Breadboard

A breadboard is a construction base for prototyping of electronics. Originally it was literally a bread board, a polished piece of wood used for slicing bread. In the 1970s the solderless breadboard (AKA plugboard, a terminal array board) became available and nowadays the term "breadboard" is commonly used to refer to these. "Breadboard" is also a synonym for "prototype".

Because the solderless breadboard does not require soldering, it is reusable. This makes it easy to use for creating temporary prototypes and experimenting with circuit design. For this reason, solderless breadboards are also extremely popular with students and in technological education. Older breadboard types did not have this property. A stripboard (veroboard) and similar prototyping printed circuit boards, which are used to build semi-permanent soldered prototypes or one-offs, cannot easily be reused. A variety of electronic systems may be prototyped by using breadboards, from small analog and digital circuits to complete central processing units (CPUs).


Evolution:

This 1920s TRF radio manufactured by Signal was constructed on a wooden breadboard.
In the early days of radio, amateurs nailed bare copper wires or terminal strips to a wooden board (often literally a board to slice bread on) and soldered electronic components to them.[1] Sometimes a paper schematic diagram was first glued to the board as a guide to placing terminals, then components and wires were installed over their symbols on the schematic. Using thumbtacks or small nails as mounting posts was also common.

Breadboards have evolved over time, with the term now being used for all kinds of prototype electronic devices. For example, US Patent 3,145,483,[2] filed in 1961 and granted in 1964, describes a wooden plate breadboard with mounted springs and other facilities. US Patent 3,496,419,[3] filed in 1967 and granted in 1970, refers to a particular printed circuit board layout as a Printed Circuit Breadboard. Both examples refer to and describe other types of breadboards as prior art.

The breadboards

most commonly used today is usually made of white plastic and is a pluggable (solderless) breadboard. It was designed by Ronald J. Portugal of EI Instruments Inc. in 1971.[4]

Alternatives:

Wire wrap backplane
Alternative methods to create prototypes are point-to-point construction (reminiscent of the original wooden breadboards), wire wrap, wiring pencil, and boards like the stripboard. Complicated systems, such as modern computers comprising millions of transistors, diodes, and resistors, do not lend themselves to prototyping using breadboards, as their complex designs can be difficult to lay out and debug on a breadboard.

Modern circuit designs are generally developed using a schematic capture and simulation system, and tested in software simulation before the first prototype circuits are built on a printed circuit board. Integrated circuit designs are a more extreme version of the same process: since producing prototype silicon is costly, extensive software simulations are performed before fabricating the first prototypes. However, prototyping techniques are still used for some applications such as RF circuits, or where software models of components are inexact or incomplete.

You could also use a square grid of pairs of holes where one hole per pair connects to its row and the other connects to its column. This same shape can be in a circle with rows and columns each spiraling opposite clockwise/counterclockw

Bus and terminal strips:

The hole pattern for a typical etched prototyping PCB (printed circuit board) is similar to the node pattern of the solderless breadboards shown above.
Solderless breadboards are available from several different manufacturers, but most share a similar layout. The layout of a typical solderless breadboard is made up from two types of areas, called strips. Strips consist of interconnected electrical terminals.

Terminal strips:
The main areas, to hold most of the electronic components.
In the middle of a terminal strip of a breadboard, one typically finds a notch running in parallel to the long side. The notch is to mark the centerline of the terminal strip and provides limited airflow (cooling) to DIP ICs straddling the centerline[citation needed]. The clips on the right and left of the notch are each connected in a radial way; typically five clips (i.e., beneath five holes) in a row on each side of the notch are electrically connected. The five clip columns on the left of the notch are often marked as A, B, C, D, and E, while the ones on the right are marked F, G, H, I and J. When a "skinny" dual in-line pin package (DIP) integrated circuit (such as a typical DIP-14 or DIP-16, which have a 0.3-inch (7.6 mm) separation between the pin rows) is plugged into a breadboard, the pins of one side of the chip are supposed to go into column E while the pins of the other side go into column F on the other side of the notch.
Bus strips:
To provide power to the electronic components.
A bus strip usually contains two columns: one for ground and one for a supply voltage. However, some breadboards only provide a single-column power distributions bus strip on each long side. Typically the column intended for a supply voltage is marked in red, while the column for ground is marked in blue or black. Some manufacturers connect all terminals in a column. Others just connect groups of, for example, 25 consecutive terminals in a column. The latter design provides a circuit designer with some more control over crosstalk (inductively coupled noise) on the power supply bus. Often the groups in a bus strip are indicated by gaps in the color marking.
Bus strips typically run down one or both sides of a terminal strip or between terminal strips. On large breadboards additional bus strips can often be found on the top and bottom of terminal strips.
Some manufacturers provide separate bus and terminal strips. Others just provide breadboard blocks which contain both in one block. Often breadboard strips or blocks of one brand can be clipped together to make a larger breadboard.

