Time Dependent Potentials and Fields – Proposal

My project will be based on the theoretical content of Chapter 10 of Griffiths, which deals with the previous equations for the electric and magnetic potentials (and fields) of charges, but with the additional dimension of time. My goal is not to rewrite Griffiths’ chapter, but rather to gain a deeper understanding of the equations he presents through graphical analysis. Central to the difference between the old equations and the new time-dependent ones is the concept of retarded time, which is defined as

(1)   \begin{equation*} t_r \equiv t - \frac{|\vec{r} - \vec{r}\ '|}{c} \end{equation*}

 

(David J. Griffiths, Introduction to Electrodynamics)

Where t is the actual time, the r vector represents the position of a given point in the system, the r’ vector represents the position of the charge, and c is the speed of light a vacuum. The second term represents a delay- how long it takes for the electromagnetic information to travel from the charge to a given point in space. Thus retarded time embodies the idea that electromagnetic messages take time to propagate through space, and in fact propagate at the speed of light. This means that the present position of a particle becomes incidental. At any given time there exists an effective position, illustrated below:

 

For a more in depth discussion of effective position (in three dimensions), see page 430 of Griffiths. In any case, our definition of retarded time leads to a time-dependent generalization of the potential equations:

(2)   \begin{equation*} V(\vec{r},t) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\rho(\vec{r}\ ',t_r)}{|\vec{r}-\vec{r}\ '|}d \tau' \end{equation*}

 

(3)   \begin{equation*} \vec{A}(\vec{r}, t) = \frac{\mu_0}{4 \pi} \int \frac{\vec{J}(\vec{r} \ ', t_r)}{|\vec{r} - \vec{r} \ ' |} d \tau' \end{equation*}

 

(David J. Griffiths, Introduction to Electrodynamics)

Which are nearly identical to previous definitions of the potentials, with the exception that both the charge density and the current density now depend on the retarded time. These potentials can be used to derive Jefimenko’s Equations, which describe the non-static electric and magnetic fields of an arbitrary continuous charge distribution. I have omitted them here; since they rely so heavily on vector notation, at this point I’m unsure whether Mathematica’s plotting capabilities will be useful in obtaining graphical representations of them or the above potentials. I intend to explore manipulating them into a form easier to graph, or perhaps examine them in the plane z = 0. Additionally, choosing a simplistic charge distribution may result in an expression that could be graphed more readily.

In any case, I want to focus more on the electrodynamics of moving point charges. Their potentials, which can be expressed without an integral, are referred to as the Liénard-Wiechert Potentials:

(4)   \begin{equation*} (i)\ \ V(\vec{r}, t) = \frac{1}{4\pi \varepsilon_0} \frac{q c}{\vec{v}(c|\vec{r} - \vec{r} \ '| -(\vec{r} - \vec{r} \ ')} \ ,\ (ii) \ \ A(\vec{r}, t) = \frac{\vec{v}}{c^2}V(\vec{r},t) \end{equation*}

(David J. Griffiths, Introduction to Electrodynamics)

I believe these equations will prove easier manipulate and graph, and may apply restrictions such as the assumption that the dot product in the denominator is zero. Additionally I’ll examine these equations in only two spacial dimensions (various vertical and horizontal planes).

The following equation can be derived from Jefimenko’s (see Griffiths’ Example 10.3), but I have made a few assumptions of my own: that the velocity is constant and that the position vector r is perpendicular to the velocity vector v. Additionally, I’ve expressed the position r in terms of x and y, and let z = 0.

(5)   \begin{equation*} V(x,y,t) = \frac{q c}{\sqrt{(c^2t)^2+(c^2-v^2)(x^2+y^2-c^2t)}} \end{equation*}

 

Using Mathematica and plugging in a few values, I was able to generate a preliminary animation:

This is the kind of result I’d like to obtain for a number of other cases, magnetic as well as electric. I think it will be easy to formulate constraints for linear and quadratic charge accelerations, as well as circular motion, and applying them to the Liénard-Wiechert Potentials will hopefully yield workable results. Ideally I’ll have a number of equations and animations and be able to select and pursue the most interesting for presentation.

