For electromagnetic waves, the most familiar example of a polarizer, Polaroid, was invented by Edwin Land over 50 years ago, partly in experiments done in the attic of the Jefferson Physical Laboratory, where he worked as an undergraduate at Harvard. The idea of polaroid is to make material that conducts electricity (poorly) in one direction, but not in the other. Then the electric field in the conducting direction will be absorbed (the energy going to resistive loss), while the electric field in the nonconductive direction will be unaffected. One way of doing this is to make sheets of polymer (polyvinyl alcohol) stretched (to align the polymer molecules along a preferred axis) and doped with iodine (to allow conduction along the polymer molecules).2

Most breadboards have a notch or a groove that runs down the center, through the middle of the terminal strips. This line down the middle serves a number of functions.

Where you connect your electronic components on your breadboard is important, because that controls what other components they are able to connect to.

A “polarizer” is a device that allows light polarized in a particular direction (the “easy transmission axis” of the polarizer) to pass through with very little absorption, but absorbs most of the light polarized in the perpendicular direction. Thus an unpolarized light beam, passing though the polarizer, emerges polarized along the easy axis.

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A bus strip lets you connect the breadboard to a power supply so that the other electronic components on the breadboard can be powered. To give your breadboard power, you’ll use the bus strips to connect to a power supply.

This project from Build Electronic Circuits uses a breadboard to connect a microcontroller to a temperature sensor, which then uses code to change the color of a lamp based on the temperature.

Polarization is defined in terms of the pattern traced out in the transverse plane by the electric field vector as a function of time. Light is called natural ...

This should convince you that in general if the fast axis is in the \(\theta\) direction, the quarter wave plate looks like \[Q_{\theta}=P_{\theta}+i P_{\theta+\pi / 2} .\]

You can find a clue to the nature of optical activity by considering what it looks like if you look at it in a mirror. If you reflect the system illustrated in Figure \( 12.7\) in the \(x\)-\(z\) plane, by changing the sign of all the \(y\) coordinates, the angle \(\theta\) changes to −\(\theta\). Thus the corn syrup that you see in a mirror must be fundamentally different from the corn syrup in your kitchen. This is not so strange. After all, your right hand looks like a left hand when you look at it in a mirror. The corn syrup must have the same property and have a definite “handedness.” In fact, because of the tetrahedral bonding of the carbon atoms of which they are built, the sugar molecules in the corn syrup can and do have such a handedness.

Breadboard simulator

First, make sure that your push button is a “through hole” style component, meaning that it has the right metal parts to be plugged into the breadboard.

It is this frequency dependence that produces the interesting patterns of color that you see when you put cellophane or a stressed piece of plastic between polarizers.

Polarization offers many opportunities to get confused when you think of the light wave in terms of photons. Let us imagine turning down the intensity of the light to the point where one photon at a time is going through the polarizers and consider first the deceptively simple situation of light moving in the \(z\) direction through crossed polarizers in the \(x\)-\(y\) plane. Suppose that the first polarizer transmits light polarized in the \(x\) direction, and the second transmits light polarized in the \(y\) direction. This is deceptively simple because it seems that we can interpret what is going on simply in terms of photons. The situation is depicted in Figure \( 12.8\). This seems simple enough to interpret in terms of photons. The unpolarized light in region \(I\) is composed equally of photons polarized in the \(x\) direction and in the \(y\) direction (goes the wrong “classical” argument). Those polarized in the \(x\) direction get through the first polarizer, so half the photons are still around in region \(II\), where the intensity is reduced by half. Then none of these get through the second polarizer, so that the intensity in region \(III\) is zero.

For example, the transparent polymer material cellophane is made into thin sheets by stretching. Because of the stretching, the polymer strands tend to be oriented along the stretch direction. The dielectric constant in this material depends on the direction of the electric field. It is easier for charges to move along the polymer strands than across them. Thus the dielectric constant is larger for electric fields in the stretch direction.

where \(P_{\pm}\) are matrices that pick out the left- and right-circularly polarized components, respectively. They satisfy \[P_{\pm}\left(\begin{array}{c} 1 \\ \pm i \end{array}\right)=\left(\begin{array}{c} 1 \\ \pm i \end{array}\right), \quad P_{\pm}\left(\begin{array}{c} 1 \\ \mp i \end{array}\right)=0 .\]

The markings on a breadboard are very important, since they help you identify which holes on the terminal strip will allow you to connect your electronic components.

