Digital circuits only care about two voltages: for example, (known as “on”) and (known as “off”).
Logic devices, which are active circuit elements, interpret voltages according to a simple set of mathematical rules known as Boolean logic. The most basic logic devices are the AND, OR, and NOT gates:
For an AND gate, the output will always be at an electric potential of (off) unless both the inputs are at (on), in which case the output will be at (on) as well.
For an OR gate, the output will always be at an electric potential of (off) unless either of the inputs are at (on), in which case the output will be at (on) as well.
For a NOT gate, the output will always be the opposite of the input. Thus, if the input is (on), the output will be (off) and vice-versa.
Alternating current changes direction of current flow. The frequency is the number of times the current reverses direction in a second. Household AC is . In AC circuits the current is impeded but not stopped by elements like capacitors and inductors.
Capacitive Reactance is a measure of how a capacitor impedes the current flow from a given voltage in an AC circuit and is inversely proportional to capacitance. Inductive Reactance is a measure of how an inductor in an AC circuit impedes the current flow from a given voltage and is directly proportional to inductance.
The total impedance of an AC circuit depends on resitance, capacitive reactance and inductive reactance.
If the capacitive reactance and inductive reactance are both zero or unequal the voltage and current are out of phase. That is they peak at different times in the cycle. The phase angle measures the lag or lead of current over voltage.
Key Equations
Examples
Example 1
Question: Consider the op-amp circuit diagram shown here. Note the fixed-voltage leads are omitted for clarity. (This is typical.)
f) By what factor is the op-amp amplifying the input voltage?
e) What is the output voltage now?
Now let’s adjust the input voltage at point to .
d) What must the output voltage be?
c) What current must be flowing through the resistor?
b) What current is flowing through the resistor?
a) If the op-amp is “doing its job,” what is the electric potential at point ?
Let’s begin with an input voltage at point of .
Answer:
a) The op-amp is supposed to make the two input voltages as close to equal as possible, or in other words, Therefore if the input voltage at point is , then the input voltage at point should also be .
b) We will use Ohm's Law to find the current going through the resistor.
c) Recall that no current ever flows into an op-amp. Therefore, the current must be the same as the current running through the resistor, which is .
d) We will again use Ohm's law. First we must find the total resistance and then we can plug in the known values to solve for the voltage.
e) We now want to find the output voltage given an input voltage of at point . Though the numbers are different, it is the same process as solving it when the input voltage was . First we must find the current and the total resistance. We then use these values to solve for the output voltage. The total resistance remains unchanged. So the new output voltage is
f) This is a simple division problem. The output voltage divided by the input voltage will give us the factor by which the output voltage is greater than the input voltage.
Advanced Topics Problem Set
You purchase a circular solenoid with turns, a radius of , and a length of . Calculate the inductance of your solenoid in Henrys.
A current of A is passing through your solenoid. The current is turned down to zero over the course of seconds. What voltage is induced in the solenoid?
What is the voltage drop across an inductor if the current passing through it is not changing with time? Does your answer depend on the physical makeup of the inductor? Explain.
Consider the transistor circuit diagram shown here. The resistor is a light bulb that shines when current passes through it. If the base is raised to a voltage of , will the light bulb shine?
If the base is lowered to a voltage of , will the light bulb shine?
Why are transistors sometimes called electronic switches?
Consider the op-amp circuit diagram shown here. Note the fixed-voltage leads are omitted for clarity. (This is typical.) Let’s begin with an input voltage at point of . If the op-amp is “doing its job,” what is the electric potential at point ?
What current is flowing through the resistor?
Recall that no current ever flows into an op-amp. What current must be flowing through the resistor?
What must the output voltage be?
Now let’s adjust the input voltage at point to . What is the output voltage now?
f. By what factor is the op-amp amplifying the input voltage? g. What are some practical applications for such a device?
Consider the logic circuit shown here. If , , and are all off, what is the state of ?
If , , and are all on, what is the state of ?
Fill out the entire “logic table” for this circuit.
State of State of State of State of
on on on
on on off
on off on
on off off
off on on
off off on
off on off
off off off
A series circuit contains the following elements: a resistor, a inductor, two capacitors and a capacitor. Voltage is provided by a generator operating at . Draw a schematic diagram of the circuit.
Calculate the total capacitance of the circuit.
