hydraulic pump

Hydraulic pump Rexroth A4VSO250

Gearpump with external teeth

Gearpump with internal teeth

A gerotor (image does not show intake or exhaust)

Fixed displacement vane pump

Principle of screw pump

Axial piston pump, swashplate principle

Radial piston pump

Hydraulic pumps are used in hydraulic drive systems and can be hydrostatic or hydrodynamic.
Hydrostatic pumps are positive displacement pumps while hydrodynamic pumps can be fixed displacement pumps, in which the displacement (flow through the pump per rotation of the pump) cannot be adjusted, or variable displacement pumps, which have a more complicated construction that allows the displacement to be adjusted.
Gear pumps (with external teeth) (fixed displacement) are simple and economical pumps. The swept volume or displacement of gear pumps for hydraulics will be between about 1 cm³ (0.001 litre) and 200 cm³ (0.2 litre). These pumps create pressure through the meshing of the gear teeth, which forces fluid around the gears to pressurize the outlet side. For lubrication, the gear pump uses a small amount of oil from the pressurized side of the gears, bleeds this through the (typically) hydrodynamic bearings, and vents the same oil either to the low pressure side of the gears, or through a dedicated drain port on the pump housing. Some gear pumps can be quite noisy, compared to other types, but modern gear pumps are highly reliable and much quieter than older models. This is in part due to designs incorporating split gears, helical gear teeth and higher precision/quality tooth profiles that mesh and unmesh more smoothly, reducing pressure ripple and related detrimental problems. Another positive attribute of the gear pump, is that catastrophic breakdown is a lot less common than in most other types of hydraulic pumps. This is because the gears gradually wear down the housing and/or main bushings, reducing the volumetric efficiency of the pump gradually until it is all but useless. This often happens long before wear causes the unit to seize or break down.
Total head and flow are the main criteria that are used to compare one pump with another or to select a centrifugal pump for an application. Total head is related to the discharge pressure of the pump. Why can't we just use discharge pressure? Pressure is a familiar concept, we are familiar with it in our daily lives. For example, fire extinguishers are pressurized at 60 psig (413 kPa), we put 35 psig (241 kPa) air pressure in our bicycle and car tires.For good reasons, pump manufacturers do not use discharge pressure as a criteria for pump selection. One of the reasons is that they do not know how you will use the pump. They do not know what flow rate you require and the flow rate of a centrifugal pump is not fixed. The discharge pressure depends on the pressure available on the suction side of the pump. If the source of water for the pump is below or above the pump suction, for the same flow rate you will get a different discharge pressure. Therefore to eliminate this problem, it is preferable to use the difference in pressure between the inlet and outlet of the pump.

The manufacturers have taken this a step further, the amount of pressure that a pump can produce will depend on the density of the fluid, for a salt water solution which is denser than pure water, the pressure will be higher for the same flow rate. Once again, the manufacturer doesn't know what type of fluid is in your system, so that a criteria that does not depend on density is very useful. There is such a criteria and it is called TOTAL HEAD, and it is defined as the difference in head between the inlet and outlet of the pump.

You can measure the discharge head by attaching a tube to the discharge side of the pump and measuring the height of the liquid in the tube with respect to the suction of the pump. The tube will have to be quite high for a typical domestic pump. If the discharge pressure is 40 psi the tube would have to be 92 feet high. This is not a practical method but it helps explain how head relates to total head and how head relates to pressure. You do the same to measure the suction head. The difference between the two is the total head of the pump.

Figure 25

The fluid in the measuring tube of the discharge or suction side of the pump will rise to the same height for all fluids regardless of the density. This is a rather astonishing statement, here's why. The pump doesn’t know anything about head, head is a concept we use to make our life easier. The pump produces pressure and the difference in pressure across the pump is the amount of pressure energy available to the system. If the fluid is dense, such as a salt solution for example, more pressure will be produced at the pump discharge than if the fluid were pure water. Compare two tanks with the same cylindrical shape, the same volume and liquid level, the tank with the denser fluid will have a higher pressure at the bottom. But the static head of the fluid surface with respect to the bottom is the same. Total head behaves the same way as static head, even if the fluid is denser the total head as compared to a less dense fluid such as pure water will be the same. This is a surprising fact.

Hydraulic Specialists, Inc. known as HSI was founded in 1979. With over 200 years of combined hydraulic experience and is proudly celebrating its 32nd year in business.  HSI remanufactures hydraulic components such as pumps, motors, valves, presses and power units.  We handle all lines of OEM's such as Oilgear, Vickers, Rexroth, Racine, Denison, HPM, Eaton, Hagglunds, Sunstrand to name a few.  We offer our customers an alternative to going back to the OEM which tend to have long lead times and high pricing.  We remanufacture current units but our niche is in the obsolete units. HSI also has an "Open Door" Policy... If you would like to come and tour our facilities or watch your unit being evaluated, assembled or tested we invite you to do so....