If you are using jumper wires, use the different colors of the wires to color code your project and more easily keep track of how things are connected.

This is just the rotation matrix \(R_{\theta}\), of (12.34)! \(R_{\theta}\) rotates both components of any light by an angle \(\theta\).

A typical breadboard will have two bus strips: a column for ground, which is marked in blue or black coloring, and a column for power, also called voltage, which is marked in red.

Here are two amusing devices that you can make with these optical elements (or matrices). Consider the combination of first a polarizer at \(45^{\circ}\) and then a quarter wave plate, as shown in Figure \( 12.6\). By forming the matrix product, \(Q_{0} P_{\pi / 4}\), you can see that this produces counterclockwise circularly polarized light from anything with a component of polarization in the \(\pi / 4\) direction. The argument goes like this. The product is \[Q_{0} P_{\pi / 4}=\left(\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\right)\left(\begin{array}{cc} 1 / 2 & 1 / 2 \\ 1 / 2 & 1 / 2 \end{array}\right)=\left(\begin{array}{cc} 1 / 2 & 1 / 2 \\ i / 2 & i / 2 \end{array}\right) .\]

A jumper wire is a short piece of wire with hard metal points on the end which plug into the holes on a breadboard. These allow you to make connections on the breadboard and start building a circuit.

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You can use a breadboard for any project that involves electronic circuits. They are easy to connect to things like a light emitting diode (LED) lights and batteries, as well as microcontrollers such as Arduino boards.

Build Circuit provides this simple breadboard project that lets you build a basic circuit that can test a remote by lighting up an LED when a remote switch is pressed.

You’ll find long rows of holes, called “strips.” Each breadboard has two types of “strips” – bus strips and terminal strips. Bus strips let you connect the board and its electronic components to a power source. Terminal strips let you actually plug various electronic components in and connect them to each other.

The discussion of (12.39) shows that in general, a wave plate will only be a quarter wave plate for light of a definite frequency.

If you need help finding, understanding, or developing a breadboard diagram, check out a tool called Fritzing. Fritzing lets you see and set up breadboard diagrams on a computer, so you can test things out and see how a circuit board fits together.

For frequencies such that \(e^{-i \Delta \phi}\) is −1, the light is polarized in the −\(45^{\circ}\) direction, and gets \(e^{-i \Delta \phi}\) through the second polarizer without further attenuation. But for frequencies such that e is 1, the light is still absorbed by the second polarizer. Intermediate frequencies are partially absorbed.

Breadboards are designed to work with any electronic component that has metal leads or pins that can be plugged into the holes on a breadboard. These are called “through hole” or “thru hole” components.

Because of the handedness of the sugar molecules, the index of refraction of the corn syrup actually depends on the handedness of the light. It is slightly different for left- and right-circularly polarized light. This happens because the \(\vec{E}\) field of a circularly polarized beam twists slightly as it traverses each sugar molecule and sees a slightly different electronic structure depending on the direction of the twist. Then, because the indices of refraction are slightly different, the left- and right-circularly polarized components of the light get different phase factors (\(k \ell\)) in passing through a thickness, \(\ell\), of the syrup.

Each horizontal row on one terminal strip is connected – meaning that anything you plug in on that row will be electrically connected to anything else plugged in on that row.

A solder board requires you to actually use a soldering iron to connect electronic components to the board. Solder boards are better for projects that are permanent and need to hold up to being installed somewhere or used multiple times.

Then, identify based on your project’s instructions where the push button should be plugged in. Many push button components are meant to straddle the center groove on the breadboard, with their pins plugged in to holes on either side.

Bread Boards electrical

For the transverse oscillations of a string, a polarizer is simply a slit that allows the string to oscillate in one transverse direction but not in the perpendicular direction.

as it must, since the first polarizer produces polarized light and the second one transmits it perfectly. \(P_{\theta}\) acting on a vector transmits the component in the \(\theta\) direction. This is easiest to visualize if \(\theta = 0\) or \(\pi / 2\). The matrices \[P_{0}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad P_{\pi / 2}=\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right) ,\]

This cool electronics project from Build Circuit uses a breadboard and a simple circuit to generate a small tune when a button is pressed.

Breadboards make it easy to build connections and develop prototypes for your electronics projects. Once you’ve gotten set up with a breadboard, you can use them to build cool electronics projects and even connect them with an Arduino or a Raspberry Pi!