In a more robust variant, one or more breadboard strips are mounted on a sheet of metal. Typically, that backing sheet also holds a number of binding posts. These posts provide a clean way to connect an external power supply. This type of breadboard may be slightly easier to handle. Several images in this article show such solderless breadboards.

push switch

Low-current, momentary action pushbutton switches, such as PCB-mount 'tactile' types, are cheap, and available in an abundance of different styles. Latching types, on the other hand, are often larger, more expensive, and available only in a relatively limited range of styles. This can be a problem if you need a small, inexpensive switch for latching power to a load. The solution is to convert a pushbutton's momentary action into a latching function.

Previous Design Ideas have proposed solutions based on discrete components (Ref. 1) and IC-based circuits (Ref.2 and Ref.3). The circuit outlined below, however, requires just two transistors and a handful of passive components to achieve the same result.

The circuit in Figure 1(a) is configured to latch power to a low-side (ground-referred) load. It works in 'toggle' mode; that is, the first switch closure applies power to the load, the second removes power, and so on.

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Figure 1  Circuit converts momentary action push switch into latching power switch.

To understand how the circuit operates, assume that the DC power supply, +V_S, has just been applied, capacitor C1 is initially uncharged, and Q1 is off. The P-channel MOSFET, Q2, is held in its off state by R1 and R3, which work in series to pull the gate up to +V_S, such that V_GS is zero. The circuit is now in its 'unlatched' state, where the load voltage, V_L, at the OUT (+) terminal is zero.

If the normally-open push switch is momentarily closed, C1 – being uncharged – pulls Q2's gate to 0V, thus turning on the MOSFET. The load voltage at OUT (+) now rises immediately toward +V_S , and Q1 receives base bias via R4 and turns on. Under these conditions, Q1 saturates and pulls Q2's gate low via R3, thus holding the MOSFET on when the switch has opened. The circuit is now in its 'latched' state, where both transistors are on, the load is energized, and C1 charges up to +V_S via R2.

When the switch is momentarily closed for a second time, the voltage on C1 (by now approximately equal to +V_S) is transferred to Q2's gate. Since Q2's gate-source voltage is now roughly zero, the MOSFET turns off and the load voltage falls to zero. Q1's base-emitter voltage also falls to zero and the transistor turns off. Therefore, when the switch is released, there is nothing to hold Q2 on, and the circuit reverts to its 'unlatched' state, where both transistors are off, the load is de-energized, and C1 discharges via R2.

Resistor R5 across the output terminals is an optional component that acts as a pull-down. When the switch is released, C1 discharges via R2 into the load. If the load impedance is very high (i.e., similar in magnitude to R2), or if it contains active devices such as LEDs, the load voltage at the instant Q2 turns off may be large enough to bias Q1 on via R4, thereby preventing the circuit from turning off properly. The presence of R5 pulls the OUT (+) terminal down to 0V when Q2 turns off, thus ensuring that Q1 turns off rapidly, and allowing the circuit to revert to its unlatched state in a proper manner.

Provided the transistors are correctly rated, the circuit will work over a wide voltage range and is well suited to driving loads such as relays, solenoids, LEDs, and so on. However, beware that certain DC fans and motors continue to rotate when their drive power is removed. This rotation can generate an EMF large enough to bias Q1 on, thereby preventing the circuit from switching off. You can eliminate this problem by inserting a blocking diode in series with the output, as shown in Figure 1(b). You must also include R5 to ensure Q1 turns off properly.

The complementary circuit outlined in Figure 2 is intended for 'high-side' loads connected to the positive supply rail such as the relay shown in this example.

Figure 2  Complementary circuit intended for high-side loads.