Hydraulic pump sits at idle dumping over a dump valve to unload the pump. Clamp command is given and the dump valve is energized to prevent oil flowing directly to the tank. Once the clamps have been completed and pressure is at the maximum, the pressure switch actuates and shuts off the clamp solenoid and dump valve. Therefore, unloading the oil from the pump to tank. The clamps maintain pressure because of a P.O. check installed in the valve stack. If the P.O. check leaks or there is excessive leakage the hydraulic pump can be set up to perform a recharge though this is often not required. If a recharge is required the PLC will control the logic and dual setting pressure switches are needed for safety check. The first pressure switch set point will control the recharge point and the second will be the pressure at which will fault the machine if reached during the machining cycle. If the system recharges more then lets say 3 times in a minute, the system alarms out due to excessive cycling which can all be set up in the PLC. Unclamp and clamp circuits can both work under this concept though a recharge is not often needed for the unclamp. Single actuating and double actuating fixtures can be different to save costs, but this type of design setup should work for just about any fixture you would ever want to put on it.

Variable flow Compensator hydraulic systems

This type of system is generally a higher flow pump and much more costly. This system generally requires an oil cooler and flow control valves in the valve stacks, due to higher flow rates. These type of systems are also generally lower pressure systems, typically 2000 PSI or less.
Basic operation
These pumps have basically pistons attached to a plate. When there is a need for flow, the plate shifts increasing the stroke of the pistons and therefore it outputs more flow. This system always runs at full system pressure. When there is no oil flow there is no load because the plate with the pistons is vertical and there is no load. A relief valve is provided for safety but should never see any oil going over it in normal operation if set correctly. This type of system works well because it keeps constant pressure on the system and does not rely on P.O. checks holding pressure at the clamps. With this system you should still have P.O. checks incase the pump shuts off during the machining cycle.

Air over oil Pump systems

These systems are the cheapest of all systems. Depending on the brand and design you may have problems. They are very simple low volume/flow systems. They generally work well when using 2-4 valve stacks that are sequenced and do not require high volume of oil.
Basic operation
Air is supplied to the system. Air will cycle an intensifier type cylinder and push oil from a reservoir into the hydraulic system. Pressure is adjusted by adjusting the incoming air flow.

High Pressure coolant systems buying considerations, efficiency, and coolant system applications

High pressure coolant can be very beneficial for almost any job, some much more then others. The benefits of the high pressure coolant usually out weigh the initial cost when it comes to tooling savings, cycle time and finish. Tooling will last longer, speeds will be increased especially when drilling or boring. High pressure coolant will keep the part from heating up by transferring the heat into the coolant, away from the tool and part. Therefore, holding tighter tolerances, keeping your chips consistent and heat free. Learn to turn, mill, and drill faster with high pressure coolant.

Benefits of high pressure coolant

• Often drastic cycle time reduction (20-70%)
  
  
• Increased speeds in hole production
  
  
• Eliminate heat related failure of insert
  
  
• Reduced chip welding and "built-up edge" in aluminum machining
  
  
• Better chip control in Low Carbon Steels
  
  
• Improved quality and speed of Threaded Holes
  
  
• Reduce or eliminate work hardening from peck drilling
  
  
• Improved tapping in tough materials
  
  
• Increased life of expensive Custom Tooling
  
  
• Improved Auto-loading by eliminating chip problems
  
  
• Increased tool life in Abrasive Materials
  
  
  

  The basic hydraulic pump system

  The basic unit will contain a .5-1.0 GPM pump which will be sufficient for the majority of fixtures unless you are using larger volume of clamps or multiple operations at once. The size of the motor depends on the pressure needed - 1,3,5(HP) horse power. The pump reservoir size is usually a 1, 3 or 5 gallon. The pump system should also contain an oil low level and HI-temperature switch for protection. Pump specs for volume can be calculated by figuring out the total volume of the clamps and figuring the speed at which the chamber fills up assuming .5 gal per minute.
  The hydraulic valving for a double acting fixture (can also be used for single acting)
  
• Directional control valves- Poppet (Least expensive,preferred for low flow small systems) or spool type (can handle more contaminants in oil)
  
• P.O. check valves on A+B
  
• Tapping Block for pressure switches on A and B
  
• Pressure switches on A+B (Sometimes only A port I'll explain later)
  
• Dump valve
  
• Pressure reducing valve (not usually needed)
  
• Flow controls (not usually needed- used usually with small volume work supports)
  

Hydraulic pressure switch considerations

Hydraulic pressure switches may not be necessary on B port if it is a single actuating fixture. If it is a single actuating fixture with a robot load, I would recommend them to confirm that there is no residual pressure left on the clamp side. Then causing the robot to collide with the clamps.
Also, if it is a manual operation only the PLC may run the unclamp solenoid for a specified amount of time instead of waiting for the unclamp pressure switch to be made. However; if there is any safety concerns during the unclamp cycle where the operator can be injured, this may not be a good idea.

