County Kilkenny

County Kilkenny () is a county in Ireland. It is in the province of Leinster and is part of the South-East Region. It is named after the city of Kilkenny. Kilkenny County Council is the local authority for the county. As of the 2016 census the population of the county was 99,232. The county was based on the historic Gaelic kingdom of Ossory ("Osraighe"), which was co-terminus with the Diocese of Ossory.
Geography and political subdivisions.
Kilkenny is the 16th-largest of Ireland's 32 counties by area, and the 21st largest in terms of population. It is the third-largest of Leinster's 12 counties in size, the seventh-largest in terms of population, and has a population density of 48 people per km2. Kilkenny borders five counties - Tipperary to the west, Waterford to the south, Carlow and Wexford to the east, and Laois to the north. Kilkenny city is the county's seat of local government and largest settlement, and is situated on the River Nore in the centre of the county. Kilkenny city is approximately from Waterford, from Dublin and from Cork.
Physical geography.
The River Nore flows through the county and the River Suir forms the border with County Waterford. Brandon Hill is the highest point with an elevation of . Most of the county has a hilly surface of moderate elevation with uplands in the north-east, the north-west and the South of the county; the middle is lower by comparison.
The county is located at and has an area of . The county extends from 52 degrees 14 minutes to 52 degrees 52 minutes north latitude, and from 6 degrees 56 minutes to 7 degrees 37 minutes west longitude. The north–south length of the county is ; and its greatest breadth from east to west, is about and its narrowest part is about from where it widens irregularly towards the north.
Kilkenny extends southward from Laois to the valley of the Suir and eastward from the Munster–Leinster border to the River Barrow. The River Nore bisects the county and the River Barrow and River Suir are natural boundaries to the east and south of the county. County Kilkenny is bordered by Laois, Carlow, Wexford, Waterford and Tipperary.
Rivers.
The main rivers in County Kilkenny, called the Three Sisters, are the Nore, Barrow and Suir. The River Nore bisects the county and flows through Ballyragget, Kilkenny City and the villages of Bennettsbridge, Thomastown and Inistioge. The River Suir forms the border between County Waterford and County Kilkenny. River Barrow forms a boundary to the east of the county.
Although County Kilkenny lies inland it has access to the sea via Belview Port, Port of Waterford, on the Suir Estuary and via New Ross on the River Barrow. On the River Barrow, from the villages of Goresbridge and Graiguenamanagh, there is a navigable river with traditional barges to the River Shannon or to Dublin Bay. Kilkenny's river network helps drain the land giving the county a highly fertile lower central plain.
Kings River and the Dinan are used for canoeing and kayaking as they contain stretches of peaceful waterways and a number of weirs and rapids. The wide and meandering Rivers Nore and Barrow are used for angling, boating and water sports.
Protected areas.
Special Areas of Conservation, under the European Union Habitats Directive, in County Kilkenny includes Hugginstown Fen south-west of Ballyhale, The Loughans near Urlingford, Cullahill Mountain on the Castlecomer Plateau near Johnstown, Spahill and Clomantagh Hill which form part of an escarpment which links the Slieveardagh Hills with the Castlecomer Plateau, Galmoy Fen north of Johnstown, Lower River Suir south of Thurles, the freshwater stretches of the Barrow/River Nore and Thomastown Quarry, near Thomastown.
In 2005 Coan Bogs was defined as a "Natural Heritage Area" under section 18 of the Wildlife Act 2000. The blanket bog consists of two small areas of upland blanket bog located near Castlecomer in the townlands of Coan East and Smithstown. Bedrock geology for both areas is shale overlain locally by glacial till and blanket bog vegetation is well developed.
Fiddown Island is a 62.6 ha state-owned "Nature Reserve" established in 1988. Near Fiddown along the River Suir it consists of a long narrow island of marsh/woodland covered in willow scrub and bordered by reed swamps - it is the only known site of its type in Ireland.
Wildlife.
The wildlife of County Kilkenny is part of the county's biodiversity and is an environmental, economic, amenity and resource.
Fauna of County Kilkenny includes hedgehogs, otters, badgers, red foxes, and bats such as Leisler's bat, Daubenton's bat, the brown long-eared bat and the common pipistrelle. There is also sika deer, fallow deer, stoat, red squirrel and pygmy shrew. The bird nesting period is from 1 March until 1 September. Woodlands, trees and hedgerows form a network of habitats, ecological 'corridors' essential for wildlife to flourish and move between habitats.
