How To Find The Magnitude Of The Magnetic Field
Electric Currents and Magnetic Fields
An electric current volition produce a magnetic field, which can be visualized every bit a series of circular field lines around a wire segment.
Learning Objectives
Describe shape of a magnetic field produced by an electrical electric current flowing through a wire
Fundamental Takeaways
Key Points
- A wire carrying electrical current will produce a magnetic field with airtight field lines surrounding the wire.
- Another version of the correct hand rules can be used to determine the magnetic field direction from a current—point the thumb in the management of the current, and the fingers curl in the management of the magnetic field loops created by it. See.
- The Biot-Savart Law can be used to make up one's mind the magnetic field strength from a current segment. For the unproblematic instance of an infinite straight current-carrying wire information technology is reduced to the form [latex]\text{B}=\frac{\mu _{0}\text{I}}{ii\pi \text{r}}[/latex].
- A more fundamental law than the British indian ocean territory-Savart law is Ampere 's Police, which relates magnetic field and electric current in a general mode. It is written in integral class equally [latex]\oint \text{B}\cdot \text{d}\mathscr{\text{l}}=\mu _{0}\text{I}_{\text{enc}}[/latex], where Ienc is the enclosed electric current and μ0 is a constant.
- A current-carrying wire feels a force in the presence of an external magnetic field. Information technology is found to be [latex]\text{F}=\text{Bi}\mathscr{\text{l}}\text{sin}\theta[/latex], where ℓ is the length of the wire, i is the current, and θ is the angle betwixt the current direction and the magnetic field.
Primal Terms
- Biot-Savart Law: An equation that describes the magnetic field generated by an electric electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electrical current. The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital police and Gauss's police for magnetism.
- Ampere'southward Law: An equation that relates magnetic fields to electric currents that produce them. Using Ampere's law, one can determine the magnetic field associated with a given current or electric current associated with a given magnetic field, providing there is no time irresolute electric field present.
Electric Current and Magnetic Fields
Electric current produces a magnetic field. This magnetic field can exist visualized as a pattern of circular field lines surrounding a wire. One way to explore the direction of a magnetic field is with a compass, as shown by a long straight current-carrying wire in. Hall probes tin can determine the magnitude of the field. Some other version of the right hand dominion emerges from this exploration and is valid for any electric current segment—point the thumb in the direction of the current, and the fingers curl in the direction of the magnetic field loops created past it.
Magnetic Field Generated past Current: (a) Compasses placed near a long direct current-carrying wire indicate that field lines class circular loops centered on the wire. (b) Right hand rule 2 states that, if the right hand thumb points in the direction of the current, the fingers curl in the direction of the field. This dominion is consistent with the field mapped for the long straight wire and is valid for any electric current segment.
Magnitude of Magnetic Field from Current
The equation for the magnetic field forcefulness (magnitude) produced by a long directly current-carrying wire is:
[latex]\text{B}=\frac{\mu _{0}\text{I}}{2\pi \text{r}}[/latex]
For a long straight wire where I is the current, r is the shortest distance to the wire, and the abiding 0=4π10−7 T⋅m/A is the permeability of complimentary infinite. (μ0 is one of the bones constants in nature, related to the speed of light. ) Since the wire is very long, the magnitude of the field depends only on altitude from the wire r, non on position along the wire. This is one of the simplest cases to calculate the magnetic field strenght from a current.
The magnetic field of a long directly wire has more implications than one might first doubtable. Each segment of current produces a magnetic field like that of a long directly wire, and the full field of whatsoever shape current is the vector sum of the fields due to each segment. The formal statement of the management and magnitude of the field due to each segment is called the British indian ocean territory-Savart law. Integral calculus is needed to sum the field for an capricious shape current. The Biot-Savart law is written in its consummate form as:
[latex]\text{B}=\frac{\mu _{\text{o}}\text{I}}{iv\pi}\int \frac{\text{d}\mathscr{\text{l}}\times \hat{\text{r}}}{\text{r}^{two}}[/latex]
where the integral sums over the wire length where vector dℓ is the direction of the current; r is the distance between the location of dℓ, and the location at which the magnetic field is being calculated; and r̂ is a unit vector in the management of r. The reader may utilise the simplifications in calculating the magnetic field from an infinite direct wire as above and see that the Biot-Savart law reduces to the showtime, simpler equation.
Ampere'southward Police force
A more cardinal police force than the British indian ocean territory-Savart constabulary is Ampere's Law, which relates magnetic field and current in a general mode. In SI units, the integral form of the original Ampere'due south circuital law is a line integral of the magnetic field around some closed curve C (arbitrary just must exist closed). The curve C in turn bounds both a surface Southward through which the electric current passes through (again arbitrary but not airtight—since no three-dimensional volume is enclosed by S), and encloses the current. You tin think of the "surface" as the cross-sectional area of a wire conveying electric current.
The mathematical statement of the law states that the total magnetic field around some path is directly proportional to the current which passes through that enclosed path. It tin be written in a number of forms, i of which is given below.
[latex]\oint \text{B}\cdot \text{d}\mathscr{\text{l}}=\mu _{0}\iint_{\text{S}}^{ } \text{J}\cdot \text{dS}=\mu _{0}\text{I}_{\text{enc}}[/latex]
where the magnetic field is integrated over a curve (circumfrence of a wire), equivalent to integrating the current density (in amperes per square meter, Am-2) over the cross section area of the wire (which is equal to the permeability constant times the enclosed current Ienc) . Ampere'due south constabulary is always valid for steady currents and can be used to calculate the B-field for certain highly symmetric situations such as an space wire or an infinite solenoid. Ampere's Law is also a component of Maxwell's Equations.
Strength on a Current-Conveying Wire
The force on a current carrying wire (every bit in ) is similar to that of a moving charge as expected since a charge conveying wire is a collection of moving charges. A electric current-carrying wire feels a forcefulness in the presence of a magnetic field. Consider a usher (wire) of length ℓ, cross section A, and charge q which is due to electric current i. If this conductor is placed in a magnetic field of magnitude B which makes an angle with the velocity of charges (current) in the conductor, the forcefulness exerted on a single accuse q is
Force on a Current-Conveying Wire: The correct hand rule can be used to decide the direction of the forcefulness on a current-carrying wire placed in an external magnetic field.
[latex]\text{F}=\text{qvBsin}\theta[/latex]
And then, for North charges where
[latex]\text{Due north}=\text{n}\mathscr{\text{fifty}}\text{A}[/latex]
the forcefulness exerted on the usher is
[latex]\text{f}=\text{FN}=\text{qvBn}\mathscr{\text{l}}\text{Asin}\theta =\text{Bi}\mathscr{\text{l}}\text{sin}\theta[/latex]
where i = nqvA. The correct hand dominion can give you the direction of the strength on the wire, as seen in the above figure. Annotation that the B-field in this case is the external field.
Permanent Magnets
Permanent magnets are objects made from ferromagnetic material that produce a persistent magnetic field.
Learning Objectives
Requite examples and counterexamples of permanent magnets
Fundamental Takeaways
Cardinal Points
- Permanent magnets are objects made from magnetized textile and produce continual magnetic fields. Everyday examples include refrigerator magnets used to hold notes on a refrigerator door.
- Materials that tin can be magnetized, which are also the ones that are strongly attracted to a magnet, are chosen ferromagnetic. Examples of these materials include iron, nickel, and cobalt.
- The counterexample to a permanent magnet is an electromagnet, which but becomes magnetized when an electrical current flows through it.
- Magnets always have a north pole and a south pole, and then if 1 were to split a permanent magnet in half, two smaller magnets would be created, each with a north pole and south pole.
- Permanent magnets are made from ferromagnetic materials that are exposed to a strong external magnetic field and heated to align their internal microcrystalline structure, making them very hard to demagnetize.
Key Terms
- permanent magnet: A fabric, or piece of such material, which retains its magnetism even when not subjected to whatsoever external magnetic fields.
- ferromagnetic: Of a fabric, such as iron or nickel, that is easily magnetized.
- electromagnet: A magnet which attracts metals but when electrically activated.
Permanent Magnets
Overview
Recall that a magnet is a material or object that generates a magnetic field. This magnetic field is invisible but is responsible for the most notable holding of a magnet: a force that pulls on other ferromagnetic materials, such every bit iron, and attracts or repels other magnets.
Types of Magnets
A permanent magnet is an object fabricated from a textile that is magnetized and creates its ain persistent magnetic field. An everyday example is a refrigerator magnet used to hold notes on a refrigerator door. Materials that can be magnetized, which are as well the ones that are strongly attracted to a magnet, are called ferromagnetic. These include iron, nickel, cobalt, some alloys of rare earth metals, and some naturally occurring minerals such as lodestone. Although ferromagnetic materials are the only ones attracted to a magnet strongly enough to be ordinarily considered magnetic, all other substances answer weakly to a magnetic field, past ane of several other types of magnetism. The counterexample to a permanent magnet is an electromagnet, which only becomes magnetized when an electrical current flows through information technology.
Instance of a Permanent Magnet: An case of a permanent magnet: a "horseshoe magnet" made of alnico, an atomic number 26 alloy. The magnet is made in the shape of a horseshoe to bring the 2 magnetic poles shut to each other, to create a stiff magnetic field there that tin pick upwards heavy pieces of atomic number 26.
Polarity
All magnets take two poles, one called the due north pole and one called the south pole. Like poles repel and unlike polls attract (in illustration to positive and negative charges in electrostatics). North and s poles e'er exist in pairs (at that place are no magnetic monopoles in nature), and then if ane were to split a permanent magnet in half, two smaller magnets would be created, each with a northward pole and south pole.
North and S Poles E'er Come in Pairs: Due north and south poles always occur in pairs. Attempts to split up them consequence in more pairs of poles. If we continue to separate the magnet, we will eventually get down to an iron cantlet with a north pole and a southward pole—these, too, cannot exist separated.
Manufacturing Permanent Magnets
Ferromagnetic materials tin exist divided into magnetically "soft" materials like annealed iron, which can exist magnetized simply do not tend to stay magnetized, and magnetically "hard" materials, which do. Permanent magnets are made from "hard" ferromagnetic materials such as alcino and ferrite that are subjected to special processing in a powerful magnetic field during manufacture, to marshal their internal microcrystalline structure, making them very hard to demagnetize.
When a magnet is brought virtually a previously unmagnetized ferromagnetic fabric, information technology causes local magnetization of the material with unlike poles closest. (This results in the attraction of the previously unmagnetized material to the magnet. ) On the microscopic scale, in that location are regions in the unmagnetized ferromagnetic textile that act like small bar magnets. In each region the poles of private atoms are aligned. Nonetheless, before magnetization these regions are small and randomly oriented throughout the unmagnetized ferromagnetic objects, so there is no internet magnetic field. In response to an external magnetic field like the one practical in the above effigy, these regions grow and become aligned. This organisation tin can become permanent when the ferromagnetic cloth is heated and then cooled.
Making a Ferromagnet: An unmagnetized piece of iron is placed between ii magnets, heated, and then cooled, or simply tapped when common cold. The iron becomes a permanent magnet with the poles aligned as shown: its due south pole is adjacent to the north pole of the original magnet, and its due north pole is adjacent to the south pole of the original magnet. Note that there are bonny forces between the magnets.
Magnetic Field Lines
Magnetic field lines are useful for visually representing the strength and direction of the magnetic field.
Learning Objectives
Chronicle the strength of the magnetic field with the density of the magnetic field lines
Key Takeaways
Key Points
- The magnetic field direction is the same direction a compass needle points, which is tangent to the magnetic field line at any given indicate.
- The strength of the B-field is inversely proportional to the distance between field lines. It is exactly proportional to the number of lines per unit area perpendicular to the lines.
- A magnetic field line tin never cantankerous another field line. The magnetic field is unique at every betoken in space.
- Magnetic field lines are continuous and unbroken, forming closed loops. Magnetic field lines are defined to begin on the north pole of a magnet and terminate on the south pole.
Key Terms
- B-field: A synonym for the magnetic field.
- magnetic field lines: A graphical representation of the magnitude and the direction of a magnetic field.
Magnetic Field Lines
Einstein is said to have been fascinated by a compass every bit a kid, perhaps musing on how the needle felt a force without direct concrete contact. His ability to think securely and clearly most activeness at a distance, particularly for gravitational, electric, and magnetic forces, subsequently enabled him to create his revolutionary theory of relativity. Since magnetic forces human action at a altitude, nosotros define a magnetic field to stand for magnetic forces. A pictorial representation of magnetic field lines is very useful in visualizing the strength and direction of the magnetic field. The direction of magnetic field lines is divers to be the direction in which the northward end of a compass needle points. The magnetic field is traditionally called the B-field.
Visualizing Magnetic Field Lines: Magnetic field lines are defined to have the direction that a small compass points when placed at a location. (A) If small compasses are used to map the magnetic field around a bar magnet, they will signal in the directions shown: away from the north pole of the magnet, toward the south pole of the magnet (think that Earth's due north magnetic pole is really a south pole in terms of definitions of poles on a bar magnet. ) (B) Connecting the arrows gives continuous magnetic field lines. The strength of the field is proportional to the closeness (or density) of the lines. (C) If the interior of the magnet could be probed, the field lines would exist establish to form continuous closed loops.
Mapping the magnetic field of an object is uncomplicated in principle. Showtime, measure out the force and direction of the magnetic field at a large number of locations (or at every indicate in infinite). And so, mark each location with an pointer (chosen a vector ) pointing in the management of the local magnetic field with its magnitude proportional to the force of the magnetic field (producing a vector field). You tin "connect" the arrows to form magnetic field lines. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength.
Magnetic field lines are like the contour lines (constant distance) on a topographic map in that they represent something continuous, and a different mapping scale would testify more or fewer lines. An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using unproblematic concepts such as the "number" of field lines through a surface. These concepts can be quickly translated to their mathematical form. For example, the number of field lines through a given surface is the surface integral of the magnetic field.
Bar Magnet and Magnetic Field Lines: The direction of magnetic field lines represented by the alignment of iron filings sprinkled on paper placed higher up a bar magnet.
Diverse phenomena accept the upshot of "displaying" magnetic field lines every bit though the field lines are concrete phenomena. For example, iron filings placed in a magnetic field line upward to grade lines that correspond to "field lines. " Magnetic fields' lines are also visually displayed in polar auroras, in which plasma particle dipole interactions create visible streaks of lite that line up with the local management of Earth's magnetic field.
Small compasses used to test a magnetic field volition not disturb it. (This is coordinating to the fashion we tested electric fields with a modest test charge. In both cases, the fields represent only the object creating them and not the probe testing them. ) Figure 15051 shows how the magnetic field appears for a current loop and a long straight wire, as could be explored with modest compasses. A small compass placed in these fields will align itself parallel to the field line at its location, with its north pole pointing in the direction of B. Notation the symbols used for field into and out of the paper. We'll explore the consequences of these various sources of magnetic fields in further sections.
Mapping Magnetic Field Lines: Pocket-sized compasses could be used to map the fields shown here. (A) The magnetic field of a circular current loop is similar to that of a bar magnet. (B) A long and straight wire creates a field with magnetic field lines forming circular loops. (C) When the wire is in the plane of the newspaper, the field is perpendicular to the paper. Note that the symbols used for the field pointing inward (similar the tail of an arrow) and the field pointing outward (similar the tip of an arrow).
Extensive exploration of magnetic fields has revealed a number of hard-and-fast rules. Nosotros use magnetic field lines to represent the field (the lines are a pictorial tool, not a physical entity in and of themselves). The backdrop of magnetic field lines tin can exist summarized past these rules:
- The management of the magnetic field is tangent to the field line at any signal in space. A minor compass will indicate in the direction of the field line.
- The force of the field is proportional to the closeness of the lines. It is exactly proportional to the number of lines per unit of measurement surface area perpendicular to the lines (called the areal density).
- Magnetic field lines can never cross, meaning that the field is unique at whatever point in space.
- Magnetic field lines are continuous, forming closed loops without beginning or end. They get from the north pole to the south pole.
The terminal property is related to the fact that the n and south poles cannot be separated. It is a singled-out departure from electric field lines, which begin and terminate on the positive and negative charges. If magnetic monopoles existed, then magnetic field lines would begin and end on them.
Geomagnetism
Earth's magnetic field is acquired by electric currents in the molten outer core and varies with time.
Learning Objectives
Explain the origin of the World's magnetic field and its importance for the life on World
Key Takeaways
Key Points
- Earth is largely protected from the solar air current, a stream of energetic charged particles emanating from the lord's day, past its magnetic field. These particles would strip away the ozone layer, which protects Globe from harmful ultraviolet rays.
- Earth'due south magnetic field is generated past a feedback loop in the liquid outer cadre: Current loops generate magnetic fields; a irresolute magnetic field generates an electric field; and the electric and magnetic fields exert a force on the charges that are flowing in currents (the Lorentz forcefulness).
- The geomagnetic field varies with time. Currents in the ionosphere and magnetosphere crusade changes over short time scales, while dramatic geomagnetic reversald (where the North and Southward poles switch locations) occur at plain random intervals ranging from 0.i to l one thousand thousand years.
Key Terms
- dynamo: A mechanism past which a celestial body such as Earth or a star generates a magnetic field over astronomical timescales via a rotating, convecting, and electrically conducting fluid.
Geomagnetism
The Structure of Globe'southward Magnetic Field
Earth is largely protected from the solar wind, a stream of energetic charged particles emanating from the sunday, by its magnetic field, which deflects about of the charged particles. These particles would strip abroad the ozone layer, which protects World from harmful ultraviolet rays. The region above the ionosphere, and extending several tens of thousands of kilometers into space, is called the magnetosphere. This region protects Earth from cosmic rays that would strip abroad the upper temper, including the ozone layer that protects our planet from harmful ultraviolet radiation. The magnetic field forcefulness ranges from approximately 25 to 65 microteslas (0.25 to 0.65 1000; by comparing, a strong refrigerator magnet has a field of about 100 G). The intensity of the field is greatest near the poles and weaker near the equator. An isodynamic nautical chart of Earth'southward magnetic field, shows a minimum intensity over Southward America while in that location are maxima over northern Canada, Siberia, and the declension of Antarctica south of Australia. Virtually the surface of Earth, its magnetic field can be closely approximated by the field of a magnetic dipole positioned at the center of Earth and tilted at an bending of about 10° with respect to the rotational axis of Earth.
Concrete Origin
Earth's magnetic field is mostly caused by electric currents in the liquid outer core, which is composed of highly conductive molten iron. A magnetic field is generated by a feedback loop: Current loops generate magnetic fields (Ampère'south law); a irresolute magnetic field generates an electrical field (Faraday's law); and the electric and magnetic fields exert a force on the charges that are flowing in currents (the Lorentz forcefulness). These effects tin be combined into a partial differential equation called the magnetic induction equation:
[latex]\frac{\partial \mathbf{\text{B}}}{\partial \text{t}} = \eta \mathbf{\nabla}^2 \mathbf{\text{B}} + \mathbf{\nabla}\times (\mathbf{\text{u}} \times \mathbf{\text{B}})[/latex]
In this equation u is the velocity of the fluid, B is the magnetic field, and eta is the magnetic diffusivity. The beginning term on the right hand side of the induction equation is a diffusion term. If Earth'due south dynamo shut off, the dipole part would disappear in a few tens of thousands of years. The motion of the molten outer fe core is sustained by convection, or motion driven by buoyancy. The temperature increases toward the heart of Earth, and the college temperature of the fluid lower down makes it buoyant. The Coriolis issue, acquired by the overall planetary rotation, tends to organize the flow into rolls aligned along the north-southward polar axis.
Origin of Earth's Magnetic Field: A schematic illustrating the human relationship between motion of conducting fluid, organized into rolls by the Coriolis strength, and the magnetic field the move generates.
Electric currents induced in the ionosphere generate magnetic fields (ionospheric dynamo region). Such a field is e'er generated almost where the atmosphere is closest to the sun, causing daily alterations that can deflect surface magnetic fields by as much as one caste. Typical daily variations of field strength are most 25 nanoteslas (nT), with variations over a few seconds of typically around 1 nT.
Time Variations
The geomagnetic field changes on time scales from milliseconds to millions of years. Shorter time scales generally arise from currents in the ionosphere (ionospheric dynamo region) and magnetosphere, and some changes tin can be traced to geomagnetic storms or daily variations in currents. Changes over fourth dimension scales of a yr or more than mostly reverberate changes in World'due south interior, peculiarly the atomic number 26-rich core. Ofttimes, World's magnetosphere is hit by solar flares causing geomagnetic storms, provoking displays of aurorae. At present, the overall geomagnetic field is becoming weaker; the present strong deterioration corresponds to a 10 to 15 percent decline over the concluding 150 years and has accelerated in the past several years. Geomagnetic intensity has declined almost continuously from a maximum 35 percentage to a higher place the modernistic value accomplished approximately ii,000 years ago. World's magnetic North Pole is drifting from northern Canada toward Siberia with a presently accelerating rate—10 km per yr at the outset of the 20th century, upward to 40 km per year in 2003, and since so has merely accelerated.
Although Earth's field is generally well approximated by a magnetic dipole with its axis near the rotational axis, there are occasional dramatic events where the N and South geomagnetic poles merchandise places. These events are called geomagnetic reversals. Evidence for these events can exist establish worldwide in basalts, sediment cores taken from the ocean floors, and seafloor magnetic anomalies. Reversals occur at plain random intervals ranging from less than 0.1 one thousand thousand years to as much every bit l million years. The nearly recent such event, called the Brunhes-Matuyama reversal, occurred almost 780,000 years ago.
Source: https://courses.lumenlearning.com/boundless-physics/chapter/magnetism-and-magnetic-fields/
Posted by: santanafaccons.blogspot.com
0 Response to "How To Find The Magnitude Of The Magnetic Field"
Post a Comment