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Hydro electricity

The Gordon Dam in Tasmania is a large hydro facility, with an installed capacity of 430 MW.

Renewable energy
Biofuel Biomass Geothermal Hydroelectricity Solar energy Tidal power Wave power Wind power

Hydroelectricity is the term referring to electricity generated by hydropower; the production of electrical power through the use of the gravitational force of falling or flowing water. It is the most widely used form of renewable energy. Once a hydroelectric complex is constructed, the project produces no direct waste, and has a considerably lower output level of the greenhouse gas carbon dioxide (CO₂) than fossil fuel powered energy plants. Worldwide, an installed capacity of 777 GWe supplied 2998 TWh of hydroelectricity in 2006.^[1] This was approximately 20% of the world's electricity, and accounted for about 88% of electricity from renewable sources

Hydropower has been used since ancient times to grind flour and perform other tasks. In the mid-1770s, French engineer Bernard Forest de Bélidor published Architecture Hydraulique which described vertical- and horizontal-axis hydraulic machines. By the late 19th century, the electrical generator was developed and could now be coupled with hydraulics. The growing demand for the Industrial Revolution would drive development as well. In 1878 the world's first hydroelectric power scheme was developed at Cragside in Northumberland, England by William George Armstrong. It was used to power a single light bulb in his art gallery. The old Schoelkopf Power Station No. 1 near Niagara Falls in the U.S. side began to produce electricity in 1881. The first Edison hydroelectric power plant, the Vulcan Street Plant, began operating September 30, 1882, in Appleton, Wisconsin, with an output of about 12.5 kilowatts. By 1886 there were 45 hydroelectric power plants in the U.S. and Canada. By 1889 there were 200 in the U.S. alone.

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Hydroelectricity is electricity that is made by the movement of water. It is usually made with dams that block a river or collect water that is pumped there. When the water is "let go" the huge pressure behind the dam forces the water down shafts that lead to a turbine, this causes the turbine to turn, and electricity is produced.

The energy of falling water has been used by humans for thousands of year

Advantages of hydroelectricity:

The way the electricity is produced does not harm the environment as much as fossil fuels like oil or coal. Hydroelectricity is immense and safe and produces no waste. Hydroelectricity can be made very quickly. This makes it useful for times when demand is high. Water that has been stored in a dam can be "let go" when needed, so the energy needed can be made quickly. Also hydroelectricity can not run out as long as there is a good water supply. Once the dam is built the electricity is free, no waste or pollution produced and electricity can be generated constantly also there is then a lot of extra energy to save and use.

Disadvantages of hydroelectricity

The building of large dams to hold the water can damage the environment. In 1983 Australian government stopped the Tasmanian state government from building a dam on the Gordon River in Tasmania after a huge public protest. The dam would have flooded the beautiful Franklin River. The Three Gorges Dam in China will be the world's largest hydroelectricity project. The dam has flooded a huge area, meaning that 1.2 million people have had to be moved. Scientists are concerned about many problems with the dam, such as pollution, silt, and the danger of the dam wall breaking.




Hydro-electricity uses the energy of running water to make electrical energy. Hydro-electric stations are built where there is running water.
Most hydro-electric stations are located in dams where water is stored. But hydro-electric stations can also be built on rivers and near waterfalls.
The flow of the water from the dam can be controlled by opening and closing gates or pipes to the dam. The dam wall also creates a high water level and this increases the pressure in the pipes which takes the water to the turbine
To make hydro-electric power, water from rain or melting snow is collected and stored in a lake or a dam. A large pipe carries the water from the dam or lake to the turbine. The pressure of the water pushes against the blades of the turbine and make it spin. The rotating turbine is connected to a generator which makes the electrical energy or electricity.
The electricity then travels through transformers and transmission lines to homes and factories.
Hydro-electricity is a renewable source of energy.
Hydro-electricity is clean power. It produces no waste, and doesn't produce any greenhouse gases.
The most famous hydro-electric power station in Australia is the Snowy Mountains Hydro-Electric Scheme. Go here to read about it: 
Hydro comes from a Greek word meaning water.

electric car

The REVAi/G-Wiz i electric car charging from an on-street station in London.