The object \(P_{\theta}\) is called a “projection operator,” because it projects the vector onto the direction parallel to \(u_{\theta}\). It satisfies \[P_{\theta} P_{\theta}=P_{\theta},\]

1. Kohärenz geh (Zusammenhang): Kohärenz. coherence no Pl.

The effects of wave plates and polarizers and the like can be summarized by multiplication of the \(Z\) vector by 2×2 matrices. For example, a perfect polarizer with an axis at an angle \(\theta\) from the 1 axis can be represented by \[P_{\theta}=\left(\begin{array}{cc} \cos ^{2} \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^{2} \theta \end{array}\right) .\]

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every plane wave is polarized. However, in an “unpolarized” beam, the light wave consists of a range of angular frequencies with different polarizations. As a result of the interference of the different harmonic components of the wave, the polarization wanders more or less randomly as a function of time and space, and on the average, no particular polarization is picked out. A simple example of what this looks like is animated in program 12-2, where we plot an electric field of the form \[\begin{aligned} &E_{x}(t)=\cos \left(\omega_{1} t+\phi_{1}\right)+\cos \left(\omega_{2} t+\phi_{2}\right), \\ &E_{y}(t)=\cos \left(\omega_{3} t+\phi_{3}\right)+\cos \left(\omega_{4} t+\phi_{4}\right), \end{aligned}\]

Connecting components to a board is very simple and doesn’t take any specialized tools. All you have to do is push the metal pins or leads of your electronic components into the holes on your breadboard.

That means that each of the holes in a terminal strip has its own unique position, which can be identified with its column letter and row number.

“Optical activity” is a property of many organic and some inorganic compounds. An optically active material rotates the polarization of light without absorbing either component of the polarization. A familiar example of such a material is corn syrup, a thick aqueous solution of sugar that you probably have in your kitchen. If you put a rectangular container of corn syrup between polarizers, as shown in Figure \( 12.7\), and rotate the second polarizer until the intensity of the light getting through is a maximum, you will find that direction of the second polarizer is not the same as that of the first. The plane of the polarization has been rotated by some angle \(\theta\). The rotation angle, \(\theta\), is proportional to the thickness of the container, the length of the region of syrup that the light goes through.

Note that in general the phase difference, \(\Delta \phi\), depends on the frequency of the light. Even if \(n_{x}\) and \(n_{y}\) depend on frequency, it would be a bizarre accident if that dependence canceled the \(\omega\) dependence from the explicit factor of \(\omega\) in (12.39).

They are also great for any electronic circuit that you don’t want to make permanent, such as a test project or prototype. If you make a mistake or want to change something, you can unplug things from the breadboard and move them around or try something else. This makes breadboards excellent for things like prototyping, testing, and any electronics projects for beginners.

Inside each breadboard are metal clips that catch an electronic component whenever it gets plugged in. These metal clips are arranged in lines that correspond to the rows and columns on the terminal strips, so that you can see and control which electronic components are connected.

The answer is actually quite interesting. Back in the 1970s, when people wanted to create their own circuits, they used wooden boards to build them on. In fact, the standard wooden breadboard found in nearly every kitchen seemed perfect for assembling homemade circuits!

Most of the area in a breadboard is taken up by terminal strips. Terminal strips are made up of small holes, or perforations, where you can plug in your electronic components.

One specific type of electronic component you can use with a board is a push button. Including a push button in your electronics project lets you provide “input” to your circuit based on whether the button is being pressed.

Each column on a breadboard is labeled, usually with a letter. Check the top of a terminal strip to see how the columns are labeled. Each row on a breadboard is also labeled, usually with a number.

breadboard中文

Clearly, the optical activity of corn syrup cannot depend on crystal structure, because the stuff is a perfectly uniform liquid, completely invariant under rotations in three-dimensional space. It can have no special axes, or any such thing. Optical activity must work very differently from birefringence.

After seeing a big plastic rectangle full of tiny holes, you might be wondering: why on earth do we call that thing a breadboard?

In the opposite order, \(P_{\pi / 4} Q_{0}\) is an analyzer for circularly polarized light. It annihilates counterclockwise light and converts clockwise polarized light to light linearly polarized in the \(\pi / 4\) direction.

This page titled 12.3: Wave Plates and Polarizers is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Howard Georgi via source content that was edited to the style and standards of the LibreTexts platform.