Note that Q1 has been replaced with a PNP transistor, and Q2 is now an N-channel MOSFET. The circuit operates in a similar way to the one described above. Here, R5 acts as a pull-up resistor which pulls the OUT (-) terminal up to +VS when Q2 turns off, thus ensuring that Q1 turns off quickly. As in the previous circuit, R5 is optional and only necessary for the types of load mentioned previously.

Note that in both circuits, the time constant produced by C1-R2 provides for debouncing of the push switch contacts. Normally, a value of 0.25s to 0.5s should be adequate. Smaller time constants may lead to erratic behaviour, whereas a larger time constant increases the waiting time between switch closures necessary to ensure that C1 charges and discharges properly. With C1 = 330nF and R2 = 1MΩ as shown, the time constant is nominally 0.33s. This is usually sufficient to debounce the contacts and to allow the load power to be toggled after a couple of seconds or so.

Both circuits are intended to latch and unlatch in response to brief, momentary switch closures. However, they have each been designed to ensure correct operation even if the push switch is held closed for any length of time. Consider the circuit in Figure 2 when Q2 is on. When the switch is pressed to unlatch the circuit, the gate is pulled down toward 0V (since C1 is uncharged) and the MOSFET switches off, allowing the junction of R1-R2 to rise toward +V_S via R5 and the load impedance. At the same time, Q1 also switches off, such that Q2's gate is pulled to 0V via the series combination of R3 & R4. If the switch is released immediately, C1 will simply charge up toward +V_S via R2. However, if the switch is kept closed, Q2's gate voltage will be defined by the potential divider formed mainly by R2 and R3+R4. If we assume that the OUT (-) terminal is roughly equal to +V_S when the circuit is unlatched, Q2's gate-source voltage is given by: V_GS = (+V_S) × (R3 + R4)/(R2 + R3 + R4) = 0.02(+V_S). Even if +V_S is as high as 30V, the resulting gate-source voltage of around 0.6V will be too low to switch the MOSFET on again. Consequently, both transistors remain off until the switch contacts open.

The circuit in Figure 2 is latched on by momentarily closing the push switch when C1 has charged up to +V_S , which causes OUT (-) to drop to 0V as Q2 immediately turns on, rapidly followed by Q1. A momentary switch closure would allow C1 to discharge to zero via R2 after the contacts open. However, if the switch is held closed, Q2's gate voltage will be defined by the potential divider formed by R2 and R3. Since Q1 is saturated, the junction of R3-R4 at Q1's collector will be pulled up to +V_S, and the junction of R1-R2 will be pulled down to 0V via Q2. Therefore, with the switch held closed, Q2's gate-source voltage is given by: V_GS = (+V_S) × R2/(R2 + R3) = 0.99(+V_S). Consequently, provided the supply voltage is at least equal to Q2's gate-source threshold voltage, both Q2 and Q1 will remain on until the switch contacts open.

Both circuits provide an inexpensive way of deriving a latching function from a momentary switch and, just like a mechanical latching switch, the quiescent (unlatched) power dissipation is zero

Working Principle of Transformer

working principle of transformer: Electrical power transformer is a static device which transforms electrical energy from one circuit to another without any direct electrical connection and with the help of mutual induction between two windings. It transforms power from one circuit to another without changing its frequency but may be in different voltage level. This is a very short and simple definition of transformer, as we will go through this portion of tutorial related to electrical power transformer, we will understand more clearly and deeply "what is transformer ?" and basic theory of transformer. Working Principle of Transformer: The working principle of transformer is very simple. It depends upon Faraday's law of electromagnetic induction. Actually, mutual induction between two or more winding is responsible for transformation action in an electrical transformer. Faraday's Laws of Electromagnetic Induction According to these Faraday's laws, "Rate of change of flux linkage with respect to time is directly proportional to the induced EMF in a conductor or coil". Basic Theory of Transformer Say you have one winding which is supplied by an alternating electrical source. The alternating current through the winding produces a continually changing flux or alternating flux that surrounds the winding. If any other winding is brought nearer to the previous one, obviously some portion of this flux will link with the second. As this flux is continually changing in its amplitude and direction, there must be a change in flux linkage in the second winding or coil. According to Faraday's law of electromagnetic induction, there must be an EMF induced in the second. If the circuit of the later winding is closed, there must be an current flowing through it. This is the simplest form of electrical power transformer and this is the most basic of working principle of transformer. For better understanding, we are trying to repeat the above explanation in a more brief way here. Whenever we apply alternating current to an electric coil, there will be an alternating flux surrounding that coil. Now if we bring another coil near the first one, there will be an alternating flux linkage with that second coil. As the flux is alternating, there will be obviously a rate of change in flux linkage with respect to time in the second coil. Naturally emf will be induced in it as per Faraday's law of electromagnetic induction. This is the most basic concept of the theory of transformer. The winding which takes electrical power from the source, is generally known as primary winding of transformer. Here in our above example it is first winding. The winding which gives the desired output voltage due to mutual induction in the transformer, is commonly known as secondary winding of transformer. Here in our example it is second winding. transformer core winding