An electric car is an automobile which is propelled by electric motor(s), using electrical energy stored in batteries or another energy storage device. Electric cars were popular in the late-19th century and early 20th century, until advances in internal combustion engine technology and mass production of cheaper gasoline vehicles led to a decline in the use of electric drive vehicle. The energy crises of the 1970s and 80s brought a short lived interest in electric cars, but in the mid 2000s took place a renewed interest in the production of electric cars due mainly to concerns about rapidly increasing oil prices and the need to curb greenhouse gas emissions. As of early 2011 series production models available in some countries include the Tesla Roadster, REVAi, Buddy, Mitsubishi i MiEV, Th!nk City, and Nissan Leaf.

Electric cars have several potential benefits as compared to conventional internal combustion automobiles that include a significant reduction of urban air pollution as they do not emit harmful tailpipe pollutants from the onboard source of power at the point of operation (zero tail pipe emissions); reduced greenhouse gas emissions from the onboard source of power depending on the fuel and technology used for electricity generation to charge the batteries; and less dependence on foreign oil, which for the United States, other developed and emerging countries is cause of concerns about their vulnerability to price shocks and supply disruption. Also for many developing countries, and particularly for the poorest in Africa, high oil prices have an adverse impact on their balance of payments, hindering their economic growth.

For electric vehicles other than battery powered road vehicles, see electric vehicle. For passenger electric vehicles, see Electric car.

Citroën Berlingo Electrique vans of the ELCIDIS goods distribution service in La Rochelle, France

A battery electric vehicle, or BEV, is a type of electric vehicle (EV) that uses chemical energy stored in rechargeable battery packs. BEVs use electric motors and motor controllers instead of, or in addition to, internal combustion engines (ICEs) for propulsion.
A battery-only electric vehicle or all-electric vehicle derives all its power from its battery packs and thus has no internal combustion engine, fuel cell, or fuel tank. Sometimes, all-electric vehicles are referred to as BEVs (although a plug-in hybrid is also a battery electric vehicle).

electric charge

he coulomb, symbol C, is the SI unit of electric charge, and is defined in terms of the ampere: 1 coulomb is the amount of electric charge (quantity of electricity) carried by a current of 1 ampere flowing for 1 second. It is also about 6.241506×1018 times the charge of an electron. It is named after Charles-Augustin de Coulomb (1736-1806).
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two negatively charged objects. Positively charged objects and negatively charged objects experience an attractive force. The SI unit of electric charge is the coulomb (C), although in electrical engineering it is also common to use the ampere-hour (Ah). The study of how charged substances interact is classical electrodynamics, which is accurate insofar as quantum effects can be ignored.
The SI unit of quantity of electric charge is the coulomb, which is equivalent to about 6.242×10¹⁸ e (e is the charge of a proton). Hence, the charge of an electron is approximately −1.602×10⁻¹⁹ C. The coulomb is defined as the quantity of charge that has passed through the cross section of an electrical conductor carrying one ampere within one second. The symbol Q is often used to denote a quantity of electricity or charge. The quantity of electric charge can be directly measured with an electrometer, or indirectly measured with a ballistic galvanometer.
After finding the quantized character of charge, in 1891 George Stoney proposed the unit 'electron' for this fundamental unit of electrical charge. This was before the discovery of the particle by J.J. Thomson in 1897. The unit is today treated as nameless, referred to as "elementary charge", "fundamental unit of charge", or simply as "e". A measure of charge should be a multiple of the elementary charge e, even if at large scales, charge seems to behave as a real quantity. In some contexts it is meaningful to speak of fractions of a charge; for example in the charging of a capacitor, or in the fractional quantum Hall effect.

Charge

• there are two kinds of charge, positive and negative
• like charges repel, unlike charges attract
• positive charge comes from having more protons than electrons; negative charge comes from having more electrons than protons
• charge is quantized, meaning that charge comes in integer multiples of the elementary charge e
• charge is conserved

Probably everyone is familiar with the first three concepts, but what does it mean for charge to be quantized? Charge comes in multiples of an indivisible unit of charge, represented by the letter e. In other words, charge comes in multiples of the charge on the electron or the proton. These things have the same size charge, but the sign is different. A proton has a charge of +e, while an electron has a charge of -e.
Electrons and protons are not the only things that carry charge. Other particles (positrons, for example) also carry charge in multiples of the electronic charge. Those are not going to be discussed, for the most part, in this course, however.
Putting "charge is quantized" in terms of an equation, we say:
q = n e
q is the symbol used to represent charge, while n is a positive or negative integer, and e is the electronic charge, 1.60 x 10^-19 Coulombs.