Flora of County Kilkenny includes the endangered autumn crocus, also rare species such as the bog orchid, the Killarney fern and the tufted salt-marsh grass. There are also vulnerable species like lesser snapdragon, meadow barley, small-white orchid, opposite-leaved pondweed, betony, red hemp nettle, narrow-leaved helleborine, lanceolate spleenwort, annual knawel and basil thyme.
Woodlands.
Surveys of woodlands of County Kilkenny include "A Woodland Survey of Kilkenny", undertaken in 1997 to identify woodlands in the county and "A survey of broadleaved woodlands in three special Areas of Conservation, Barrow-Nore, River Unshin and Lough Forbes (2000)" which covers some of County Kilkenny, and "National Survey of Native Woodlands (NSNW) 2003-2008" which was one of the largest ecological surveys to be completed in Ireland and did its fieldwork in the county in 2003.
Estimates of the woodlands in County Kilkenny include the "Forest Inventory Planning System (1998)" which estimated Kilkenny had 2251 ha of mixed woodland and broadleaf woodland not dominated by beech, representing 1.09% of the county and the "National Forest Inventory 2004 and 2006" which estimated that 4430 ha of the county is native forested land, Scots pine was regarded as non-native, representing 2.15% of the county.
The best quality sites in the county in terms of their conservation value include Kylecorragh Wood (SAC) along the River Nore, Grenan Wood (SAC), Knockadrina, Garryricken North (NHA/SNR), Ballytobin/Ballaghtobin, Brownstown Wood (NHA/SAC), Kyleadohir Wood (NHA/SNR), Brown's Wood, Thomastown, Ballykeeffe Wood, Cullentragh, Rossenarra, Newrath and Garryricken South. The most threatened forested area in the county is Greatwood.
The main governmental organisations responsible for ensuring the development of forestry within Kilkenny are the National Parks and Wildlife Service (Department of the Environment, Community and Local Government) and the Forest Service (Department of Agriculture, Food and the Marine). There are legislative measures which recognise the importance of trees woodlands and hedgerows and provide for their protection including the "Forestry Act (1946)" and "Wildlife (Amendment) Act (2000)". These stop trees being cut down unless a notice of intention is given and prohibit the cutting of hedges within the bird nesting period. The main threats come mainly from invasive species, heavy grazing and damaging activities such as non-native planting, native felling and dumping. There are instances of felling (both legal and illegal) of ash trees for the production of hurleys.
There are around 180 significant trees in County Kilkenny included in the Tree Register of Ireland, compiled by the Tree Council of Ireland, based on characteristics such as age, height, diameter, historical or folklore connections. The tallest tree in Kilkenny is a silver fir of 5.39 x 43m located at Woodstock Gardens in Inistioge and the tree with the largest girth is a common lime of 9.01 x 27m located at Coolmore House in Thomastown. At present there under the "Planning and Development Act (2000)" there are 4 Tree Preservation Orders (TPO) in County Kilkenny for the protection of trees, groups of trees and woods of amenity value. These include Oldcourt in Inistioge, Keatingstown, Barna in Freshford, Sawney's Wood in Castlecomer and the Castle Gardens in Kilkenny City.
Important trees in county Kilkenny include two cedars of Lebanon at "Kildalton Agricultural College" in Piltown, a Monterey cypress, beech, silver fir and two coastal redwood at Woodstock Gardens in Inistioge. In Thomastown there is a common lime at "Coolmore House", an English/common oak at "Stoneen Kilfawe", an English/common oak at Mount Juilliet Estate and a Wellingtonia, giant sequoia, Monterey cypress, sweet/Spanish chestnut and a holm oak/evergreen oak at "Kilfane Glebe". There is also a Turkey oak at "Threecastles House" in Kilkenny, an ash and two English/common oak at "Ballytobin House" in Callan, a common lime, Wellingtonia giant sequoia and two Monterey cypress at Shankill Castle in Paulstown, an English/common oak at "Ballykeefe House" in Kilmanagh, an English/common oak at Fanningstown House in Owning and a beech at "Castlecomer Golf Club" in Castlecomer.
Climate.
The climate of County Kilkenny, like the climate of Ireland, is a changeable oceanic climate with few extremes. It is defined as a temperate oceanic climate, or "Cfb" on the Köppen climate classification system. Kilkenny lies in plant Hardiness zone
9. Weatherwise, the county is generally representative of wide river valleys in the region with low temperatures on cloudless nights, and is significant in that it records some of the highest summer and lowest winter temperatures in Ireland. The highest air temperature ever recorded in Ireland was , at Kilkenny Castle on 26 June 1887.
