is actually the displacement needed so we have. W= F (Rb- Rn) By the coulombs law we find F=(9a^1)/4πE0r and we multiply it to (1/rn-1/rb) such that is the independent variables. Mathematical Proofs: Let A curve represent the path of the charge such that this curve is an equation in the parametric from and in it lies the vector properties of that particle we have; F (x,y)= V∙r (t) To prove that v is a conservative vector, we let the anti-derivative of f (x,y) exists in a form f (x,y)= f (x,y) ἱ + g (x,y) ἱ in a condition where the df/dy=dy/dx We see that if thus partial derivative represents some gradients or shape representing charge the change is homogeneous (early trancedantals III) Let F=πg/4πeo r Partial derivative with respect to r. dw/dr=∇/r Separable equation, df=∇/r dr Integrate, ⨜df=∇/r dr = ∇ ln/ru- ral w= ∇ (1/rb-1/ra) dw=du In a surface of any conductor which requires doing work on the charge it depends on the potential difference denoted by the difference by (Rb-Rn). What will happen if we distribute the potential undiminished such tent (rb-ra) is a maximum per unit charge thus we need to physically write this physical principle to a mathematical equation we have: V=(Wa-b)/t=kg/r You already know that voltage is the amount of work done on a charge. Hence we equivalently say that the charges are in motion, but how much, how can we quantify? Suppose that there is a cylindrical conductor having the cross sectional circular area A and the length L which is equal to volt. If there are n electrons that are inside the area at t= 0 And some electrons get to pass A when t . . . . .tn, the number of electrons must then be equal to the volume x e in the time frame tf is equal to Q= neVdt A since the definition of current is Q/t we decide this by t: I= Q/t=neAVdt Where Vd is the drift velocity of the electrons due to the collisions. Vd=aI=eE/m (d/vf)=(eE∝)/mvf T= ∝/vf Where vf= (frequency/ speed0 of the electron by fermi distribution. And ∝ is the mean force path travelled by the electron. I=neA ((eE∝)/mvf) I=ne^2 A∝E/mvf Because t=v/L We may rearrange the above equation to yield, (ne^2A∝)/mvf (v/l)=I (ex) The Ohms Law is in the form I=V/R equation (ex) is equal to the ohms law formula. If we regard: R=(ne^2∝)/mvf A Since in the Ohms Law R is in the reciprocal or inverse relationship we take the reciprocal so that: =mvfL/(ne^2∝A) This term is equal to resistivity of the material: ρ=mvfL/(ne^2 )∝A Of thread that are flexible as they move back and forth, how it would look like. Can we even say that the motion of the electron in the atomic orbital has some force related to it? That the motion of the atoms creates a magnetic moment in a collective manner, magnetic moment adds up to produce magnetic domains of the magnetic material. This can be enhanced and be turned into a permanent magnet. In the sentences above it is implied that magnetism and electricity are related. It was proven when we say that the motion of the electron (current) in a specified area creates a magnetic moment. This is a verified when arrested accidentally discover observed that when a compass is placed near an electric current carrying wire it would be deflected. As a magnet reacts in a repel- repel- attract manner. It should have a force related on it. That may be in the form of the magnetic and electric force. When we study vector mechanics we can find that Ө exist in between them using the cross product when a charge particle travels in a place where there is a magnetic field the speed does not change but the direction is altered so there is a change in velocity. When it travels in an electric field speed and direction may vary. We can wish to create the strongest magnet. The earth is the biggest magnet in the planet but definitely not the strongest. As we may wish to pile up bar magnets, and make turns of wires arranged arbitrary to produce a strong electromagnet. Magnets and coils of wire together with the mechanical motion can transform mechanical energy to electrical energy as stated by Faraday, that the change in the magnetic field, area, frequency, may cause electric current as it was known as electromagnetic induction. The converse is also true that a charging electric field and induce magnetism. The main contribution is in power generation that enables us to stop up voltage without dramatically increasing the current using transformers. The heat of the study of magnetism is when Maxwell was able to conjoined the two concept (Electricity and Magnetism) into one known as electromagnetism using four fundamental equations although Maxwell did not single handedly derived all of them, he was the one who fully understand their use. It was when he theorized that light is a form of electromagnetic wave and was able to calculate the speed using his predicted constants for electric permeability and magnetic permeability off free spaces. Modern communication owes a lot from him although he is not the one who created the first radio; he is the one that created the first camera. Communication: When pulled together, it is achieved, relies on the use and manipulation of electromagnetic waves. Atom electromagnetic and Optics: From classical view, light is treated as a ray that we regard to strike a plane mirror or a lens but this is not the case for which in every interaction there is something move to explain in its relation. As the travel of light concerns a media inside the earth from gas to water to glasses and lenses and each material has a set of their electrical permittivity , and magnetic constants in a way that this governs the interacts with matter dictated by (km)which is the relative permittivity/ permeability. V=1/√(ϵμ)=1/√kkm=C/√kkm And c/v=n=√kkm≅k We see that even the index of refraction is an after effect of electromagnetic phenomena. Same is true with reflection that by quantum theory it shows that Өr and Өἱ must be equal such that momentum is conserved but classically it is proved using snells law. Force Fields: Forces may emanate (in t) from different things and may classify according what they affect. Like for example a massive body that exerts a gravitational force in the area of accretion region. And the force may come from a point charge in spaces. The charge releases a force that affects other charges in an analogous manner that we are attracted by quantity. As what we know gravitational pull/ magnitude depends on mass, but what might be the determinant factor contributing to the strength of an electric field? Let us try to imagine a particle charge in space and quantify it using the conventional unit of a coulomb ©. And we imagine that there are vector arrows representing a force E. The longer the arrows, the greater its magnitude which is definitely depending on the amount of charge ©. We say that there is a force that acts on q thus,F/g is equal to the electric field, we haveE=F/g. We may wish to ask ourselves, how fast an elevator travels if there is a one ampere in a wire. Having the cross sectional area of 1.0 mm2. Let, n=N/V=(8.5x〖10〗^28 electrons)/m^3 And, 1.0 mm2= 1.0 x 10-6 m2 vd=1/nAe vd=1.0A/((8.5 x〖10〗^28 electrons m^(-3) )(1.0x〖10〗^(-6) m^2 )(1.6x〖10〗^19 C)) Vd= 7.4x 10-4 m/s= 0.00074 m/s If the wire is 1 meter and we wish to operate appliances with an electron having this value of vd. We find, v=d/t t=d/v t=1.0m/(0.00074 m/s) t=1351.35 s t=22.5 minutes before the appliance works But this is not the case, when you turn on the switch the light instantaneously turns on what might be the reason behind this? The answer might be after you close the switch the electric field is set in the circuit that (tends) cause the free electrons to drift almost simultaneously. We state the following conclusions (resistance) That resistivity increase as L increases. It decreases as area increases. Resistivity is the property of a material. Ohm’s Law tells you the relationship between voltage and the resistance in a material. Magnetic forces are a consequence of the electrons motion it is easy to write about the electrons motion but definitely difficult that the origin of such a fundamental force like the magnetic force is motivating in another force. The forces are electric force, the electron velocity and the minute Newtonian force to suffice the electrons mass. In the atom the electron sweeps an area equal 2πr^2 where r is the radius from the nucleus. Suppose there is a long wire symmetric at x axis (0, 0) and is parallel to the y axis, determine the magnetic field at point ρ. -a θ= πrɸ sin〖θ=y/r〗 since r= √(x^2 )+y^2 ;dy=dl We use trigonometric substitutions. The originality condition sets the principle what affects in the interaction of electric, and magnetic vector so with the velocity vector of the atom. Forces and moments of a plain area We think of the path of electron around the atom to cover an area A and we try to determine the forces that lie orthogonal and tangential to this. We may ever ask if there is a force passing per unit area and what is its behavior? We begin by using the formula. A= 2∏r2 if this is A plane one dimensional circle we try to find of the moment of inertia like as if a thin circular plate. X2+y2= r2 ⨜⨜▒█((kr^2 )dydxdA@) ⨜ykr^2 dAdx k⨜^x▒〖ykr^2 dA〗 Is in motion we call kxyr^2 Let us link, moment of inertia (resist spiring) (or male at spring) with momentum. As defined momentum is a vector force brought by mass and velocity that by law of inertia would continue moving cause has velocity then we regard the computed valve to be a scalar function. kπ/8 r ∙mv In the case of the electron since the area Hπ/8 πr^3 (mvr) where:L=mvr =kπ/8 πr^3 L What if this motion kinetic energy is connected to Electric Force and since e is current in motion we call it electric current. We do the cross product. kπ r^3 (L) I kπ/4 r^3 L X (ev/2πr) ((π^2 )r Lev)/2πr =r^2 Lev Where r2 is equal to the torque and the product of the two is equal to the magnetic force or perhaps the magnetic moment of the electron. In the classical view such a formula is correct but it is not the case in the quantum world because the magnetic moment of the electron does not grow without bound as r or v reaches – or + we will show how quantization occurs. Furthermore we will show the importance of the area being bisected. The Quantum Numbers and their Importance The current carrying wires We see that it current passes in an area it sets a magnetic moment giving rise to the total magnetic force 𝓑. kπrLev/8 2/8 kπr^2 mvev Iμmv/4 And a line segment L is cut into an infinitive line dL. Such that the number of charge Q increases as dL increases ∆L and furthermore A is the X area of such a conductor. dQ= nqAdL N is the number of moving charge per unit volume, the charge in B is equal to ∆B= dB brought by dL and dq we substitute: dB=μ/4π nqAvdDl/r^2 ř Since nqAvd= I dB=μo/4π Idlxr/r^2 We integrate to find the total B ⨜dB= (µo)/4π ⨜▒█(IdLxř/r^2 @) The Magnetic Force of a circular Loop If there is a current encircling the area of a circle show that B is directed inward such that B is more concentrated at the center. Suppose we have a circle having a radius k and is the path of the current I. the perimeter 2πr is path of the current. The interaction of current having and entering the system is equal so that equilibrium is met 𝓔ἱ= 𝓔0 However the direction of the currents are not the same using the right hand rule it shows that forces cancels each other out. +y and –y have equal magnitude but opposite direction such that the circular loop is symmetrical on the center so by ≅ 0. When there a N turns of wires. The change of B ∆B is directly proportional to N and the µ= NI a2 π . This is there are in turns of loops we have. Bx=(μ0μ/(2π(x^2+a^2 )^3 ))/2 Amperes Law= E The magnetic fields of the area that enclose a current source. The magnetic field around a current carrying wire is related by the formula. B=d0I/2πr And such tent B strength is the same for r but dl is no a constant which is actually equal to the perimeter of the surface enclosing the concept carrying wire. Then if we take the summation of ∇ df and regard dl to be a dine integral since the valve of B with not change whether on what path of integration is taken according to Greens theorem it can be considered as a closed smooth current therefore we may threat it as a line integral. Gauss Law= B ∇=B ∇∙dl= ⨜B∙dl Since B= k constant ≈ One takes it out from the integral: ⨜dl The Electromagnetic Induction Principles: Michael Faraday discovers the principles of Induction. His experiments concluded that a changing flux in a loop produce a flow of electrons. Hence a voltage or potential difference 𝓔 is produced. The interaction of atoms to external magnetic fields is not yet fully understands during this time. However, let us imagine a change T enveloped by an area since it has an electric field and this E flows outside to the area then the total flux is proportional to amount of change enclosed there in but if tent area is enclosed a dipole the flux is zero because the change is zero. In the case of magnets it is always dipole (H,s) so: B∙dA=0 (Gauss Law for Magnets) This leaves us uneasy that even Dirac postulate tent a monopole should exists. The area of the coil exposed to the magnetic field contributes to the amount of change to the magnetic flux. ɸB=BAcos ɸ that is if the exposure of A depends on ɸ dɸB=B∙dA We may say that A changes with respect to time. So equally the magnetic flux changes with time. dDB/dt=ε This produce a current which is opposite in direction of E and we introduce the sign (-) representing the currents direction if dB/dt is increasing then it is positive. When it is decreasing then the sign is negative. -dB/dt=ε Faraday Law Suppose we have a conductor with length L, and may move with certain velocity v, only that VLB. Calculate the amount of motional electromotive force. We recall that: E=v/L so V=EL The rate at which the conductor moves along the magnetic field is a contributing factor fordɸB/dt, the product of BVL gives the total motioned. Electromotive Force: E= VBL What if the conductor is derived in n lengths dl? The total electromotive force is the (B) summation of B field. Parallel to dl the cross product of (VxB) dl responsible for the total E. ε= ⨕▒〖(vxB)∙dl〗 The Change in Current: Can the change in current change the magnetic flux? ɸB= μnAI A=BA If the amount of charge passing per unit volume changes we say I=dq/dt=I dI=dy/dt Such thing is true, hence we have ɸB= μnA dI/dt How the electrons are moving, what makes it move? We assumed that it was current alone. ε=dɸB/dt= -μ0nA dI/dt We mention the changing magnetic flux induced by the change in current. The situation. Such that an electromotive force is also present in the second loop B. The sound conductor is made up of the same material but differs with the number of turns and area enclosed, even in fact has no source of magnetic nor electric field on its own but the galvanometer tells us that induce this current. The magnetic field of the soleniod is like that of the bar magnet brought by dI/dt. Let us propose that a vector area exist, surface that is an open ball extending infinitely in space and that at certain distance from the solenoid coil has an appreciable effect. Because we know that there is current in the second conductor we denote it by I. we set imaginary B moving out. Since the magnetic field travels in space the magnetic permeability has an effect to this ἱɸB ∝ which is why there is always a power logs in a system of mutual induction. If thus ἱɸB is theone sucepted by the end coil we include it in the calculation and formulate the Angel’s Law of Induction. Mutual concentration of fields using greens theorem. Conserve or not conserve is not the question because by reference, relativistically we cannot know which is which. Postulate. The magnetic fields cannot grow without bound. If the magnetic field or any vector is present its speed component is not greater than C. If it is a scalar producing vector like gradient, such can’t lead to vectors which are greater than C. By1. Max= ε1+ε2 Bax= ε1+ε2 So when either ε1 or ε2=0 Bmax= ε1 or B=ε2 The emergence of induction and the mutual existence of two vectors, in the physical, space, let a vector be defined by a field which is compared by any curve whose only relationship even by supper imposition principle is governed by B and E. ε∙B=0 Condition of Simultaneously obeyed in one Equation: Λ- Simultaneously function We consider a unit volume with cross sectional area A and a length dt been covered by the simultaneously vector function that by relativity can either be purely magnetic of electric at same relative point of observance. The relative and vector axial translation is a space advancing vector free energy propulsion determined by t if we postulate tent B= E. Then the projection of the vector towards the other can be of the form. So that it is never zero. But maybe in unity having the dual nature. The emitter of the waves has something to do with the frequency we can say that energy travel but not matter. But there is energy equivalence in matter.
Posted on: Thu, 30 Jan 2014 03:44:19 +0000
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