The Law of Conservation of Charge

The Law of conservation of charge states that the net charge of an isolated system remains constant.
If a system starts out with an equal number of positive and negative charges, there¹s nothing we can do to create an excess of one kind of charge in that system unless we bring in charge from outside the system (or remove some charge from the system). Likewise, if something starts out with a certain net charge, say +100 e, it will always have +100 e unless it is allowed to interact with something external to it.
Charge can be created and destroyed, but only in positive-negative pairs.
Table of elementary particle masses and charges:

Electrostatic charging

Forces between two electrically-charged objects can be extremely large. Most things are electrically neutral; they have equal amounts of positive and negative charge. If this wasn¹t the case, the world we live in would be a much stranger place. We also have a lot of control over how things get charged. This is because we can choose the appropriate material to use in a given situation.
Metals are good conductors of electric charge, while plastics, wood, and rubber are not. They¹re called insulators. Charge does not flow nearly as easily through insulators as it does through conductors, which is why wires you plug into a wall socket are covered with a protective rubber coating. Charge flows along the wire, but not through the coating to you.
Materials are divided into three categories, depending on how easily they will allow charge (i.e., electrons) to flow along them. These are:

• conductors - metals, for example
• semi-conductors - silicon is a good example
• insulators - rubber, wood, plastic for example

Most materials are either conductors or insulators. The difference between them is that in conductors, the outermost electrons in the atoms are so loosely bound to their atoms that they¹re free to travel around. In insulators, on the other hand, the electrons are much more tightly bound to the atoms, and are not free to flow. Semi-conductors are a very useful intermediate class, not as conductive as metals but considerably more conductive than insulators. By adding certain impurities to semi-conductors in the appropriate concentrations the conductivity can be well-controlled.
There are three ways that objects can be given a net charge. These are:

1. Charging by friction - this is useful for charging insulators. If you rub one material with another (say, a plastic ruler with a piece of paper towel), electrons have a tendency to be transferred from one material to the other. For example, rubbing glass with silk or saran wrap generally leaves the glass with a positive charge; rubbing PVC rod with fur generally gives the rod a negative charge.
   
2. Charging by conduction - useful for charging metals and other conductors. If a charged object touches a conductor, some charge will be transferred between the object and the conductor, charging the conductor with the same sign as the charge on the object.
   
3. Charging by induction - also useful for charging metals and other conductors. Again, a charged object is used, but this time it is only brought close to the conductor, and does not touch it. If the conductor is connected to ground (ground is basically anything neutral that can give up electrons to, or take electrons from, an object), electrons will either flow on to it or away from it. When the ground connection is removed , the conductor will have a charge opposite in sign to that of the charged object. 

The parallel between gravity and electrostatics

An electric field describes how an electric charge affects the region around it. It's a powerful concept, because it allows you to determine ahead of time how a charge will be affected if it is brought into the region. Many people have trouble with the concept of a field, though, because it's something that's hard to get a real feel for. The fact is, though, that you're already familiar with a field. We've talked about gravity, and we've even used a gravitational field; we just didn't call it a field.
When talking about gravity, we got into the (probably bad) habit of calling g "the acceleration due to gravity". It's more accurate to call g the gravitational field produced by the Earth at the surface of the Earth. If you understand gravity you can understand electric forces and fields because the equations that govern both have the same form.
The gravitational force between two masses (m and M) separated by a distance r is given by Newton's law of universal gravitation:

A similar equation applies to the force between two charges (q and Q) separated by a distance r:

The force equations are similar, so the behavior of interacting masses is similar to that of interacting charges, and similar analysis methods can be used. The main difference is that gravitational forces are always attractive, while electrostatic forces can be attractive or repulsive. The charge (q or Q) plays the same role in the electrostatic case that the mass (m or M) plays in the case of the gravity.
A good example of a question involving two interacting masses is a projectile motion problem, where there is one mass m, the projectile, interacting with a much larger mass M, the Earth. If we throw the projectile (at some random launch angle) off a 40-meter-high cliff, the force on the projectile is given by:
F = mg
This is the same equation as the more complicated equation above, with G, M, and the radius of the Earth, squared, incorporated into g, the gravitational field.
So, you've seen a field before, in the form of g. Electric fields operate in a similar way. An equivalent electrostatics problem is to launch a charge q (again, at some random angle) into a uniform electric field E, as we did for m in the Earth's gravitational field g. The force on the charge is given by F = qE, the same way the force on the mass m is given by F = mg.
We can extend the parallel between gravity and electrostatics to energy, but we'll deal with that later. The bottom line is that if you can do projectile motion questions using gravity, you should be able to do them using electrostatics. In some cases, you¹ll need to apply both; in other cases one force will be so much larger than the other that you can ignore one (generally if you can ignore one, it'll be the gravitational force).

ampere's law

The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source. Ampere's Law states that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

In the electric case, the relation of field to source is quantified in Gauss's Law which is a very powerful tool for calculating electric fields.