When this acts on an arbitrary vector you get circularly polarization unless the vector is annihilated by \(P_{\pi / 4}\). \[Q_{0} P_{\pi / 4}\left(\begin{array}{l} \psi_{1} \\ \psi_{2} \end{array}\right)=\frac{\psi_{1}+\psi_{2}}{2}\left(\begin{array}{l} 1 \\ i \end{array}\right) .\]

In particular, the phase difference, between \(x\) and \(y\) polarized light in going through the plate is \[\Delta \phi=\frac{n_{x}-n_{y}}{c} \omega \ell .\]

The best way to find electronics projects that will let you use your board is to check out project ideas on various websites! Here are some great examples of electronics projects that use breadboards:

To do that, you’ll need to reference your circuit diagram, if you are working with one. If you have written directions for your electronics project, that can also tell you which row and column each electronic component needs to connect to.

A wave plate in which the phase difference is \(\pi / 2\) is called a “quarter wave plate.” For a wave plate in which the phase difference is between 0 and \(\pi\), it is conventional to call the axis with the smaller phase the “fast axis.” A quarter wave plate with fast axis along the 1 axis is represented by \[Q_{0}=\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right) .\]

Breadboard Kit

A solderless board is the plastic board we’ve been discussing, and it’s the type of breadboard you’ll use for most Thimble.io projects. This type of breadboard lets you plug in and unplug electronic components without having to do any soldering.

One reason that polarization is important is that the polarization state of an electromagnetic wave can be easily manipulated. Two of the most important devices for such manipulation are polarizers and wave plates.

Binding posts are not automatically connected to the breadboard, so if you want to use them, you’ll first need to use jumper wires to connect the binding posts to the bus strips. To connect wires to binding posts, first unscrew the post until you can see the hole that goes through it. Thread your wire through that hole, then screw the post back down.

We can now use our matrix language to see how this leads to optical activity. Up to an irrelevant overall phase, we can choose the phase produced on the left-circularly polarized light to be −\(\theta\) and that on the right-circularly polarized light to be \(\theta\). Then we can represent the action of the syrup on an arbitrary wave by the matrix \[e^{-i \theta} P_{+}+e^{i \theta} P_{-} ,\]

Bus strips are usually found at the outer edges of a breadboard or in between the terminal strips, and are almost always narrower than the terminal strips.

When connecting a power supply, always make sure you follow the breadboard diagram you’re using exactly. If your board doesn’t seem to be connected to the power supply, check to make sure you have plugged things in to the right bus strip and haven’t confused positive and negative ones.

However, you also have to make sure that your components are connected properly, and not just plugged in. That means making sure that everything is plugged in to the correct row and column.

BreadboardsAmazon

where the phases are random and the frequencies are chosen at random in a small range around a central frequency. You can watch the \(\vec{E}\) field wandering in the \(x\)-\(y\) plane, eventually filling it up. The narrower the range of frequencies in the wave, the more slowly the polarization wanders. In the example in program 12-2, the range of frequencies is of the order of 10% of the central frequency, so the polarization wanders rapidly. But for a beam with a fairly well-defined frequency, the polarization will be nearly constant over many cycles of the wave. The time over which the polarization is approximately constant is called the coherence time of the wave. For a plane wave of definite frequency, the coherence time is infinite.

Consider, now, putting such a wave plate between two crossed polarizers, oriented at \(\pm 45^{\circ}\), as shown in Figure \( 12.5\). Without the wave plate, no light would get through because the first polarizer transmits only light polarized at \(45^{\circ}\), described by the \(Z\) vector \[Z=\left(\begin{array}{l} 1 / \sqrt{2} \\ 1 / \sqrt{2} \end{array}\right)\]

The same effect may arise because of the inherent structure of a transparent crystal. An example is the naturally occurring mineral, calcite, a crystalline form of calcium carbonate, \(CaCO_{3}\). Crystals of calcite have the fascinating property of splitting a beam of unpolarized light into its two polarization states. Birefringence can even be produced mechanically, by stressing a transparent material, squeezing the electronic structure in one direction.

How to use a breadboard

Any electronics project you do with a breadboard will have its own requirements, so you’ll need to check your project’s breadboard diagram and instructions to identify where to connect the components.

A breadboard connects to a power supply through the bus terminals, which are also sometimes called “rails.” Most board bus terminals include a positive, or voltage, bus and a negative, or ground, bus. Positive bus strips are marked with the color red and a plus sign, and negative bus strips are marked with the color blue or black and a minus sign.