The above mentioned form of transformer is theoretically possible but not practically, because in open air very tiny portion of the flux of the first winding will link with second; so the current that flows through the closed circuit of later, will be so small in amount that it will be difficult to measure. The rate of change of flux linkage depends upon the amount of linked flux with the second winding. So, it is desired to be linked to almost all flux of primary winding to the secondary winding. This is effectively and efficiently done by placing one low reluctance path common to both of the winding. This low reluctance path is core of transformer, through which maximum number of flux produced by the primary is passed through and linked with the secondary winding. This is the most basic theory of transformer. Main Constructional Parts of Transformer

The three main parts of a transformer are, Primary Winding of transformer - which produces magnetic flux when it is connected to electrical source. Magnetic Core of transformer - the magnetic flux produced by the primary winding, that will pass through this low reluctance path linked with secondary winding and create a closed magnetic circuit. Secondary Winding of transformer - the flux, produced by primary winding, passes through the core, will link with the secondary winding. This winding also wounds on the same core and gives the desired output of the transformer.

Silicon

Silicon is a chemical element with symbol Si and atomic number 14. It is a tetravalent metalloid, more reactive than germanium, the metalloid directly below it in the table. Controversy about silicon's character dates to its discovery; it was first prepared and characterized in pure form in 1823. In 1808, it was given the name silicium (from Latin: silex, hard stone or flint), with an -ium word-ending to suggest a metal, a name which the element retains in several non-English languages. However, its final English name, first suggested in 1817, reflects the more physically similar elements carbon and boron.

Silicon is the eighth most common element in the universe by mass, but very rarely occurs as the pure free element in nature. It is most widely distributed in dusts, sands, planetoids, and planets as various forms of silicon dioxide (silica) or silicates. Over 90% of the Earth's crust is composed of silicate minerals, making silicon the second most abundant element in the Earth's crust (about 28% by mass) after oxygen.[9]

Most silicon is used commercially without being separated, and indeed often with little processing of compounds from nature. These include direct industrial building-use of clays, silica sand and stone. Silicate goes into Portland cement for mortar and stucco, and when combined with silica sand and gravel, to make concrete. Silicates are also in whiteware ceramics such as porcelain, and in traditional quartz-based soda-lime glass and many other specialty glasses. More modern silicon compounds such as silicon carbide form abrasives and high-strength ceramics. Silicon is the basis of the widely used synthetic polymers called silicones.

Elemental silicon also has a large impact on the modern world economy. Although most free silicon is used in the steel refining, aluminium-casting, and fine chemical industries (often to make fumed silica), the relatively small portion of very highly purified silicon that is used in semiconductor electronics (< 10%) is perhaps even more critical. Because of wide use of silicon in integrated circuits, the basis of most computers, a great deal of modern technology depends on it.

Silicon is an essential element in biology, although only tiny traces of it appear to be required by animals.[10] However, various sea sponges as well as microorganisms like diatoms and radiolaria secrete skeletal structures made of silica. Silica is often deposited in plant tissues, such as in the bark and wood of Chrysobalanaceae and the silica cells and silicified trichomes of Cannabis sativa, horsetails and many grasses.

pnp transistor

The PNP Transistor

The PNP Transistor is the exact opposite to the NPN Transistor device we looked at in the previous tutorial. Basically, in this type of transistor construction the two diodes are reversed with respect to the NPN type giving a Positive-Negative-Positive type of configuration, with the arrow which also defines the Emitter terminal this time pointing inwards in the transistor symbol.

Also, all the polarities for a PNP transistor are reversed which means that it “sinks” current into its Base as opposed to the NPN Transistor which “sources” current through its Base. The main difference between the two types of transistors is that holes are the more important carriers for PNP transistors, whereas electrons are the important carriers for NPN transistors.