Ampere's Law Applications	Index Ampere's law Currents as magnetic field sources

The magnetic field at a distance r from a very long straight wire, carrying a steady current I, has a magnitude equal to

(31.)

and a direction perpendicular to r and I. The path integral along a circle centered around the wire (see Figure 31.1) is equal to

(31.2)

Here we have used the fact that the magnetic field is tangential at any point on the circular integration path.





Figure 31.1. Magnetic field generated by current. Any arbitrary path can be thought of as a collection of radial segments (r changes and [theta] remains constant) and circular segments ([theta] changes and r remains constant). For the radial segments the magnetic field will be perpendicular to the displacement and the scaler product between the magnetic field and the displacement is zero. Consider now a small circular segment of a trajectory around the wire (see Figure 31.2). The path integral along this circular segment is equal to
(31.3)






Figure 31.2. Path integral along a small circular path. Equation (31.3) shows that the contribution of this circular segment to the total path integral is independent of the distance r and only depends on the change in the angle [Delta][theta]. For a closed path, the total change in angle will be 2[pi], and eq.(31.3) can be rewritten as
(31.4)

This expression is Ampere's Law:

" The integral of B around any closed mathematical path equals u₀ times the current intercepted by the area spanning the path "

Example: Problem 31.5

Six parallel aluminum wires of small, but finite, radius lie in the same plane. The wires are separated by equal distances d, and they carry equal currents I in the same direction. Find the magnetic field at the center of the first wire. Assume that the currents in each wire is uniformly distributed over its cross section.

A schematic layout of the problem is shown in Figure 31.3. The magnetic field generated by a single wire is equal to

(31.5)

where r is the distance from the center of the wire. Equation (31.5) is correct for all points outside the wire, and can therefore be used to determine the magnetic field generated by wire 2, 3, 4, 5, and 6. The field at the center of wire 1, due to the current flowing in wire 1, can be determined using Ampere's law, and is equal to zero. The total magnetic field at the center of wire 1 can be found by vector addition of the contributions of each of the six wires. Since the direction of each of these contributions is the same, the total magnetic field at the center of wire 1 is equal to

(31.6)






Figure 31.3. Problem 31.5

31.2. The solenoid

A solenoid is a device used to generate a homogeneous magnetic field. It can be made of a thin conducting wire wound in a tight helical coil of many turns. The magnetic field inside a solenoid can be determined by summing the magnetic fields generated by N individual rings (where N is the number of turns of the solenoid). We will limit our discussion of the magnetic field generated by a solenoid to that generated by an ideal solenoid which is infinitely long, and has very tightly wound coils.
The ideal solenoid has translational and rotational symmetry. However, since magnetic field lines must form closed loops, the magnetic field can not be directed along a radial direction (otherwise field lines would be created or destroyed on the central axis of the solenoid). Therefore we conclude that the field lines in a solenoid must be parallel to the solenoid axis. The magnitude of the magnetic field can be obtained by applying Ampere's law.





Figure 31.4. The ideal solenoid. Consider the integration path shown in Figure 31.4. The path integral of the magnetic field around this integration path is equal to
(31.7)

where L is the horizontal length of the integration path. The current enclosed by the integration path is equal to N ^. I₀ where N is the number of turns enclosed by the integration path and I₀ is the current in each turn of the solenoid. Using Ampere's law we conclude that

(31.8)

or

(31.9)

where n is the number of turns of the solenoid per unit length. Equation (31.9) shows that the magnetic field B is independent of the position inside the solenoid. We conclude that the magnetic field inside an ideal solenoid is uniform.

Example: Problem 31.14

A long solenoid of n turns per unit length carries a current I, and a long straight wire lying along the axis of this solenoid carries a current I'. Find the net magnetic field within the solenoid, at a distance r from the axis. Describe the shape of the magnetic field lines.

The magnetic field generated by the solenoid is uniform, directed parallel to the solenoid axis, and has a magnitude equal to

(31.10)

The magnetic field if a long straight wire, carrying a current I' has a magnitude equal to

(31.11)

and is directed perpendicular to the direction of r and I'. The direction of B_wire is therefore perpendicular to the direction of B_sol. The net magnetic field inside the solenoid is equal to the vector sum of B_wire and B_sol. Its magnitude is equal to

(31.12)

The angle a between the direction of the magnetic field and the z-axis is given by

(31.13)

Example: Problem 31.15

A coaxial cable consists of a long cylindrical copper wire of radius r₁ surrounded by a cylindrical shell of inner radius r₂ and outer radius r₃ (see Figure 31.5). The wire and the shell carry equal and opposite currents I uniformly distributed over their volumes. Find formulas for the magnetic field in each of the regions r < r₁, r₁ < r < r₂, r₂ < r < r₃, and r > r₃.