Most breadboards have multiple bus terminals, usually one on either side of the board. These different strips are not connected, so if you want to connect both sides, you’ll need to use a jumper wire.

Are you hoping to get started with electronics projects, but you don’t have a breadboard, soldering iron, or other materials needed to built your own circuits? Don’t worry! We’ll teach you everything you need to know about how to use electronics! But first, let’s go over how to use a breadboard.

Here, Instructables lists ten awesome, easy to do electronics projects that can get you started with your breadboard. Most of them include an LED circuit, which lets you see exactly how things are working by blinking, flashing, or turning off a light. Learn how to make a light blink, a bell ring, and even how to detect static electricity!

This project from Build Electronic Circuits lets you use a breadboard to connect a radar device to a microcontroller to detect a person’s breath and use that input to fade or brighten an LED light.

Still, the name stuck, so even though a plastic rectangle full of small holes isn’t great for making sandwiches on, we still call it a breadboard! Some people still make circuits on real wooden breadboards the old-fashioned way – you can see an example here.

Inside, metal strips create connections between the electronic components that you plug in. These connections create an electronic circuit, which you can then use to control any of your electronics projects!

Breadboards are an excellent way to get started building circuits and making your own electronics projects. You can easily move things around, test things out, and try different components.

A breadboard lets you experiment with your own circuits, but it also has the right labeling to help you follow along with a board diagram someone else has provided.

Some boards, especially larger breadboards, come with binding posts, which are another way to connect your breadboard to an external power source. Binding posts look like pegs, or pins, attached to the platform that the breadboard is on.

Terminal strips are connected in specific ways based on their rows and columns. It’s important to understand the layout of the terminal strip on the breadboard you’re working with. Make sure you check the labeling of your breadboard before plugging things in!

Breadboards are used for electronics projects that need to be assembled without a fully equipped electronics workshop. Because the inner workings of a board already provides connections between the electronic components, you can create an electronic circuit without needing to solder or do anything else besides simply plugging things in.

A wave plate in which the phase difference is \(\pi\) is called a “half wave plate.” A half wave plate is obtained by replacing the \(i\) in (12.45)-(12.47) by -1. Thus, \[H_{\theta}=P_{\theta}-P_{\theta+\pi / 2} .\]

Another aspect of breadboard types is their size. Both types of boards come in a variety of sizes, from very small ones for miniature projects to bigger ones that let you build large, complex circuits and give you lots of room to work.

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Next, you’ll want to make sure that your push button is connected to the other components. Check your project’s breadboard diagram to see what else you need, and where to put it. You might use something like a resistor or jumper wires to connect your button to an external power supply or other components.

This list from Elprocus includes ten awesome breadboard projects, including a kitchen timer, a water tank level indicator, and smart fan that turns on and off based on the temperature.

However, using a pre-made breadboard with connections built into it already made things a lot easier, so people stopped using wooden breadboards once these awesome tools became available.

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You can use a breadboard for any electronic project that uses circuits. These simple aspects of electronic engineering let you build and control all sorts of things, from lights to sensors, and can be connected to a microcontroller to do even more.

Jumper wires are not technically part of the breadboard itself, but they’re an important part of any circuit or other electronics project that uses a breadboard.

“Wave plates” are optical elements that change the relative phase of the two components of \(Z\). Wave plates are possible because there are materials in which the index of refraction depends on the polarization. This property is called “birefringence.” It can happen in various ways.

The center groove on a breadboard allows certain types of integrated circuits called dual in-line packages to be connected in a way that straddles that line. It also shows where the terminal strips have been divided and which columns are connected, and it also allows breadboards to be easily stacked on top of each other for storage or larger projects.

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Coming out of the first polarizer, the vector, \(Z\), looks like (12.40) for all the frequency components in the white light. But when the wave plate is inserted in between, a frequency dependent phase difference is added, so that the \(Z\) vector coming out of the wave plate (up to an irrelevant overall phase) looks like \[Z=\left(\begin{array}{c} 1 / \sqrt{2} \\ e^{-i \Delta \phi} / \sqrt{2} \end{array}\right) .\]

One might wonder about the reason for the handedness of the sugar molecules. In fact, there are physical processes, the weak interactions that give rise to \(\beta\)-radioactivity, that look different when reflected in a mirror3 and thus in principle could distinguish between left-handed and right-handed molecules. However, these interactions are most likely irrelevant to the handedness of corn syrup. Probably, the reason is biology rather than physics. Long ago, when the beginnings of life emerged from the primordial soup, purely by accident, the right-handed sugars were used. From then on, the handedness was maintained by the processes of reproduction.