Then, PNP transistors use a small base current and a negative base voltage to control a much larger emitter-collector current. In other words for a PNP transistor, the Emitter is more positive with respect to the Base and also with respect to the Collector.

The construction of a “PNP transistor” consists of two P-type semiconductor materials either side of an N-type material as shown below.

A PNP Transistor Configuration

(Note: Arrow defines the emitter and conventional current flow, “in” for a PNP transistor.)

 

The construction and terminal voltages for an NPN transistor are shown above. The PNP Transistor has very similar characteristics to their NPN bipolar cousins, except that the polarities (or biasing) of the current and voltage directions are reversed for any one of the possible three configurations looked at in the first tutorial, Common Base, Common Emitter and Common Collector.

PNP Transistor Connection

The voltage between the Base and Emitter ( VBE ), is now negative at the Base and positive at the Emitter because for a PNP transistor, the Base terminal is always biased negative with respect to the Emitter.

Also the Emitter supply voltage is positive with respect to the Collector ( VCE ). So for a PNP transistor to conduct the Emitter is always more positive with respect to both the Base and the Collector.

The voltage sources are connected to a PNP transistor are as shown. This time the Emitter is connected to the supply voltage VCC with the load resistor, RL which limits the maximum current flowing through the device connected to the Collector terminal. The Base voltage VB which is biased negative with respect to the Emitter and is connected to the Base resistor RB, which again is used to limit the maximum Base current.

To cause the Base current to flow in a PNP transistor the Base needs to be more negative than the Emitter (current must leave the base) by approx 0.7 volts for a silicon device or 0.3 volts for a germanium device with the formulas used to calculate the Base resistor, Base current or Collector current are the same as those used for an equivalent NPN transistor and is given as.

 

We can see that the fundamental differences between a NPN Transistor and a PNP Transistor is the proper biasing of the transistors junctions as the current directions and voltage polarities are always opposite to each other. So for the circuit above: Ic = Ie – Ib as current must leave the Base.

Generally, the PNP transistor can replace NPN transistors in most electronic circuits, the only difference is the polarities of the voltages, and the directions of the current flow. PNP transistors can also be used as switching devices and an example of a PNP transistor switch is shown below.

A PNP Transistor Circuit

 

The Output Characteristics Curves for a PNP transistor look very similar to those for an equivalent NPN transistor except that they are rotated by 180o to take account of the reverse polarity voltages and currents, (the currents flowing out of the Base and Collector in a PNP transistor are negative). The same dynamic load line can be drawn onto the I-V curves to find the PNP transistors operating points.

Transistor Matching

Complementary Transistors

You may think what is the point of having a PNP Transistor, when there are plenty of NPN Transistors available that can be used as an amplifier or solid-state switch?. Well, having two different types of transistors “PNP” and “NPN”, can be a great advantage when designing power amplifier circuits such as the Class B Amplifier.

Class-B amplifiers uses “Complementary” or “Matched Pair” (that is one PNP and one NPN connected together) transistors in its output stage or in reversible H-Bridge motor control circuits were we want to control the flow of current evenly through the motor in both directions.

A pair of corresponding NPN and PNP transistors with near identical characteristics to each other are calledComplementary Transistors for example, a TIP3055 (NPN transistor) and the TIP2955 (PNP transistor) are good examples of complementary or matched pair silicon power transistors. They both have a DC current gain, Beta, ( Ic/Ib ) matched to within 10% and high Collector current of about 15A making them ideal for general motor control or robotic applications.

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Also, class B amplifiers use complementary NPN and PNP in their power output stage design. The NPN transistor conducts for only the positive half of the signal while the PNP transistor conducts for negative half of the signal.

This allows the amplifier to drive the required power through the load loudspeaker in both directions at the stated nominal impedance and power resulting in an output current which is likely to be in the order of several amps shared evenly between the two complementary transistors.

Identifying the PNP Transistor

We saw in the first tutorial of this transistors section, that transistors are basically made up of two Diodes connected together back-to-back.

We can use this analogy to determine whether a transistor is of the PNP type or NPN type by testing its Resistance between the three different leads, Emitter, Base andCollector. By testing each pair of transistor leads in both directions with a multimeter will result in six tests in total with the expected resistance values in Ohm’s given below.