The magnetic field lines are circles, centered on the symmetry axis of the coaxial cable. First consider an integration path with r < r₁. The path integral of B along this path is equal to

(31.14)

The current enclosed by this integration path is equal to

(31.15)

Applying Faraday's law we can relate the current enclosed to the path integral of B

(31.16)

Therefore, the magnetic field is B is equal to

(31.17)






Figure 31.5. Problem 31.15. In the region between the wire and the shell, the enclosed current is equal to I and the path integral of the magnetic field is given by eq.(31.14). Ampere's law states then that
(31.18)

and the magnetic field is given by

(31.19)

In the third region (r₂ < r < r₃) the path integral of the magnetic field along a circular path with radius r is given by eq.(31.14). The enclosed current is equal to

(31.20)

The magnetic field is therefore equal to

(31.21)

The current enclosed by an integration path with a radius r > r₃ is equal to zero (since the current in the wire and in the shell are flowing in opposite directions). The magnetic field in this region is therefore also equal to zero.

31.3. Motion of charges in electric and magnetic fields

The magnetic force acting on particle with charge q moving with velocity v is equal to

(31.22)

This force is always perpendicular to the direction of motion of the particle, and will therefore only change the direction of motion, and not the magnitude of the velocity. If the charged particle is moving in a uniform magnetic field, with strength B, that is perpendicular to the velocity v, then the magnitude of the magnetic force is given by

(31.23)

and its direction is perpendicular to v. As a result of this force, the particle will carry out uniform circular motion. The radius of the circle is determined by the requirement that the strength magnetic force is equal to the centripetal force. Thus

(31.24)

The radius r of the orbit is equal to

(31.25)

where p is the momentum of the charged particle. The distance traveled by the particle in one revolution is equal to

(31.26)

The time T required to complete one revolution is equal to

(31.27)

The frequency of this motion is equal to

(31.28)

and is called the cyclotron frequency. Equation (31.28) shows that the cyclotron frequency is independent of the energy of the particle, and depends only on its mass m and charge q.
The effect of a magnetic field on the motion of a charged particle can be used to determine some of its properties. One example is a measurement of the charge of the electron. An electron moving in a uniform magnetic field will described a circular motion with a radius given by eq.(31.25). Suppose the electron is accelerated by a potential V₀. The final kinetic energy of the electron is given by

(31.29)

The momentum p of the electron is determined by its kinetic energy

(31.30)

The radius of curvature of the trajectory of the electron is thus equal to

(31.31)

Equation (31.31) shows that a measurement of r can be used to determine the mass over charge ratio of the electron.
Another application of the effect of a magnetic field on the motion of a charged particle is the cyclotron. A cyclotron consists of an evacuated cavity placed between the poles of a large electromagnet. The cavity is cut into two D-shaped pieces (called dees) with a gap between them. An oscillating high voltage is connected to the plates, generating an oscillating electric field in the region between the two dees. A charged particle, injected in the center of the cyclotron, will carry out a uniform circular motion for the first half of one turn. The frequency of the motion of the particle depends on its mass, its charge and the magnetic field strength. The frequency of the oscillator is chosen such that each time the particle crosses the gap between the dees, it will be accelerated by the electric field. As the energy of the ion increases, its radius of curvature will increase until it reaches the edge of the cyclotron and is extracted. During its motion in the cyclotron, the ion will cross the gap between the dees many times, and it will be accelerated to high energies.
Up to now we have assumed that the direction of the motion of the charged particle is perpendicular to the direction of the magnetic field. If this is the case, uniform circular motion will result. If the direction of motion of the ion is not perpendicular to the magnetic field, spiral motion will result. The velocity of the charged particle can be decomposed into two components: one parallel and one perpendicular to the magnetic field. The magnetic force acting on the particle will be determined by the component of its velocity perpendicular to the magnetic field. The projection of the motion of the particle on the x-y plane (assumed to be perpendicular to the magnetic field) will be circular. The magnetic field will not effect the component of the motion parallel to the field, and this component of the velocity will remain constant. The net result will be spiral motion.

31.4. Crossed electric and magnetic fields

A charged particle moving in a region with an electric and magnetic field will experience a total force equal to

(31.32)

This force is called the Lorentz force.





Figure 31.6. Charged particle moving in crossed E and B fields. Consider a special case in which the electric field is perpendicular to the magnetic field. The motion of a charged particle in such a region can be quit complicated. A charged particle with a positive charge q and velocity v is moving in this field (see Figure 31.6). The direction of the particle shown in Figure 31.6 is perpendicular to both the electric field and the magnetic field. The electric force acting on the particle is directed along the direction of the electric field and has a magnitude equal to


(31.33)
The magnetic force acting on the charge particle is directed perpendicular to both v and B and has a magnitude equal to



(31.34)
The net force acting on the particle is the sum of these two components and has a magnitude equal to



(31.35)
If the charged particle has a velocity equal to



(31.36)
then the net force will be equal to zero, and the motion of the particle will be uniform linear motion. A device with crossed electric and magnetic fields is called a velocity selector. If slit are placed in the appropriate positions, it will transport only those particles that have a velocity defined by the magnitudes of the electric and magnetic fields.