There are no vertical connections on a terminal strip. Horizontal rows on either side of the center groove are also not connected to each other.

You can check that the matrices are \[P_{\pm}=\frac{1}{2}\left(\begin{array}{cc} 1 & \mp i \\ \pm i & 1 \end{array}\right) .\]

All of the electronic components that you connect to your breadboard need electricity to work! In order to power your board, you’ll need to connect it to a power supply.

The holes in a breadboard are connected by metal clips that span five holes, horizontally. These metal clips allow each row of five holes to be connected.

Breadboard OS

Breadboard labels help you identify which holes in the terminal strips are connected. On most breadboards, they are connected horizontally in sets of five. That means the first five holes in Row 1 are connected, but none of those holes are connected with Row 2, or with holes on the other side of the center groove.

But compare this with the apparently similar situation in which the second polarizer transmits light polarized at \(45^{\circ}\) in the \(x\)-\(y\) plane, as shown in Figure \( 12.9\). Now the wave description tells us that the intensity in region \(III\) is reduced by another factor of 2 from that in region \(II\). This is impossible to interpret in terms of classical particles. To see this, it is only necessary to turn down the intensity so that only one photon comes through at a time. Then the first polarizer is OK. As before, if the photon is polarized in the \(x\) direction, it get through. But now what happens at the second polarizer. The photon cannot split up. Either it gets through or it doesn’t. To be consistent with the wave description, in which the intensity is reduced by another factor of two, the transmission at the second polarizer must be a probabilistic event. Half the time the photon gets through. Half the time it is absorbed. There is no way for the

Figure \( 12.5\): Initially unpolarized light passing through a pair of crossed polarizers with a wave plate in between.

In any beam of light, at any given point and time, the electric field points in a particular direction. Likewise, because any plane electromagnetic wave with a definite angular frequency can be described by (12.20) and (12.21), \[\begin{aligned} \vec{E} &=\operatorname{Re}\left(\vec{e}(\vec{k}) e^{i \vec{k} \cdot \vec{r}-i \omega t}\right) \\ \vec{B} &=\operatorname{Re}\left(\vec{b}(\vec{k}) e^{i \vec{k} \cdot \vec{r}-i \omega t}\right) \end{aligned}\] \[\vec{b}(\vec{k})=\frac{1}{\omega} \vec{k} \times \vec{e}(\vec{k})=\frac{n}{c} \hat{k} \times \vec{e}(\vec{k}) \quad \text { and } \quad \hat{k} \cdot \vec{e}(\vec{k})=0.\]

Breadboard near me

Then (12.52) becomes \[e^{-i \theta} \frac{1}{2}\left(\begin{array}{cc} 1 & -i \\ i & 1 \end{array}\right)+e^{i \theta} \frac{1}{2}\left(\begin{array}{cc} 1 & i \\ -i & 1 \end{array}\right)=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) .\]

However the birefringence is produced, we can make a wave plate by orienting the material so that the \(x\) and \(y\) directions correspond to different indices of refraction, \(n_{x}\) and \(n_{y}\), and then making a slice of the material in the form of a plate in the \(x\)-\(y\) plane, with some thickness \(\ell\) in the \(z\) direction. Now an electromagnetic wave traveling in the \(z\) direction through the plate has different \(k\) values depending on its polarization: \[k=\left\{\begin{array}{l} \frac{n_{x}}{c} \omega \text { for polarization in the } x \text { direction } \\ \frac{n_{y}}{c} \omega \text { for polarization in the } y \text { direction } \end{array}\right.\]

If you want to get started with breadboards and other awesome engineering and computer science projects, check out our monthly subscription service that pairs online classes with hands-on STEM kits. All ages welcome!

Always make sure all your wires are securely connected – a loose connection will make your power supply unreliable and prevent your circuit from working.

When you plug a component into a breadboard, make sure it’s plugged in all the way. Push it in as far as it will go. This prevents shaky or unreliable connections between the components and the metal clips. After you’ve plugged something in to your breadboard, you should be able to pick the breadboard up and turn it upside down without anything falling out!

You can also use color coding to ensure that you’re plugging things in correctly. For example, many battery packs and other power supply options have red and black wires to indicate which bus strip they are meant to be plugged in to – use the red wire for positive, and the black one for negative.