• 1. Emitter-Base Terminals – The Emitter to Base should act like a normal diode and conduct one way only.
• 2. Collector-Base Terminals – The Collector-Base junction should act like a normal diode and conduct one way only.
• 3. Emitter-Collector Terminals – The Emitter-Collector should not conduct in either direction.

Transistor Resistance Values for a PNP Transistor and a NPN Transistor

Between Transistor Terminals		PNP	NPN
Collector	Emitter	RHIGH	RHIGH
Collector	Base	RLOW	RHIGH
Emitter	Collector	RHIGH	RHIGH
Emitter	Base	RLOW	RHIGH
Base	Collector	RHIGH	RLOW
Base	Emitter	RHIGH	RLOW

 

Then we can define a PNP Transistor as being normally “OFF” but a small output current and negative voltage at its Base ( B ) relative to its Emitter ( E ) will turn it “ON” allowing a much large Emitter-Collector current to flow. PNP transistors conduct when Ve is much greater than Vc.

In other words, a Bipolar PNP Transistor will ONLY conduct if both the Base and Collector terminals are negative with respect to the Emitter

In the next tutorial about Bipolar Transistors instead of using the transistor as an amplifying device, we will look at the operation of the transistor in its saturation and cut-off regions when used as a solid-state switch. Bipolar transistor switches are used in many applications to switch a DC current “ON” or “OFF”, from LED’s which require only a few milliamps of switching current at low DC voltages, or motors and relays which may require higher currents at higher voltages.

current divider rule

current divider rule

In electronics, a current divider is a simple linear circuit that produces an output current (IX) that is a fraction of its input current (IT). Current division refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.

The formula describing a current divider is similar in form to that for the voltage divider. However, the ratio describing current division places the impedance of the considered branches in the denominator, unlike voltage division where the considered impedance is in the numerator. This is because in current dividers, total energy expended is minimized, resulting in currents that go through paths of least impedance, therefore the inverse relationship with impedance. On the other hand, voltage divider is used to satisfy Kirchhoff's Voltage Law. The voltage around a loop must sum up to zero, so the voltage drops must be divided evenly in a direct relationship with the impedance.

To be specific, if two or more impedances are in parallel, the current that enters the combination will be split between them in inverse proportion to their impedances (according to Ohm's law). It also follows that if the impedances have the same value the current is split equally.

Current divider:

A general formula for the current IX in a resistor RX that is in parallel with a combination of other resistors of total resistance RT is (see Figure 1):

I_X = \frac{R_T}{(R_X)+(R_T)}I_T \ 

where IT is the total current entering the combined network of RX in parallel with RT. Notice that when RT is composed of a parallel combination of resistors, say R1, R2, ... etc., then the reciprocal of each resistor must be added to find the total resistance RT:

 \frac {1}{R_T} = \frac {1} {R_1} + \frac {1} {R_2} + \frac {1}{R_3} + ... \ . 

General case[edit]

Although the resistive divider is most common, the current divider may be made of frequency dependent impedances. In the general case the current IX is given by:

I_X = \frac{Z_T} {Z_X+Z_T}I_T \ ,[1]

Using Admittance[edit]

Instead of using impedances, the current divider rule can be applied just like the voltage divider rule if admittance (the inverse of impedance) is used.

I_X = \frac{Y_X} {Y_{Total}}I_T

Take care to note that YTotal is a straightforward addition, not the sum of the inverses inverted (as you would do for a standard parallel resistive network). For Figure 1, the current IX would be

I_X = \frac{Y_X} {Y_{Total}}I_T = \frac{\frac{1}{R_X}} {\frac{1}{R_X} + \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}}I_T

physics all equation

Mechanics

velocity

v̅ =  Δs Δt

v =  ds dt

acceleration

a̅ =  Δv Δt

a =  dv dt

equations of motion

v = v0 + at x = x0 + v0t + ½at2 v2 = v02 + 2a(x − x0) v̅ = ½(v + v0)

newton's 2nd law

∑ F = m a

∑ F =  dp dt

weight

W = m g

dry friction

ƒ = μN

centrip. accel.

ac =  v2 r

ac = − ω2 r

momentum

p = m v

impulse

J = F̅ Δt

J =  ⌠ ⌡ F dt

impulse-momentum

F̅ Δt = m Δv

⌠ ⌡ F dt = Δp

work

W = F̅Δs cos θ

W =  ⌠ ⌡ F · ds

work-energy

F̅Δs cos θ = ΔE

⌠ ⌡ F · ds = ΔE

kinetic energy

K = ½mv2

general p.e.