The Met Éireann Kilkenny Weather Observing Station, 2 km north-west of Kilkenny City centre, on the Duningstown Road, opened in May 1957, and observations ceased in April 2008. A climatological station is currently in operation within 1 km of the old site, and as of March 2010, was providing live weather data to the general public and climate data to Met Éireann.
The maximum temperature recorded at the Met Station was on 2 August 1995. Extremes recorded at the station include the highest air temperature of on 29 June 1976, the lowest air temperature of on 2 January 1979 and the lowest ground temperature of on 12 January 1982. The maximum daily sunshine was 16.3 hours on 18 June 1978. The warmest and sunniest month on record in Kilkenny was August 1995 with a total of 274.9 hours sunshine and very high temperatures throughout. The maximum daily sunshine was 16.3 hours on 18 June 1978. The overall trend in temperatures has been on the rise with a marked increase from 1988 onwards. Annual temperatures are running over 0.5 degrees above 20th-century levels.
The maximum daily rainfall recorded at Kilkenny station was on 17 July 1983. The late 1950s and early 1960s were wet but rainfall had been steady throughout the century. 2002 was a very wet year and since 2005 annual rainfall has been increased steadily, with 2009 being the wettest year since records commenced in 1958. At the centre of the county, Kilkenny is in a sheltered location, over 60 km inland and is surrounded by hills over 200m, which ensures that it is not a windy location. The highest wind gust of 77 knots, from a south-west direction, was recorded on 12 January 1974.
Geology.
The geology of Kilkenny includes the Kiltorcan Formation which is early Carboniferous in age. The Formation is located around Kiltoncan Hill near Ballyhale in the Callan and Knocktopher areas. It forms the uppermost part of the Old Red Sandstone and is the distinctive Upper Devonian–Lower Carboniferous unit in southern Ireland. It contains non-red lithologies, green mudstones, siltstones, fine sandstones and yellow sandstones. There is a fossil assemblage containing Cyclostigma and Archaeopteris and Archaeopteris hibernica.
Most of the county is principally limestone of the upper and lower group, corresponding with the rest of Ireland. A large area in the north and east contains beds of coal, surrounded by limestone strata, alternated with shale, argillaceous ironstone, and sandstone. This occurs eastward of the Nore around Castlecomer, along the border with Laois. It is generally accompanied by culm, which was used extensively for burning lime.
The natural environment and resources of County Kilkenny includes its rivers, wildlife (mammals, birds, plants), woodlands, hedgerows, and diverse landscapes and geological features. The main land use is grassland, dairy farming and tillage farming especially around Kilkenny City and in the fertile central plain of the Nore Valley. Conifer forests are found on the upland areas.
Habitats of international and national importance, are designated under European Union and national legislation. The four categories of designated site in effect in County Kilkenny are Special Areas of Conservation, Natural Heritage Areas, Statutory Nature Reserves and Wildfowl Sanctuaries. The 2008-2014 county development plan recorded 36 designated natural heritage sites of international and national importance in County Kilkenny, covering approximately 4.5% of the county.
Mountains and hills.
County Kilkenny is comparably low compared to other mountain ranges in Ireland with the highest peak being Brandon Hill (), at above sea level. The majority of rest of the county is hilly except for the centre of the county, just south of the city of Kilkenny, which is comparably lower than the rest of the county. There are highlands in the North-East, the North-West and in the South of the county.
In the north of the county the uplands of the Castlecomer district, the Castlecomer Plateau includes the Culla Hills to the west of the Nore Valley with Castlecomer Hills and Slieve Marcy to the east. These hills are divided by the valley of the Dinan river which joins the Nore from the east. The highest point of the Castlecomer Hills is , and located North-West of the town of Castlecomer and near the border with Laois. The highest point of the Culla Hills is in Laois at , but its undulating slopes spread over a considerable area of Kilkenny in the north-westerly area.
In the west of the county the Slieveardagh Hills and Booley Hills extend westwards to County Tipperary. The highest point in the Slieveardagh Hills is Knocknamuck at . The Booley Mountains partly divide from Tory Hill to the valley of the Kilmacow River which flows into the Suir.
In the south of the county is Brandon Hill, at and is near the Barrow and Graiguenamanagh. The area enclosed between the rivers Nore and Barrow towards their point of joining is elevated. Along the west of the Barrow and Nore is mostly covered by hills of nearly equal elevation except along the left bank of the river Suir. Here there is a rich area of land between the river and the hills.
Baronies.