Calculate the capacitive reactance.
Calculate the impedance.
Calculate the peak current.
f. Calculate the phase angle. g. Resonance occurs at the frequency when peak current is maximized. What is that frequency?
Answers to Selected Problems
a. b.
Zero
a. Yes b. No c. Because they turn current flow on and off.
4a. b. c.
d.
e.
f.
5a. b.
c.
6b. c.
d.
e.
f.
g.
Chapter 18: Light Version 2
The Big Idea
Light is a wave of changing electric and magnetic fields. Light waves are caused by disturbances in an electromagnetic field, like the acceleration of charged particles (such as electrons). Light has a dual nature; at times, it acts like waves, while at other times it acts like particles, called photons. Light travels through space at the maximum speed allowed by the laws of physics, called the speed of light. Light has no mass, but it carries energy and momentum. Fermat's principle states that light will always take the path that takes the least amount of time (not distance).
Fermat’s Principle governs the paths light will take and explains the familiar phenomena of reflection, refraction, diffraction, scattering and color absorption and dispersion. Light rarely travels in a straight line path. When photons interact with electrons in matter, the time it takes for this interaction determines the path. For example, higher frequency blue light is refracted more than red because blue interacts more frequently with electrons. Also, the path of least time is achieved when blue light bends more than red light so that it gets out of the 'slow' region faster. Fermat’s Principle explains the many fascinating phenomena of light from rainbows to sunsets to the halos around the moon.
Key Concepts
When charged particles accelerate, changing electric and magnetic fields radiate outward. The traveling electric and magnetic fields of an accelerating (often oscillating) charged particle are known as electromagnetic radiation or light.
The color of light that we observe is a measure of the frequency of the light: the smaller the frequency, the redder the light.
> The spectrum of electromagnetic radiation can be roughly broken into the following ranges:
EM wave Wavelength range Comparison size
gamma-ray and shorter atomic nucleus
ray hydrogen atom
ultraviolet (UV) small molecule
violet (visible) typical molecule
blue (visible) typical molecule
green (visible) typical molecule
red (visible) typical molecule
infrared (IR) human hair
microwave human finger
radio Larger than car antenna
Light can have any wavelength. Our vision is restricted to a very narrow range of colors between red and violet.
Fermat’s Principle makes the angle of incident light equal to the angle of reflected light. This is the law of reflection.
When light travels from one type of material (like air) into another (like glass), its effective speed is reduced due to interactions between photons and electrons. If the ray enters the material at an angle, Fermat’s Principle dictates that the light will change the direction of its motion. One way to think about this is that light takes the path of least time to get from points A to point B, thus it takes a more direct path through ‘slower’ mediums, so it can get out of the slow part faster. Light does not always travel in a straight line, it travels on the path of least time. This is called refraction.
White light consists of a mixture of all the visible colors: red, orange, yellow, green, blue, indigo, and violet (ROYGBIV). Our perception of the color black is tied to the absence of light.
Different frequencies of light (and hence different colors in the visible spectrum) will travel at slightly different speeds in materials, like glass, and thus have slightly different refracting angles. This phenomena gives rise to rainbows.
Our eyes include color-sensitive and brightness-sensitive cells. The three different color-sensitive cells (cones) can have sensitivity in three colors: red, blue, and green. Our perception of other colors is made from the relative amounts of each color that the cones register from light reflected from the object we are looking at. Our brightness-sensitive cells work well in low light. This is why things look ‘black and white’ at night.
The chemical bonds in pigments and dyes – like those in a colorful shirt – absorb light at frequencies that correspond to certain colors. When you shine white light on these pigments and dyes, some colors are absorbed and some colors are reflected. We only see the colors objects reflect.
Color Addition
Red Green Blue Perceived color
white
black
magenta
yellow
cyan
Key Applications
Total internal reflection occurs when light goes from a slow (high index of refraction) medium to a fast (low index of refraction) medium. With total internal reflection, light refracts so much it actually refracts back into the first medium. This is how fiber optic cables work: no light leaves the wire.
Rayleigh scattering occurs when light interacts with our atmosphere. The shorter the wavelength of light, the more strongly it is disturbed by collisions with atmospheric molecules. Accordingly, blue light from the Sun is preferentially scattered by these collisions into our line of sight. This is why the sky appears blue.