ΔU = −  ⌠ ⌡ F · ds

F = − ∇U

gravitational p.e.

ΔUg = mgΔh

efficiency

ℰ =  Wout Ein

power
P̅ =  ΔW Δt	P̅ = F̅v cos θ
P =  dW dt	P = F · v

angular velocity

ω̅ =  Δθ Δt

ω =  dθ dt

v = ω × r

angular acceleration

α̅ =  Δω Δt

α =  dω dt

a = α × r − ω2 r

equations of rotation

ω = ω0 + αt θ = θ0 + ω0t + ½αt2 ω2 = ω02 + 2α(θ − θ0) ω̅ = ½(ω + ω0)

2nd law for rotation

∑ τ = I α

∑ τ =  dL dt

torque

τ = rF sin θ

τ = r × F

moment of inertia

I = ∑ mr2

I =  ⌠ ⌡  r2 dm

rotational work

W = τ̅Δθ

W =  ⌠ ⌡  τ · dθ

rotational power

P = τω cos θ

P = τ · ω

rotational k.e.

K = ½Iω2

angular momentum

L = mrv sin θ L = r × p L = I ω

universal gravitation

Fg = −  Gm1m2  r̂ r2

gravitational field

g = −  Gm  r̂ r2

gravitational p.e.

Ug = −  Gm1m2 r

gravitational potential

Vg = −  Gm r

orbital speed

v = √  Gm r

escape speed

v = √  2Gm r

hooke's law

F = − k Δx

elastic p.e.

Us = ½kΔx2

s.h.o.

T = 2π √  m k

simple pendulum

T = 2π √  ℓ g

frequency

ƒ =  1 T

angular frequency

ω = 2πƒ

density

ρ =  m V

pressure

P =  F A

pressure in a fluid

P = P0 + ρgh

buoyancy

B = ρgVdisplaced

mass flow rate

I =  m t

volume flow rate

φ =  V t

mass continuity

ρ1A1v1 = ρ2A2v2

volume continuity

A1v1 = A2v2

bernoulli's equation
P1 + ρgy1 + ½ρv12 =	P2 + ρgy2 + ½ρv22

dynamic viscosity

η =  F̅/A Δvx/Δz

η =  F/A dvx/dz

kinematic viscosity

ν =  η ρ

aerodynamic drag

R = ½ρCAv2

mach number

Ma =  v c

reynolds number

Re =  ρvD η

froude number

Fr =  v √gℓ

young's modulus

F  = E  Δℓ A ℓ0

shear modulus

F  = G  Δx A y

bulk modulus

F  = K  ΔV A V0

surface tension

γ =  F ℓ

Thermal Physics

solid expansion

Δℓ = αℓ0ΔT ΔA = 2αA0ΔT ΔV = 3αV0ΔT

liquid expansion

ΔV = βV0ΔT

sensible heat

Q = mcΔT

latent heat

Q = mL

ideal gas law

PV = nRT

molecular constants

nR =Nk

maxwell-boltzmann

−  mv2 p(v) =  4v2 ⎛ ⎝ m ⎞3/2 ⎠ e 2kT √π 2kT

molecular k.e.

⟨K⟩ =  3  kT 2

molecular	speeds
vp = √  2kT m ⟨v⟩ = √8kTπm vrms = √  3kT m

heat flow rate

Φ̅ =  ΔQ Δt

Φ =  dQ dt

thermal conduction

Φ =  kAΔT ℓ

stefan-boltzmann law

Φ = εσA(T4 − T04)

wien displacement law

λmax =  b T

internal energy

ΔU = 3⁄2nRΔT

thermodynamic work

W = − ⌠ ⌡ P dV

1st law of thermo.

ΔU = Q + W

entropy

ΔS =  ΔQ T

S = k log w

efficiency

ℰreal = 1 −  QC QH

ℰideal = 1 −  TC TH

c.o.p.