The county is subdivided into twelve baronies which are in turn divided into civil parishes and townlands. There are about 800 townlands in Kilkenny. Each barony was made up of a number of parishes or parts of parishes. Both civil parishes and baronies are now largely obsolete (except for purposes such as legal transactions involving land) and are no longer used for local government purposes.
Civil parishes and townlands.
For religious administration, the county was divided into parishes. Every parish had at least one church. The barony boundaries and the parish boundaries were not connected.
From the 17th to mid-19th centuries, civil parishes were based on early Christian and medieval monastic and church settlements. The civil parishes are divided into townlands (See List of townlands in County Kilkenny). As the population grew, new parishes were created and the civil parish covered the same area as the established Church of Ireland. The Roman Catholic Church adapted to a new structure based on towns and villages. There 2,508 civil parishes in Ireland, which often break both barony and county boundaries.
Local government and politics.
Local Government.
Local government in County Kilkenny is governed by the "Local Government Acts", the most recent of which (Local Government Reform Act 2014) established a unitary structure of local government. This single-tier structure consists of Kilkenny County Council. Until 2014, the second tier of local government consisted of Kilkenny Borough Council which was a town council. The city of Kilkenny was allowed to use the title of "Borough Council" instead of "Town Council", but Kilkenny Borough Council had no additional responsibilities. Since the enactment of the aforementioned Act on 1 June 2014, which abolished the borough, the county council is solely responsible for local services.
For the purposes of local government, Kilkenny is divided into 4 local electoral areas - Callan–Thomastown (6), Castlecomer (6), Kilkenny city (7) and Piltown (5) - which return a total of 24 elected members to Kilkenny County Council. The 2019 Kilkenny local elections had a voter turnout of 53%, a decrease of 4.1% on the 2014 election. The highest turnout was in Callan–Thomastown (56.4%) and the lowest was in Kilkenny city (50.4%).
In the 2019 local elections, Fianna Fáil and Fine Gael emerged as the two dominant parties, holding 20 of the 24 seats between them. The Labour Party hold 2 seats, the Green Party hold 1 seat, and there is 1 independent. Sinn Féin, which had held 3 seats, failed to gain a seat in the 2019 election. Council elections are held every 5 years, with the next election due to be held in May 2024.
National Elections.
For elections to Dáil Éireann, Kilkenny is part of the Carlow–Kilkenny constituency which returns 5 TDs. The present form of the constituency was created for the 1948 general election. Kilkenny has been represented through several parliamentary constituencies in the past. From 1918 to 1921 Kilkenny was part of the North Kilkenny United Kingdom Parliamentary constituency. In 1921 the Carlow–Kilkenny Dáil constituency was created and has stayed apart from between 1937 and 1948 when there was just a Kilkenny constituency.
European Elections.
For statistical purposes at European level, the county is part of the South-East Region - a NUTS III entity - which is in turn part of the level II NUTS entity - Southern. The county is located within the South constituency for elections to the European Parliament. The constituency has 5 seats and comprises the South-East, South-West and Mid-West regions of Ireland.
As the county is part of the South-East Region, some county councillors are also representatives on the Southern Regional Assembly.
History.
County Kilkenny takes its name from the city of Kilkenny. Kilkenny is the anglicised version of the Irish "Cill Chainnigh", meaning "Church (Cell) of Cainneach or Canice". This probably relates to the church and round tower, now St. Canice's Cathedral, which was built in honour of St. Canice.
The Kingdom of Osraige was one of the ancient Kingdoms of Ireland. The Kings of Osraige, the Mac Giolla Phádraig family, reigned over Osraige and Cill Chainnigh was their stronghold. The Kingdom of Ossory existed from at least the 2nd century until the 13th century AD. The current ecclesiastical dioceses of that area is still known as Ossory. The medieval Diocese of Ossory and was established in 549 AD, and its territory corresponded to the medieval Kingdom of Ossory. In historic times, Kilkenny replaced Aghaboe as the chief church in Osraige.
The kingdom was bounded by two of the Three Sisters the rivers Barrow and Suir and the northern limit was, generally, the Slieve Bloom Mountains. The Osraige —their name means people of the deer— inhabited much of modern County Kilkenny and parts of neighbouring County Laois. To the west and south, Osraige was bounded by the River Suir, to the east the watershed of the River Barrow marked the boundary with Leinster, and to the north it extended into and beyond the Slieve Bloom Mountains. The River Nore ran through the Kingdom.
Osraige formed the easternmost part of the kingdom and province of Munster until the middle of the 9th century, after which it was attached to Leinster. Osraige was largely a buffer state between Leinster and Munster. Its most significant neighbours were the Loígsi, Uí Cheinnselaig and Uí Baircche of Leinster to the north and east and the Déisi Muman, Eóganacht Chaisil and Éile of Munster to the south and west.
The name Osraige is said to be from the Usdaie, a Celtic tribe that Ptolemy's map of Ireland places in roughly the same area that Osraige would later occupy. The territory indicated by Ptolemy probably included the major late Iron Age hill-fort at Freestone Hill which produced some Roman finds. Also the interesting burial at Stoneyford which is of typical Roman type and probably dates to the 1st century AD. The Osraighe themselves claimed to be descended from the Érainn people. Others propose that the Ivernic groups included the Osraige of the Kingdom of Osraige/Ossory. The Brigantes were the only Celtic tribe to have a presence in both England and Ireland, in the latter of which they could be found around Kilkenny, Wexford and Waterford.
Pope Adrian IV gave Norman King Henry II of England permission to claim Ireland 1154. The Cisternians came to Jerpoint and Kilkenny around 1155/60. Jerpoint Abbey is founded by Donal MacGiollaPhadruig, King of Ossory 1158. In 1168, Dermot MacMurrough the King of Leinster was driven out of his kingdom by Rory O'Connor the High King of Ireland with the help of Tiernan O'Rourke. MacMurrough looked for help from Henry II and got help from a Cambro-Norman lord notable Richard de Clare, 2nd Earl of Pembroke, known to history as Strongbow. MacMurrough secured the services of Richard, promising him the hand of his daughter Aoife of Leinster and the succession to Leinster. Richard and other Marcher barons and knights by King Henry assembled an army. The army, under Raymond le Gros, took Wexford, Waterford and Dublin in 1169 and 1170, and Strongbow joined them in August 1170. The day after the capture of Waterford, he married MacMorrough's daughter, Aoife.
The Lordship of Ireland was a lordship created in the wake of the Norman invasion of Ireland in 1169–71. Kilkenny formed part of the lordship of Leinster. Strongbow became Lord of Leinster in 1171.
Following the Norman invasion, the island of Ireland was divided into thirty-two counties. The Republic of Ireland today is made up of twenty-six of the traditional thirty-two counties with the other six forming Northern Ireland. Two former counties in the Republic have been subdivided, giving a modern total of twenty-nine counties for administrative purposes rather than twenty-six.
Places of interest.
Architecture.
The architecture of County Kilkenny contains features from all eras since the Stone Age including Norman and Anglo-Irish castles, Georgian urban buildings, towns and villages with unique architectures, palladian and rococo country houses, Gothic and neo-Gothic cathedrals and buildings. In the late 20th century a new economic climate resulted in a renaissance of culture and design, with some at the cutting edge of modern architecture. County Kilkenny contains varied architecture including passage graves, ringforts, Irish round towers, castles, churches and cathedrals, abbeys and priories, bridges and roads, and townhouses of varying style.
Early architecture.
Evidence of Neolithic settlement can be found throughout the county. There are great burial mounds including the portal tombs and dolmens at Owning, Kilmogue-Harristown and Borrismore. There are passage graves at Clomantagh Hill and Knockroe. There were non-megalith single-grave burial tombs, Linkardstown-type Cists, excavated at Jerpoint West. These are late Neolithic and before the single-grave rite of the Bronze Age. A Neolithic house was identified in Granny near Waterford, making it the oldest house in County Kilkenny. The square house consisted of slot-trenches, internal floor surfaces, a hearth and wooden posts at each corner, one of the post-holes was radiocarbon-dated to 3997-3728 BC. A new form of early Neolithic pottery with a lip around the inside of the rim were found. This Granny pottery is similar to pottery found in the south-east of England.
Ogham stones are found throughout County Kilkenny. The Gowran Ogham Stone which is on display in St. Mary's Collegiate Church Gowran is an example of a Christianised ogham stone. The ogham lettering is from the 3rd or 4th century. The Christian cross was carved c. 6th century following St. Patrck's arrival in Ireland in 432 AD. The ogham stone was found on the site during the rebuilding of the chancel in 1826. The Gowran Ogham Stone has survived to the present day due to the fact that it was used as a building stone in another part of the church and lay undisturbed for centuries.
Passage graves consisted of a central burial chamber, with a long passageway to the entrance. Knockroe contains a passage tomb which is protected by listing order. From some time beginning around the Iron Age, Ireland has thousands of ringforts, or "raths". Carigeen, Danesfort, Dunbell big and Tullaroan all contain ring forts protected by preservation order.
The round stone tower is one feature of "early historic architecture" not usually found outside Ireland with only three in Scotland and one on the Isle of Man. The tower at St Canice's Cathedral in Kilkenny City is a good example. Other round towers are situated around the county such as Tullaherin round tower, one near Bennettsbridge, Kells, Johnstown, and Aghaviller near Knocktopher.
Castles.
Kilkenny Castle is a castle in Kilkenny. It was the seat of the Butler family. Formerly the family name was FitzWalter. The castle was sold to the local Castle Restoration Committee in the middle of the 20th century for £50. Shortly afterwards it was handed over to the State, and has since been refurbished and is open to visitors. Part of the National Art Gallery is on display in the castle. There are ornamental gardens on the town side of the castle, and extensive land and gardens to the front. It has become one of the most visited tourist sites in Ireland.
Foulksrath Castle is a 15th-century Anglo-Norman tower house located in Jenkinstown in County Kilkenny. It was built by the Purcell clan, who also constructed several others nearby. After over three centuries as owners, the family was reduced to living as peasants in the castle stables after it was confiscated by Oliver Cromwell and given to his officers after the Cromwellian conquest of Ireland.
Abbeys and priories.

Force between magnets

Magnets exert forces and torques on each other due to the rules of electromagnetism. The forces of attraction field of magnets are due to microscopic currents of electrically charged electrons orbiting nuclei and the intrinsic magnetism of fundamental particles (such as electrons) that make up the material. Both of these are modeled quite well as tiny loops of current called magnetic dipoles that produce their own magnetic field and are affected by external magnetic fields. The most elementary force between magnets is the magnetic dipole–dipole interaction. If all of the magnetic dipoles that make up two magnets are known then the net force on both magnets can be determined by summing up all these interactions between the dipoles of the first magnet and that of the second.
It is often more convenient to model the force between two magnets as being due to forces between magnetic poles having magnetic charges spread over them. Positive and negative magnetic charge is always connected by a string of magnetized material; isolated magnetic charge does not exist. This model works well in predicting the forces between simple magnets where good models of how the magnetic charge is distributed are available.
Magnetic poles vs. atomic currents.
The field of a magnet is the sum of fields from all magnetized volume elements, which consist of small magnetic dipoles on an atomic level. The direct summation of all those dipole fields would require three-dimensional integration just to obtain the field of one magnet, which may be intricate.
In case of a homogeneous magnetization, the problem can be simplified at least in two different ways, using Stokes' theorem. Upon integration along the direction of magnetization, all dipoles along the line of integration cancel each other, except at the magnet's end surface. The field then emerges only from those (mathematical) magnetic charges spread over the magnet's end facets. On the contrary, when integrating over a magnetized area orthogonal to the direction of magnetization, the dipoles within this area cancel each other, except at the magnet's outer surface, where they (mathematically) sum up to a ring current. This is called Ampère model. In both models, only two-dimensional distributions over the magnet's surface have to be considered, which is simpler than the original three-dimensional problem.
Magnetic-charge model: In the magnetic-charge model, the pole surfaces of a permanent magnet are imagined to be covered with so-called "magnetic charge", north pole particles on the north pole and south pole particles' on the south pole, that are the source of the magnetic field lines. The field due to magnetic charges is obtained through Coulomb's law with magnetic instead of electric charges. If the magnetic pole distribution is known, then the pole model gives the exact distribution of the magnetic field intensity H both inside and outside the magnet. The surface charge distribution is uniform, if the magnet is homogeneously magnetized and has flat end facets (such as a cylinder or prism).
Ampère model: In the Ampère model, all magnetization is due to the effect of microscopic, or atomic, circular "bound currents", also called "Ampèrian currents" throughout the material. The net effect of these microscopic bound currents is to make the magnet behave as if there is a macroscopic electric current flowing in loops in the magnet with the magnetic field normal to the loops. The field due to such currents is then obtained through the Biot–Savart law. The Ampère model gives the correct magnetic flux density B both inside and outside the magnet. It is sometimes difficult to calculate the Ampèrian currents on the surface of a magnet.
Magnetic dipole moment.
Far away from a magnet, its magnetic field is almost always described (to a good approximation) by a dipole field characterized by its total magnetic dipole moment, m. This is true regardless of the shape of the magnet, so long as the magnetic moment is non-zero. One characteristic of a dipole field is that the strength of the field falls off inversely with the cube of the distance from the magnet's center.
The magnetic moment of a magnet is therefore a measure of its strength and orientation. A loop of electric current, a bar magnet, an electron, a molecule, and a planet all have magnetic moments. More precisely, the term "magnetic moment" normally refers to a system's magnetic dipole moment, which produces the first term in the multipole expansion of a general magnetic field.
Both the torque and force exerted on a magnet by an external magnetic field are proportional to that magnet's magnetic moment. The magnetic moment is a vector: it has both a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet (inside the magnet). For example, the direction of the magnetic moment of a bar magnet, such as the one in a compass is the direction that the north poles points toward.
In the physically correct Ampère model, magnetic dipole moments are due to infinitesimally small loops of current. For a sufficiently small loop of current, "I", and area, A, the magnetic dipole moment is:
where the direction of m is normal to the area in a direction determined using the current and the right-hand rule. As such, the SI unit of magnetic dipole moment is ampere meter2. More precisely, to account for solenoids with many turns the unit of magnetic dipole moment is Ampere-turn meter2.
In the magnetic-charge model, the magnetic dipole moment is due to two equal and opposite magnetic charges that are separated by a distance, "d". In this model, m is similar to the electric dipole moment p due to electrical charges:
where "q"m is the ‘magnetic charge’. The direction of the magnetic dipole moment points from the negative south pole to the positive north pole of this tiny magnet.
Magnetic force due to non-uniform magnetic field.
Magnets are drawn along the magnetic field gradient. The simplest example of this is the attraction of opposite poles of two magnets. Every magnet produces a magnetic field that is stronger near its poles. If opposite poles of two separate magnets are facing each other, each of the magnets are drawn into the stronger magnetic field near the pole of the other. If like poles are facing each other though, they are repulsed from the larger magnetic field.
The magnetic-charge model predicts a correct mathematical form for this force and is easier to understand qualitatively. For if a magnet is placed in a uniform magnetic field then both poles will feel the same magnetic force but in opposite directions, since they have opposite magnetic charge. But, when a magnet is placed in the non-uniform field, such as that due to another magnet, the pole experiencing the large magnetic field will experience the large force and there will be a net force on the magnet. If the magnet is aligned with the magnetic field, corresponding to two magnets oriented in the same direction near the poles, then it will be drawn into the larger magnetic field. If it is oppositely aligned, such as the case of two magnets with like poles facing each other, then the magnet will be repelled from the region of higher magnetic field.
In the Ampère model, there is also a force on a magnetic dipole due to a non-uniform magnetic field, but this is due to Lorentz forces on the current loop that makes up the magnetic dipole. The force obtained in the case of a current loop model is
where the gradient ∇ is the change of the quantity m · B per unit distance, and the direction is that of maximum increase of m · B. To understand this equation, note that the dot product m · B = "mB"cos("θ"), where "m" and "B" represent the magnitude of the m and B vectors and "θ" is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictly larger m · B). B represents the strength and direction of the magnetic field. This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions having their own m then summing up the forces on each of these regions.
Magnetic-charge model.
The magnetic-charge model assumes that the magnetic forces between magnets are due to magnetic charges near the poles. This model works even close to the magnet when the magnetic field becomes more complicated, and more dependent on the detailed shape and magnetization of the magnet than just the magnetic dipole contribution. Formally, the field can be expressed as a multipole expansion: A dipole field, plus a quadrupole field, plus an octopole field, etc. in the Ampère model, but this can be very cumbersome mathematically.
Calculating the magnetic force.
Calculating the attractive or repulsive force between two magnets is, in the general case, a very complex operation, as it depends on the shape, magnetization, orientation and separation of the magnets. The magnetic-charge model does depend on some knowledge of how the ‘magnetic charge’ is distributed over the magnetic poles. It is only truly useful for simple configurations even then. Fortunately, this restriction covers many useful cases.
Force between two magnetic poles.
If both poles are small enough to be represented as single points then they can be considered to be point magnetic charges. Classically, the force between two magnetic poles is given by:
where
The pole description is useful to practicing magneticians who design real-world magnets, but real magnets have a pole distribution more complex than a single north and south. Therefore, implementation of the pole idea is not simple. In some cases, one of the more complex formulas given below will be more useful.
Force between two nearby magnetized surfaces of area "A".
The mechanical force between two nearby magnetized surfaces can be calculated with the following equation. The equation is valid only for cases in which the effect of fringing is negligible and the volume of the air gap is much smaller than that of the magnetized material, the force for each magnetized surface is:
where:
The derivation of this equation is analogous to the force between two nearby electrically charged surfaces, which assumes that the field in between the plates is uniform.
Force between two bar magnets.
The force between two identical cylindrical bar magnets placed end to end at great distance formula_6 is approximately:
where
Note that these formulations assume point-like magnetic-charge distributions instead of a uniform distribution over the end facets, which is a good approximation only at relatively great distances. For intermediate distances, numerical methods must be used.
Force between two cylindrical magnets.
For two cylindrical magnets with radius formula_9, and length formula_10, with their magnetic dipole aligned, the force can be computed analytically using elliptic integrals. In the limit formula_6, the force can be approximated by,
Where formula_13 is the magnetization of the magnets and formula_14 is the distance between them. For small values of formula_14, the results are erroneous as the force becomes large for close-to-zero distance.
If the magnet is long (formula_16), a measurement of the magnetic flux density very close to the magnet formula_17 is roughly related to formula_13 by the formula
The effective magnetic dipole moment can be written as
Where formula_21 is the volume of the magnet. For a cylinder this is formula_22, and
formula_13 is the magnetisation field of the dipole.
When formula_24 the point dipole approximation is obtained,
Which matches the expression of the force between two magnetic dipoles.
Ampère model.
French scientist André Marie Ampère found that the magnetism produced by permanent magnets and the magnetism produced by electromagnets are the same kind of magnetism.
Because of that, the strength of a permanent magnet can be expressed in the same terms as that of an electromagnet.
The strength of magnetism of an electromagnet that is a flat loop of wire through which a current flows, measured at a distance that is great compared to the size of the loop, is proportional to that current and is proportional to the surface area of that loop.
For purpose of expressing the strength of a permanent magnet in same terms as that of an electromagnet, a permanent magnet is thought of as if it contains small current-loops throughout its volume,
and then the magnetic strength of that magnet is found to be proportional to the current of each loop (in Ampere), and proportional to the surface of each loop (in square meter), and proportional to the density of current-loops in the material (in units per cubic meter), so the dimension of strength of magnetism of a permanent magnet is Ampere times square meter per cubic meter, is Ampere per meter.
That is why Ampere per meter is the correct unit of magnetism, even though these small current loops are not really present in a permanent magnet.
The validity of Ampere's model means that it is allowable to think of the magnetic material as if it consists of current-loops, and the total effect is the sum of the effect of each current-loop, and so the magnetic effect of a real magnet can be computed as the sum of magnetic effects of tiny pieces of magnetic material that are at a distance that is great compared to the size of each piece.
This is very useful for computing magnetic force-field of a real magnet ;
It involves summing a large amount of small forces and you should not do that by hand, but let your computer do that for you ;
All that the computer program needs to know is the force between small magnets that are at great distance from each other.
In such computations it is often assumed that each (same-size) small piece of magnetic material has an equally strong magnetism, but this is not always true :
a magnet that is placed near another magnet can change the magnetization of that other magnet.
For permanent magnets this is usually only a small change, but if you have an electromagnet that consists of a wire wound round an iron core, and you bring a permanent magnet near to that core, then the magnetization of that core can change drastically (for example, if there is no current in the wire, the electromagnet would not be magnetic, but when the permanent magnet is brought near, the core of the electromagnet becomes magnetic).
Thus the Ampere model is suitable for computing the magnetic force-field of a permanent magnet, but for electromagnets it can be better to use a magnetic-circuit approach.
Magnetic dipole–dipole interaction.
If two or more magnets are small enough or sufficiently distant that their shape and size is not important then both magnets can be modeled as being magnetic dipoles having a magnetic moments m1 and m2. In case of uniformly magnetized spherical magnets this model is precise even at finite size and distance, as the outside field of such magnets is exactly a dipole field.
The magnetic field of a magnetic dipole in vector notation is:
where
This is "exactly" the field of a point dipole, "exactly" the dipole term in the multipole expansion of an arbitrary field, and "approximately" the field of any dipole-like configuration at large distances.
If the coordinate system is shifted to center it on m1 and rotated such that the z-axis points in the direction of m1 then the previous equation simplifies to
where the variables r and "θ" are measured in a frame of reference with origin in m1 and oriented such that m1 is at the origin pointing in the z-direction. This frame is called Local coordinates and is shown in the Figure on the right.
The force of one magnetic dipole on another is determined by using the magnetic field of the first dipole given above and determining the force due to the magnetic field on the second dipole using the force equation given above. Using vector notation, the force of a magnetic dipole m1 on the magnetic dipole m2 is:
where r is the distance-vector from dipole moment m1 to dipole moment m2, with "r"=||r||. The force acting on m1 is in opposite direction. As an example the magnetic force for two magnets pointing in the z-direction and aligned on the z-axis and separated by the distance z is:
The final formulas are shown next. They are expressed in the global coordinate system,