Project Proposal

Project Proposal

Nuclear magnetic resonance is a technique for determining the structure of organic molecules in solution.  It takes advantage of the magnetic properties of the nucleus to sense the proximity of double bonds, electronegative atoms like oxygen, and other magnetic nuclei in the molecular structure.  The technique can identify these bond types and functional groups in a molecule.  The same principles this project will study apply in MRI (Magnetic Resonance Imaging) machines that create detailed images of a medical patient without using dangerous, ionizing radiation.  An NMR spectrometer is a scaled-down MRI designed to look at small samples of a substance dissolved in solution [1].

The magnetic field generated by an NMR spectrometer, and the effective field experienced by a nucleus will be modeled.  These field’s effects on atomic nuclei will be studied and modeled.  The shim system, which is the process by which the magnetic field is homogenized, will also be explained.

The Larmor frequency is the resonant frequency of a given nucleus and is only dependent on the strength of the external magnetic field, and the magnet strength of the nucleus.  It is this frequency that allows us to select what elements to analyze in a given test.  The following equation is used to calculate a Larmor frequency [1]:

(1)   \begin{equation*} \nu_{0}=\frac{\gamma B_{eff}}{2\pi} \end{equation*}

where \nu_{0} is the Larmor frequency, \gamma is the magnetogyric ratio, or nuclear magnet strength, and B_{eff} is the effective magnetic field strength.  The effective magnetic field strength is given by the following equation [1]:

(2)   \begin{equation*} B_{eff}=B_{0}(1-\sigma) \end{equation*}

where B_{0} is the external magnetic field strength and \sigmais the shielding factor.  When the atom is polarized and stretched by a magnetic field, a small magnetic field is generated in the opposite direction, in accordance with Lenz’s Law, that also affects the nucleus.  The shielding factor is a direct measurement of electron density around a nucleus, and is important because the electron density determines the effect of the external magnetic field on the nucleus:  more electron density causes less of an effect, and vice-versa.  The resonant frequency can then be rewritten to show the dependence of frequency on external magnetic field strength:

(3)   \begin{equation*} \nu_{0}=\frac{\gamma B_{0}(1-\sigma)}{2\pi} \end{equation*}

Most of the information from NMR comes from the chemical shift, which is the change in a nucleus’ resonant frequency as the external magnetic field’s influence on the nucleus increases or decreases.  To standardize the results of NMR experiments, we express the chemical shift in parts-per-million so it is the same regardless of how powerful the instrument’s magnet is.  The formula for chemical shift is [1]:

(4)   \begin{equation*} \delta=10^{6}(\sigma_{0}-\sigma) \end{equation*}

where \delta is the chemical shift, and \sigma_{0} is the reference shielding factor used to define the zero-point of the chemical shift scale.  It is most often the shielding factor for a compound called tetramethylsilane, or TMS, because of the peak’s consistent and reliable strength and shape.

 

There are four experiments in particular that are performed using NMR.  1H-NMR focuses on hydrogen nuclei, while 13C-NMR, DEPT-90, and DEPT-135 focus on 13C nuclei.  They are so widely used because most organic molecules are made primarily of hydrogen and carbon atoms, so much information can be gleaned these four spectra.  To that end, these four tests will be performed on                   3,3-dimethyl-2-butanol

 

 

Figure 1:  3,3-dimethyl-2-butanol  

Mathematica, Wolfram Alpha, Excel and the NMR software Topspin will be used in the study and to create figures.  The book “NMR Spectroscopy Explained” by Neil E. Jacobsen will be used as the primary theoretical tool.  Karen Wovkulich, the Chemistry Instrumentation Manager at Vassar College, will also be consulted.

References:

1)                   Jacobsen, Neil E. NMR Spectroscopy Explained: Simplified Theory, Applications and Examples for Organic Chemistry and Structural Biology. Hoboken, NJ: Wiley-Interscience, 2007. Print.

Project Outline

This project is about investigating electro-optic modulators and therefore I will focus on three main categories, theory, the device itself and its applications.