COPreal =  QC QH − QC

COPideal =  TC TH − TC

Waves & Optics

periodic waves

v = ƒλ

frequency

ƒ =  1 T

beat frequency

fbeat = fhigh − flow

intensity

I =  ⟨P⟩ A

intensity level

LI = 10 log ⎛ ⎝ I ⎞ ⎠ I0

pressure level

LP = 20 log ⎛ ⎝ ∆Pmax ⎞ ⎠ ∆P0

interference fringes

nλ = d sin θ

nλ  ≈  x d L

index of refraction

n =  c v

snell's law

n1 sin θ1 = n2 sin θ2

critical angle

sin θc =  n2 n1

image location

1  =  1  +  1 ƒ do di

image size

M =  hi  =  di ho do

spherical mirrors

ƒ ≈  r 2

Electricity & Magnetism

coulomb's law

F = k  q1q2 r2

electric field, def.

E =  FE q

electric field, around charges
E = k ∑  q  r̂ r2	E = k  ⌠ ⌡ dq  r̂ r2

field and potential

E̅ = −  ∆V d

E = − ∇V

electric potential, def.

ΔV =  ΔUE q

electric potential, around charges
V = k ∑  q r	V = k  ⌠ ⌡ dq r

capacitance

C =  Q V

plate capacitor

C =  κε0A d

cylindrical capacitor

C =  2πκε0ℓ ln (b/a)

spherical capacitor

C =  4πκε0 (1/a) − (1/b)

capacitive p.e.

U =  1  CV2 =  1 Q2  =  1  QV 2 2 C 2

electric current

I̅ =  Δq Δt

I =  dq dt

ohm's law

V = IR E = ρ J J = σE

resitivity-conductivity

ρ =  1 σ

electric resistance

R =  ρℓ A

electric power

P = VI = I2R =  V2 R

resistors in series

Rs = ∑ Ri

resistors in parallel

1  = ∑  1 Rp Ri

capacitors in series

1  = ∑  1 Cs Ci

capacitors in parallel

Cp = ∑ Ci

magnetic force, charge

FB = qvB sin θ

FB = q v × B

magnetic force, current

FB = IℓB sin θ

dFB = I dℓ × B

biot-savart law

B =  μ0I ⌠ ⌡ ds × r̂ 4π r2

solenoid

B = µ0nI

straight wire

B =  μ0I 2πr

parallel wires

FB  =  μ0 I1I2 ℓ 2π r

electric flux

ΦE = EA cos θ

ΦE =  ⌠ ⌡ E · dA

magnetic flux

ΦB = BA cos θ

ΦB =  ⌠ ⌡ B · dA

motional emf

ℰ = Bℓv

induced emf

ℰ̅ = −  ΔΦB Δt

ℰ = −  dΦB dt

gauss's law

∯  E · dA =  Q ε0

∇ · E =  ρ ε0

no one's law

∯ B · dA = 0

∇ · B = 0

faraday's law

∮E · ds = −  dΦB dt

∇ × E = −  ∂B ∂t

ampere's law

∮B · ds = μ0ε0  dΦE  + μ0I dt

∇ × B = μ0ε0  ∂E  + μ0 J ∂t

Modern Physics

time dilation

t' =  t √(1 − v2/c2)

length contraction

ℓ' = ℓ √(1 − v2/c2)

relativistic mass

m' =  m √(1 − v2/c2)

relative velocity

u' =  u + v 1 + uv/c2

relativistic energy

E =  mc2 √(1 − v2/c2)

relativistic momentum

p =  mv √(1 − v2/c2)

energy-momentum

E2 = p2c2 + m02c4

mass-energy

E = mc2

relativistic doppler effect

λ  =  ƒ0  = √  ⎛ ⎝ 1 + v/c ⎞ ⎠ λ0 ƒ 1 − v/c

photon energy

E = hf

photoelectric effect

Kmax = E − ϕ = h(ƒ − ƒ0)

photon momentum

p =  h λ

schroedinger's	equation
iℏ  ∂  Ψ(r,t) = −  ℏ2  ∇2Ψ(r,t) + V(r)Ψ(r,t) ∂t 2m
Eψ(r) = −  ℏ2  ∇2ψ(r) + V(r)ψ(r) 2m

uncertainty principle

Δpx Δx ≥  ℏ  2

ΔE Δt ≥  ℏ  2

rydberg equation

1  = −R∞  ⎛ ⎝ 1  −  1 ⎞ ⎠ λ n2 n02

half life

N = N02−t/τ