Beautiful sunsets are another manifestation of Rayleigh scattering that occurs when light travels long distances through the atmosphere. The blue light and some green is scattered away, making the sun appear red.
Lenses, made from curved pieces of glass, focus or de-focus light as it passes through. Lenses that focus light are called converging lenses, and these are the ones used to make telescopes and cameras. Lenses that de-focus light are called diverging lenses.
Lenses can be used to make visual representations, called images.
Mirrors are made from highly reflective metal that is applied to a curved or flat piece of glass. Converging mirrors can be used to focus light – headlights, telescopes, satellite TV receivers, and solar cookers all rely on this principle. Like lenses, mirrors can create images.
The focal length, , of a lens or mirror is the distance from the surface of the lens or mirror to the place where the light is focused. This is called the focal point or focus. For diverging lenses or mirrors, the focal length is negative.
When light rays converge in front of a mirror or behind a lens, a real image is formed. Real images are useful in that you can place photographic film at the physical location of the real image, expose the film to the light, and make a two-dimensional representation of the world, a photograph.
When light rays diverge in front of a mirror or behind a lens, a virtual image is formed. A virtual image is a trick, like the person you see “behind” a mirror’s surface when you brush your teeth. Since virtual images aren’t actually “anywhere,” you can’t place photographic film anywhere to capture them.
Real images are upside-down, or inverted. You can make a real image of an object by putting it farther from a mirror or lens than the focal length. Virtual images are typically right-side-up. You can make virtual images by moving the mirror or lens closer to the object than the focal length.
Waves are characterized by their ability to constructively and destructively interfere. Light waves which interfere with themselves after interaction with a small aperture or target are said to diffract.
Light creates interference patterns when passing through holes (“slits”) in an obstruction such as paper or the surface of a CD, or when passing through a thin film such as soap.
Key Equations
The product of the wavelength of the light (in meters) and the frequency of the light (in , or ) is always equal to a constant, namely the speed of light .
The index of refraction, , is the ratio of its speed in a vacuum to the slower speed it travels in a material. can depend slightly on wavelength.
Double slit interference maxima. is the order of the interference maximum in question, is the distance between slits. and is the angular position of the maximum.
Single slit interference maxima. and are defined as above and is the width of the slit.
Diffraction grating interference maxima. and are defined as above and is the distance between the lines on the grating.
Thin film interference: is the index of refraction of the film, is the thickness of the film, and is an integer. In the film interference, there is a delay (phase change) if the light is reflected from an object with an index of refraction greater than that of the incident material.
For lenses, the distance from the center of the lens to the focus is . Focal lengths for foci behind the lens are positive in sign. The distance from the center of the lens to the object in question is , where distances to the left of the lens are positive in sign. The distance from the center of the lens to the image is . This number is positive for real images (formed to the right of the lens), and negative for virtual images (formed to the left of the lens). For mirrors, the same equation holds! However, the object and image distances are both positive for real images formed to the left of the mirror. For virtual images formed to the right of the mirror, the image distance is negative
The size of an object’s image is larger (or smaller) than the object itself by its magnification, . The level of magnification is proportional to the ratio of and . An image that is double the size of the object would have magnification .
The radius of curvature of a mirror is twice its focal length
Question: Nisha stands at the edge of an aquarium deep. She shines a laser at a height of that hits the water of the pool from the edge. Draw a diagram of this situation. Label all known lengths.
a) How far from the edge of the pool will the light hit bottom?
b) If her friend, Marc, were at the bottom and shined a light back, hitting the same spot as Nisha’s, how far from the edge would he have to be so that the light never leaves the water?
Answer:
a) To solve for the di
stance from the edge we must first solve for the distance from the laser to the pool surface and then add that to the distance from the pool surface to the bottom of the pool. We can find the distance from the laser to the pool by using the Pythagorean Theorem. Now that we have the length from the laser to the pull all we need is the length from the surface of the pool to the bottom of it.
To find this value we will use the equation Once we have solved for , we will be able to use trigonometry to solve for the distance from the surface of the pool to the bottom of the pool. We know that and that . So once we solve for , we can solve for . This is the complement of , so Now we can solve for .
Now we can use trigonometry.
Now we simply need to add the two distances together to get our answer.
CK-12 People's Physics Book Version 2 Page 18