Figure 31.7. Current in a magnetic field. A technique used to determine the density and sign of charge carriers in a metal is based on the forces exerted by crossed E and B fields on the charge carriers. The diagram shown in Figure 31.7 shows a metallic strip carrying a current in the direction shown and placed in a uniform magnetic field with the direction of the magnetic field being perpendicular to the electric field (which generates the current I). Suppose the charge carriers in the material are electrons, than the electrons will move in a direction opposite to that of the current (see Figure 31.7). Since the magnetic field is perpendicular to the electric field, it is also perpendicular to the direction of motion of the electrons. As a result of the magnetic force, the electrons are deflected downwards, and an excess of negative charge will be created on the bottom of the strip. At the same time, a deficit of negative charge will be created at the top of the strip. This charge distribution will generate an electric field that is perpendicular to the external electric field and, under equilibrium conditions, the electric force produced by this field will cancel the magnetic force acting on the electrons. When this occurs, the internal electric field, E_in, is equal to the product of the electron velocity, v_d, and the strength of the magnetic field, B. As a result of the internal electric field, a potential difference will be created between the top and bottom of the strip. If the metallic strip has a width w, then the potential difference [Delta]V will be equal to


(31.37)
This effect is called the Hall effect.
The drift velocity of the electrons depend on the current I in the wire, its cross sectional area A and the density n of electrons (see Chapter 28):



(31.38)
Combining eq.(31.38) and eq.(31.37) we obtain the following expression for [Delta]V



(31.39)
A measurement of [Delta]V can therefore be used to determine n.

31.5. Forces on a wire

A current I flowing through a wire is equivalent to a collection of charges moving with a certain velocity v_d along the wire. The amount of charge dq present in a segment dL of the wire is equal to



(31.40)
If the wire is placed in a magnetic field, a magnetic force will be exerted on each of the charge carriers, and as a result, a force will be exerted on the wire. Suppose the angle between the direction of the current and the direction of the field is equal to [theta] (see Figure 31.8). The magnetic force acting on the segment dL of the wire is equal to



(31.41)
The total force exerted by the magnetic field on the wire can be found by integrating eq.(31.41) along the entire wire.






Figure 31.8. Magnetic force on wire.

Example: Problem 31.33

A balance can be used to measure the strength of the magnetic field. Consider a loop of wire, carrying a precisely known current, shown in Figure 31.9 which is partially immersed in the magnetic field. The force that the magnetic field exerts on the loop can be measured with the balance, and this permits the calculation of the strength of the magnetic field. Suppose that the short side of the loop measured 10.0 cm, the current in the wire is 0.225 A, and the magnetic force is 5.35 x 10^-2 N. What is the strength of the magnetic field ?

Consider the three segments of the current loop shown in Figure 31.9 which are immersed in the magnetic field. The magnetic force acting on segment 1 and 3 have equal magnitude, but are directed in an opposite direction, and therefore cancel. The magnitude of the magnetic force acting on segment 2 can be calculated using eq.(31.41) and is equal to



(31.42)
This force is measured using a balance and is equal to 5.35 x 10^-2 N. The strength of the magnetic field is thus equal to



(31.43)





Figure 31.9. Current loop in immersed in magnetic field.

31.6. Torque on a current loop

If a current loop is immersed in a magnetic field, the net magnetic force will be equal to zero. However, the torque on this loop will in general not be equal to zero. Suppose a rectangular current loop is placed in a uniform magnetic field (see Figure 31.10). The angle between the normal of the current loop and the magnetic field is equal to [theta]. The magnetic forces acting on the top and the bottom sections of the current loop are equal to



(31.44)
where L is the length of the top and bottom edge. The torque exerted on the current loop, with respect to its axis, is equal to



(31.45)





Figure 31.10. Current loop placed in uniform magnetic field. Using the definition of the magnetic dipole moment u, discussed in Chapter 30, eq.(31.45) can be rewritten as


(31.46)
where



(31.47)
Using vector notation, eq.(31.45) can be rewritten as



(31.48)
where the direction of the magnetic moment is defined using the right-hand rule.
The work that must be done against the magnetic field to rotate the current loop by an angle d[theta] is equal to - [tau] d[theta]. The change in potential energy of the current loop when it rotates between [theta]₀ and [theta]₁ is given by



(31.49)
A common choice for the reference point is [theta]₀ = 90deg. and U([theta]₀) = 0 J. If this choice is made we can rewrite eq.(31.50) as



(31.50)
In vector notation: