Can a Working Scale Model of the Earth's Magnetic Field Be Made
Electric Currents and Magnetic Fields
An electric current will produce a magnetic field, which tin can be visualized every bit a serial of circular field lines effectually a wire segment.
Learning Objectives
Describe shape of a magnetic field produced by an electric current flowing through a wire
Key Takeaways
Cardinal Points
- A wire carrying electrical electric current volition produce a magnetic field with closed field lines surrounding the wire.
- Some other version of the right mitt rules can exist used to determine the magnetic field direction from a current—betoken the thumb in the direction of the current, and the fingers curlicue in the direction of the magnetic field loops created by information technology. See.
- The British indian ocean territory-Savart Constabulary can be used to determine the magnetic field force from a electric current segment. For the simple case of an infinite directly current-carrying wire it is reduced to the form [latex]\text{B}=\frac{\mu _{0}\text{I}}{2\pi \text{r}}[/latex].
- A more fundamental law than the Biot-Savart police force is Ampere 's Police force, which relates magnetic field and current in a general way. It is written in integral form equally [latex]\oint \text{B}\cdot \text{d}\mathscr{\text{l}}=\mu _{0}\text{I}_{\text{enc}}[/latex], where Ienc is the enclosed current and μ0 is a abiding.
- A electric 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{50}}\text{sin}\theta[/latex], where ℓ is the length of the wire, i is the current, and θ is the angle betwixt the current management and the magnetic field.
Key Terms
- Biot-Savart Law: An equation that describes the magnetic field generated by an electric current. It relates the magnetic field to the magnitude, management, length, and proximity of the electric current. The law is valid in the magnetostatic approximation, and is consistent with both Ampère'south circuital police force and Gauss'due south constabulary for magnetism.
- Ampere's Law: An equation that relates magnetic fields to electric currents that produce them. Using Ampere'south police, i can determine the magnetic field associated with a given current or current associated with a given magnetic field, providing at that place is no time changing electric field nowadays.
Electrical Current and Magnetic Fields
Electric current produces a magnetic field. This magnetic field can be 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 past a long direct current-carrying wire in. Hall probes can decide the magnitude of the field. Another version of the right hand dominion emerges from this exploration and is valid for whatsoever electric current segment—point the thumb in the direction of the current, and the fingers coil in the management of the magnetic field loops created by it.
Magnetic Field Generated by Current: (a) Compasses placed near a long directly current-carrying wire indicate that field lines class circular loops centered on the wire. (b) Right hand dominion 2 states that, if the right hand thumb points in the management of the current, the fingers curl in the direction of the field. This rule is consistent with the field mapped for the long straight wire and is valid for any current segment.
Magnitude of Magnetic Field from Current
The equation for the magnetic field force (magnitude) produced past a long straight current-carrying wire is:
[latex]\text{B}=\frac{\mu _{0}\text{I}}{two\pi \text{r}}[/latex]
For a long direct wire where I is the electric current, r is the shortest distance to the wire, and the constant 0=4π10−vii T⋅yard/A is the permeability of free space. (μ0 is ane of the basic constants in nature, related to the speed of light. ) Since the wire is very long, the magnitude of the field depends only on distance 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 straight wire has more implications than one might first suspect. Each segment of electric current produces a magnetic field like that of a long straight wire, and the full field of any shape current is the vector sum of the fields due to each segment. The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law. Integral calculus is needed to sum the field for an capricious shape current. The British indian ocean territory-Savart law is written in its complete form as:
[latex]\text{B}=\frac{\mu _{\text{o}}\text{I}}{iv\pi}\int \frac{\text{d}\mathscr{\text{l}}\times \chapeau{\text{r}}}{\text{r}^{2}}[/latex]
where the integral sums over the wire length where vector dℓ is the direction of the electric 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 direction of r. The reader may apply the simplifications in calculating the magnetic field from an infinite straight wire as higher up and see that the British indian ocean territory-Savart constabulary reduces to the first, simpler equation.
Ampere's Constabulary
A more fundamental law than the Biot-Savart police force is Ampere'due south Law, which relates magnetic field and current in a general way. In SI units, the integral class of the original Ampere'due south circuital law is a line integral of the magnetic field around some airtight bend C (arbitrary but must be closed). The curve C in turn bounds both a surface Southward through which the electric electric current passes through (again capricious but non airtight—since no iii-dimensional volume is enclosed by S), and encloses the electric current. You tin can think of the "surface" every bit the cantankerous-sectional expanse of a wire carrying 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 can be written in a number of forms, one of which is given below.
[latex]\oint \text{B}\cdot \text{d}\mathscr{\text{50}}=\mu _{0}\iint_{\text{Due south}}^{ } \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 foursquare meter, Am-ii) over the cantankerous section area of the wire (which is equal to the permeability constant times the enclosed current Ienc) . Ampere's constabulary is always valid for steady currents and tin be used to calculate the B-field for certain highly symmetric situations such equally an space wire or an infinite solenoid. Ampere'southward Law is also a component of Maxwell's Equations.
Force on a Current-Carrying Wire
The forcefulness 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 current-carrying wire feels a force 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 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 force exerted on a unmarried charge q is
Forcefulness on a Electric current-Carrying Wire: The right manus dominion can exist used to make up one's mind the direction of the force on a current-conveying wire placed in an external magnetic field.
[latex]\text{F}=\text{qvBsin}\theta[/latex]
Then, for N charges where
[latex]\text{Northward}=\text{n}\mathscr{\text{fifty}}\text{A}[/latex]
the strength 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 manus rule tin can give you the direction of the strength on the wire, as seen in the above effigy. Annotation that the B-field in this case is the external field.
Permanent Magnets
Permanent magnets are objects made from ferromagnetic fabric that produce a persistent magnetic field.
Learning Objectives
Give examples and counterexamples of permanent magnets
Key Takeaways
Fundamental Points
- Permanent magnets are objects fabricated from magnetized cloth and produce continual magnetic fields. Everyday examples include refrigerator magnets used to agree notes on a fridge door.
- Materials that tin can exist magnetized, which are also the ones that are strongly attracted to a magnet, are called ferromagnetic. Examples of these materials include fe, nickel, and cobalt.
- The counterexample to a permanent magnet is an electromagnet, which merely becomes magnetized when an electrical current flows through it.
- Magnets ever have a due north pole and a south pole, so if i were to split a permanent magnet in half, ii smaller magnets would be created, each with a due north pole and south pole.
- Permanent magnets are made from ferromagnetic materials that are exposed to a potent external magnetic field and heated to marshal their internal microcrystalline structure, making them very hard to demagnetize.
Key Terms
- permanent magnet: A material, or piece of such material, which retains its magnetism even when not subjected to any external magnetic fields.
- ferromagnetic: Of a cloth, such as iron or nickel, that is hands magnetized.
- electromagnet: A magnet which attracts metals only when electrically activated.
Permanent Magnets
Overview
Think that a magnet is a material or object that generates a magnetic field. This magnetic field is invisible but is responsible for the nigh notable property 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 made from a textile that is magnetized and creates its own persistent magnetic field. An everyday example is a fridge magnet used to concur notes on a refrigerator door. Materials that tin can exist magnetized, which are likewise the ones that are strongly attracted to a magnet, are chosen ferromagnetic. These include fe, nickel, cobalt, some alloys of rare earth metals, and some naturally occurring minerals such every bit lodestone. Although ferromagnetic materials are the only ones attracted to a magnet strongly plenty to be commonly considered magnetic, all other substances reply weakly to a magnetic field, past one of several other types of magnetism. The counterexample to a permanent magnet is an electromagnet, which only becomes magnetized when an electrical electric current flows through it.
Example of a Permanent Magnet: An example of a permanent magnet: a "horseshoe magnet" made of alnico, an iron alloy. The magnet is fabricated in the shape of a horseshoe to bring the two magnetic poles close to each other, to create a strong magnetic field at that place that can choice up heavy pieces of iron.
Polarity
All magnets have two poles, i chosen the north pole and 1 called the south pole. Like poles repel and unlike polls concenter (in analogy to positive and negative charges in electrostatics). N and south poles ever exist in pairs (there are no magnetic monopoles in nature), and then if i were to split a permanent magnet in one-half, two smaller magnets would exist created, each with a due north pole and southward pole.
Northward and S Poles Ever Come in Pairs: North and south poles e'er occur in pairs. Attempts to divide them effect in more pairs of poles. If we continue to split the magnet, we will eventually get down to an fe atom with a north pole and a south pole—these, as well, cannot be separated.
Manufacturing Permanent Magnets
Ferromagnetic materials tin can be divided into magnetically "soft" materials like annealed iron, which can be magnetized but exercise non tend to stay magnetized, and magnetically "hard" materials, which do. Permanent magnets are made from "hard" ferromagnetic materials such equally alcino and ferrite that are subjected to special processing in a powerful magnetic field during manufacture, to align their internal microcrystalline construction, making them very hard to demagnetize.
When a magnet is brought near a previously unmagnetized ferromagnetic fabric, it causes local magnetization of the material with unlike poles closest. (This results in the attraction of the previously unmagnetized cloth to the magnet. ) On the microscopic calibration, in that location are regions in the unmagnetized ferromagnetic material that act similar small bar magnets. In each region the poles of individual atoms are aligned. However, before magnetization these regions are small and randomly oriented throughout the unmagnetized ferromagnetic objects, and then at that place is no net magnetic field. In response to an external magnetic field similar the one applied in the above figure, these regions abound and become aligned. This system can become permanent when the ferromagnetic cloth is heated and and then cooled.
Making a Ferromagnet: An unmagnetized piece of atomic number 26 is placed between two magnets, heated, so cooled, or merely tapped when cold. The iron becomes a permanent magnet with the poles aligned every bit shown: its south pole is adjacent to the n pole of the original magnet, and its north pole is adjacent to the due 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
Relate the strength of the magnetic field with the density of the magnetic field lines
Key Takeaways
Primal Points
- The magnetic field direction is the same direction a compass needle points, which is tangent to the magnetic field line at whatever given bespeak.
- The strength of the B-field is inversely proportional to the distance betwixt field lines. It is exactly proportional to the number of lines per unit area perpendicular to the lines.
- A magnetic field line tin can never cross some other field line. The magnetic field is unique at every point in infinite.
- Magnetic field lines are continuous and unbroken, forming airtight loops. Magnetic field lines are defined to begin on the n pole of a magnet and finish 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 management of a magnetic field.
Magnetic Field Lines
Einstein is said to accept been fascinated past a compass as a child, perhaps musing on how the needle felt a force without directly physical contact. His ability to remember deeply and clearly nearly action at a distance, particularly for gravitational, electric, and magnetic forces, later enabled him to create his revolutionary theory of relativity. Since magnetic forces human action at a distance, we ascertain a magnetic field to represent magnetic forces. A pictorial representation of magnetic field lines is very useful in visualizing the strength and management of the magnetic field. The direction of magnetic field lines is divers to exist 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 effectually a bar magnet, they volition betoken in the directions shown: abroad from the northward pole of the magnet, toward the south pole of the magnet (call up that Earth's 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 force 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 be found to grade continuous closed loops.
Mapping the magnetic field of an object is uncomplicated in principle. Start, measure the force and direction of the magnetic field at a big number of locations (or at every point in space). And so, mark each location with an arrow (called a vector ) pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field (producing a vector field). Yous can "connect" the arrows to form magnetic field lines. The direction of the magnetic field at whatsoever point is parallel to the management of nearby field lines, and the local density of field lines can be made proportional to its forcefulness.
Magnetic field lines are like the contour lines (constant altitude) on a topographic map in that they represent something continuous, and a different mapping scale would show more than 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 simple concepts such as the "number" of field lines through a surface. These concepts can exist quickly translated to their mathematical course. For case, 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 above a bar magnet.
Various phenomena accept the effect of "displaying" magnetic field lines every bit though the field lines are physical phenomena. For example, iron filings placed in a magnetic field line up to form lines that correspond to "field lines. " Magnetic fields' lines are likewise visually displayed in polar auroras, in which plasma particle dipole interactions create visible streaks of lite that line upwards with the local direction of Earth'due south magnetic field.
Small compasses used to test a magnetic field will not disturb it. (This is analogous to the way we tested electric fields with a small test accuse. In both cases, the fields stand for just 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 exist explored with small-scale compasses. A pocket-size compass placed in these fields will align itself parallel to the field line at its location, with its northward pole pointing in the direction of B. Note 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 electric current loop is similar to that of a bar magnet. (B) A long and directly wire creates a field with magnetic field lines forming circular loops. (C) When the wire is in the aeroplane of the newspaper, the field is perpendicular to the paper. Note that the symbols used for the field pointing in (similar the tail of an arrow) and the field pointing outward (like the tip of an arrow).
Extensive exploration of magnetic fields has revealed a number of hard-and-fast rules. We employ magnetic field lines to represent the field (the lines are a pictorial tool, not a physical entity in and of themselves). The properties of magnetic field lines tin can exist summarized by these rules:
- The direction of the magnetic field is tangent to the field line at whatsoever point in infinite. A small compass will point in the direction of the field line.
- The strength of the field is proportional to the closeness of the lines. It is exactly proportional to the number of lines per unit of measurement area perpendicular to the lines (called the areal density).
- Magnetic field lines tin never cantankerous, significant that the field is unique at any point in infinite.
- Magnetic field lines are continuous, forming closed loops without outset or end. They go from the north pole to the due south pole.
The last property is related to the fact that the north and southward poles cannot exist separated. It is a distinct difference from electric field lines, which begin and stop 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 caused by electric currents in the molten outer core and varies with time.
Learning Objectives
Explicate the origin of the Earth'due south magnetic field and its importance for the life on Earth
Key Takeaways
Key Points
- Globe is largely protected from the solar air current, a stream of energetic charged particles emanating from the sun, by its magnetic field. These particles would strip away the ozone layer, which protects World from harmful ultraviolet rays.
- Earth's magnetic field is generated by a feedback loop in the liquid outer core: Current loops generate magnetic fields; a changing magnetic field generates an electrical field; and the electric and magnetic fields exert a strength on the charges that are flowing in currents (the Lorentz force).
- The geomagnetic field varies with time. Currents in the ionosphere and magnetosphere cause changes over short fourth dimension scales, while dramatic geomagnetic reversald (where the Due north and South poles switch locations) occur at apparently random intervals ranging from 0.1 to 50 meg years.
Key Terms
- dynamo: A machinery by which a celestial body such every bit Earth or a star generates a magnetic field over astronomical timescales via a rotating, convecting, and electrically conducting fluid.
Geomagnetism
The Structure of Earth's Magnetic Field
Earth is largely protected from the solar wind, a stream of energetic charged particles emanating from the sun, by its magnetic field, which deflects most of the charged particles. These particles would strip away the ozone layer, which protects Globe from harmful ultraviolet rays. The region higher up the ionosphere, and extending several tens of thousands of kilometers into infinite, is called the magnetosphere. This region protects Earth from cosmic rays that would strip away the upper atmosphere, including the ozone layer that protects our planet from harmful ultraviolet radiation. The magnetic field strength ranges from approximately 25 to 65 microteslas (0.25 to 0.65 G; by comparison, a stiff refrigerator magnet has a field of nearly 100 One thousand). 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 South America while in that location are maxima over northern Canada, Siberia, and the declension of Antarctica south of Australia. Near the surface of Earth, its magnetic field can be closely approximated by the field of a magnetic dipole positioned at the center of World and tilted at an angle of almost 10° with respect to the rotational centrality of World.
Physical Origin
Globe'due south magnetic field is more often than not caused by electric currents in the liquid outer cadre, which is composed of highly conductive molten iron. A magnetic field is generated past a feedback loop: Current loops generate magnetic fields (Ampère's law); a changing 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 can be combined into a partial differential equation called the magnetic induction equation:
[latex]\frac{\partial \mathbf{\text{B}}}{\fractional \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 commencement term on the correct hand side of the consecration equation is a diffusion term. If Globe's dynamo close off, the dipole office would disappear in a few tens of thousands of years. The motion of the molten outer iron core is sustained past convection, or motility driven by buoyancy. The temperature increases toward the center of World, and the higher temperature of the fluid lower down makes it buoyant. The Coriolis effect, caused by the overall planetary rotation, tends to organize the flow into rolls aligned along the north-south polar axis.
Origin of World'south Magnetic Field: A schematic illustrating the relationship betwixt motion of conducting fluid, organized into rolls by the Coriolis force, and the magnetic field the motion generates.
Electric currents induced in the ionosphere generate magnetic fields (ionospheric dynamo region). Such a field is always generated virtually where the atmosphere is closest to the dominicus, causing daily alterations that can deflect surface magnetic fields by as much every bit i degree. Typical daily variations of field strength are virtually 25 nanoteslas (nT), with variations over a few seconds of typically effectually 1 nT.
Time Variations
The geomagnetic field changes on time scales from milliseconds to millions of years. Shorter time scales mostly 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 time scales of a year or more generally reflect changes in Earth'due south interior, especially the iron-rich core. Oftentimes, Earth'southward magnetosphere is hit by solar flares causing geomagnetic storms, provoking displays of aurorae. Now, the overall geomagnetic field is becoming weaker; the nowadays strong deterioration corresponds to a 10 to xv pct decline over the concluding 150 years and has accelerated in the past several years. Geomagnetic intensity has declined almost continuously from a maximum 35 per centum above the modern value achieved approximately ii,000 years ago. Globe's magnetic North Pole is drifting from northern Canada toward Siberia with a before long accelerating rate—10 km per twelvemonth at the get-go of the 20th century, upward to 40 km per year in 2003, and since then has but accelerated.
Although Globe's field is generally well approximated by a magnetic dipole with its axis almost the rotational axis, at that place are occasional dramatic events where the North and South geomagnetic poles trade places. These events are chosen geomagnetic reversals. Evidence for these events tin can be found worldwide in basalts, sediment cores taken from the ocean floors, and seafloor magnetic anomalies. Reversals occur at apparently random intervals ranging from less than 0.one million years to as much as 50 meg years. The most recent such event, chosen the Brunhes-Matuyama reversal, occurred well-nigh 780,000 years ago.
Source: https://courses.lumenlearning.com/boundless-physics/chapter/magnetism-and-magnetic-fields/
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