Theory: I will look into the mathematics and the physics involved in the three different kinds of modulation, phase, amplitude and polarization. The equations involved will be studied from several different perspectives.  I may include hand worked solutions as well as simulations or animations from Mathematica. The animations could involve light waves entering and exiting the EOM, or before and after animations of what the electro magnetic wave looks like before and after modulation. Two equations I found that will be interesting to investigate are one relating electric field strength to induced index of refraction:

(1)   \begin{equation*} \frac{1}{n^{2}}=\frac{1}{{n_{0}}^{2}}}+rE+RE^2 \end{equation*}

where n_{0} is the index of refraction with no present electric field, r and R are constants that are dependent on the crystal. Another equation allows the EoM to function as a half wave plate once a specific voltage is applied.

(2)   \begin{equation*} V_{HW}=\frac{\lambda}{2r{n_{0}}^{3}} \end{equation*}

This equation is the key to modulating polarization of the laser beam.

Actual Device: Here I will investigate the electro-optic modulator itself. I will talk about what  the device consists of, possibly making a picture with labels of the different components of the inside of one of these devices and go into the physics of the crystal itself. Here is where I will go into some detail about the electro-optic effect and piezoelectricity.  Examples of devices and their properties will be given, most likely from companies that manufacture them for research purposes. I will also talk about the physical mechanism behind the piezo-electric effect within the crystal. The device that produces the effect is some kind of very high frequency voltage generator.

Applications: In this section I will look into a few applications of electro-optic modulators. I want to find an application for each kind of modulation. I will discuss why an experimentalist would be interested in doing the different kinds of modulations and why it is useful. I may be able to find some recent data from a published paper that used one of these devices.

Scattering-Proposal (Revised)

Project Outline

For my project I would like to build on the work of a previous student, Rahul Khakurel, who also explored Rayleigh scattering.  However the form of the Rayleigh theory I will be working with is modified to include different variables.  These variables would be easier to measure in an experimental setting and can be manipulated to observe the effect on the scattering intensity.

Link: http://pages.vassar.edu/magnes/applied-quantum-mechanics-and-optics-phys-375/rahul-khakurel/

The Rayleigh theory of light intensity for scattering by a single particle is:

(1)   \begin{equation*} I_R=I_0((1+cos\theta)/2R^2)(2\pi/\lambda)^4((n^2-1)/(n^2+2))(d/2)^6 \end{equation*}

R=distance to the sample, θ=scattering angle, n=refractive index, d=diameter of the particle

The cross section of one particle is:

(2)   \begin{equation*} \sigma=(2\pi^5/3)(d^6/\lambda^4)((n^2-1)/(n^2+2))^2 \end{equation*}

 (Zare et al., Laser Experiments for Beginners)

However, I would like to look at the scattering for multiple particles suspended in a solution with water.  The equation I will be using is:

(3)   \begin{equation*} I=I_0\sigma N n_{water} l (1+cos^2\theta) \end{equation*}

I0= incident intensity, σ=scattering cross section (same as for single particle), N=particle density, nwater=refractive index of water, l=cuvette length, θ=scattering angle

All of these variables can be manipulated, except for the refractive index of water.

The Rayleigh coefficient, which is the particle cross section multiplied by the particle density of the cuvette., accounts for the likelihood of scattering given particle size and density.  The likelihood of scattering is also dependent on the distance the light travels through the sample, or the length of the cuvette.  As the light passes through the sample it is also refracted by both the particles and background medium.  (For my modeling purposes the background medium is water).  The 1+cosine term accounts for the phase shift between the incident light and the scattered light based on the angle between the incident beam and the observed scattered light.  The incident intensity is the amount of light entering the cuvette and is also the maximum intensity of light that could be scattered.

I am assuming for the purposes of modeling that the particles do not absorb any of the light and that they are spherical in shape.

Just for reference, here is the equation Rahul was using:

(4)   \begin{equation*} I=({NV^2}/{r^2\lambda^4})I_0f(n_1,n_2) \end{equation*}

N=number of spheres, V=volume, r=distance to observer, λ=wavelength, I0=incident intensity, f=function taking into account differing refractive indices

For his work he set the distance to the observer, incident intensity, and the function taking into account the differing refractive indices all equal to one.

When the particle size is no longer within the Rayleigh limit the Rayleigh-Debye equation will be used.  This equation introduces a unitless form factor to the Rayleigh theory so that the theory can be applied to particles larger than the Rayleigh limit.  (Simple alternative to Mie scattering, which accurately predicts scattering for larger particles but is quite complicated.

The Rayleigh-Debye equation:

(5)   \begin{equation*} I=I_0\sigma N n_{water} l (1+cos^2\theta)P(\theta) \end{equation*}

the form factor (Kerker, The Scattering of Light):

(6)   \begin{equation*} P(\theta)=[(3/u^3)(sin(u)-ucos(u))]^2 \end{equation*}

(7)   \begin{equation*} u=2kasin(\theta/2) \end{equation*}

(8)   \begin{equation*} k=2\pi/\lambda \end{equation*}

a=particle radius and all the other variables are the same as for the Rayleigh equation

Please Note: In the equation for the form factor the entire function MUST be squared.  In Laser Experiments for Beginnings by Richard Zare the form factor equation does not have the squared term.  However he references the book by Milton Kerker and the equation is supposed to have a squared.  This would explain the weird “tails” of x-axis negative intensity that Rahul’s graphs had.  The graphs should have forward  x-axis scattering only.

I would like to first make a graph that predicts the how the scattering intensity changes as a function of wavelength for a varying number of particle densities (but a controlled particle size within the Rayleigh limit).  This graph will be made using Mathematica and the wavelengths used will be the visible spectrum.  The scattering intensity will be measure at 90 degrees—the angle at which there is maximum scattering intensity for Rayleigh scattering.

Previously Rahul modeled the Rayleigh light scattering as the particle size changed from 2nm to 902nm for yellow light.  Since Rayleigh scattering is wavelength dependent I would like to show how scattering changes as particle size changes (within the Rayleigh limit) for other wavelengths of light in addition to yellow—like red, green, and blue—to explore whether the change in scattering becomes more for certain wavelengths.  The concentration will be held constant.  This will also be done using Mathematica.

I will then look at the scattering intensity for the range of particles just under and just over the Rayleigh limit for the same wavelengths used in the graph of particles within the Rayleigh limit using both the Rayleigh and Rayleigh-Debye factor, which introduces a correcting form factor.  This is to observe the predicted intensity around the boundary of the Rayleigh limit, and how that intensity varies due to the introduction of the form factor.

The goal of predicting the scattering intensity would be to explore Static Light Scattering.  Static Light Scattering uses the scattering intensity predicted by the Rayleigh theory (at a 90 degree scattering angle) and the concentration of the cuvette to determine the molecular weight of the particles.  Molecular weight can be measured, in theory, by creating a Debye plot (Intensity of scattered light vs. concentration) and finding the intercept at zero concentration (which is equal to 1/molecular weight).  The gradient of the Debye plot is the 2nd Viral Coefficient (A2), which can be used to determine the interaction strength between the solvent and the particles.

A2 >0 the particles will stay in stable solution

A2 < 0 the particles may aggregate

A2 = 0 the particle-solvent interactions are completely balanced by the particle-particle interactions

Applications of Static Light Scattering:

Static Light Scattering has experimental application because it could be used to determine the weight of the molecules in an unknown solution.  By comparing the experimentally determined weight to predicted molecular weights, it may be possible to determine the identity of the molecules.  And particle aggregation can be applied to proteins.  Protein aggregation can be used as an important marker in many diseases.  A “normal” solution taken from a healthy individual can be used as a standard and compared to solutions taken from patients as an early detection method (if the protein aggregates indicates disease).

There are a couple quick checks to determine if the Rayleigh theory is accurate.  The first would be to look at the predicted scattering intensity.  This intensity should not be greater than the incident intensity.  Another check would be that the predicted scattering intensity for the Rayleigh theory is greatest at 90 degrees.  For the Rayleigh-Debye equation the forward scattering should be the greatest.  Both the Rayleigh theory and Rayleigh-Debye theory can be checked experimentally by actually making a solution and measuring the scattering intensity for a given wavelength.  The experimental parameters would provide values for the variables in the Rayleigh (and Rayleigh-Debye) equation.  With those values the predicted scattering intensity could be determined and compared to the actual scattering intensity.  In order to check the Static Light Scattering, the molecular weight of a particle with a known molecular weight could be predicted and then compared to that known weight.

REFERENCES:

Kerker, Milton. The Scattering of Light and Other Electromagnetic Radiation. New York:  Academic Press, 1969

Zare, Richard N., Bertrand H. Spencer, Dwight S. Springer, Matthew P. Jacobson. Laser Experiments for Beginners. Sausalito: University Science Books, 1995.

 

Theo

Magnetic induction is the main component of energy production in wind turbines. The efficiency of this energy production is based on wire turns and the power of the magnets used. To examine this more closely I will attempt to model the efficiencies of different arrangements. I hope to look at both different magnets, and different wire arrangements in order to flesh out how efficient magnetic induction based turbines are. Chapter seven of Griffiths touches on the subject, but does not delve into its real world applications as much as I hope to.

Project Proposal: Magnetic Levitation and Maglev Trains

For my project, I will research magnetic levitation – the method by which objects are suspended with only the support of magnetic fields – and explore the application of this method in maglev trains. These trains operate by one of three maglev technologies: electromagnetic suspension (EMS), electrodynamic suspension (EDS), or magnetodynamic suspension (MDS).

I will model the magnetic fields that allow these trains to levitate and propel along the tracks, as well as the induced currents that are generated by these fields in the electrodynamic suspension (EDS) method. Mathematica, Microsoft Excel, and Wolfram Alpha are the computational tools that I intend on using to do this.

The visual aids on my blog page will include graphs modeling the magnetic fields, as well as the induced currents in the EDS method. Through an analysis of these models, I will explain the physics behind this futuristic method of transportation.

Electro Optic Modulation

An electro optic modulator is a device that contains some kind of material that displays the electro optic effect. This means something inside this device, most likely a crystal of some kind has its optical properties change when a voltage is applied across it. This change in the crystal allows the laser beam traveling through it  to also change “modulating it”. This modulation can change things such as amplitude and phase among other things. My  project is  going to be investigating into how an electro optic modulator works.

Proposal

Using Scratch Programming and Mathematica, I hope to model, investigate and illustrate the manipulation of electromagnetic waves and how this differs with respect to liquid crystal display (LCD), plasma display panel (PDP), and cathode-ray tube display (CRT) screens. I shall focus on polarization and the control of polarization effects for each screen and how these differences manifest with regards to our perception of light. Although one potential hurdle could be difficulty in selecting a proper method of investigation, I aim to develop this analysis based on my own calculations and personal observation.

Blog Proposal

Electrodynamics is concerned with answering a simple question: what is the force exerted on a test charge Q by some arbitrary configuration of charge? Griffiths does not achieve the equation that truly answers this until the tenth chapter; while powerful, the relationship is complex. My goal is to explore this overarching equation and the ideas directly preceding it, including the Liénard-Wiechert Potentials, Jefimenko’s Equations, and retarded time. In order to deal with the five dimensional nature of these equations I will make heavy use of Mathematica’s plotting and manipulation functions. My hope is to generate a number of graphical representations of these functions, some with simplifying assumptions made, and thereby expose their physical significance.

Nuclear Magnetic Resonance Spectroscopy

The magnetic field and radio frequency pulses generated in an NMR spectrometer will be modeled, and their effects on atomic nuclei will be studied.  Mathematica, Excel, Wolfram Alpha, and the NMR software Topspin are the computational tools most likely to be used.  The book “NMR Spectroscopy Explained” by Neil E. Jacobsen will be used as the primary theoretical tool.  Visual pieces will include a model of the magnetic field inside an NMR, a model of a radio frequency pulse, and one or more spectra of an as-yet-undetermined molecule that will be used to explain the physics behind the NMR process.