Explain How You Can Determine the Direction of a Magnetic Field Produced by a Wire

Learning Objectives

Past the end of this section, y'all will be able to:

• Calculate current that produces a magnetic field.
• Use the right hand dominion ii to determine the direction of current or the direction of magnetic field loops.

How much current is needed to produce a significant magnetic field, perhaps every bit strong every bit the Earth's field? Surveyors will tell you that overhead electric power lines create magnetic fields that interfere with their compass readings. Indeed, when Oersted discovered in 1820 that a current in a wire affected a compass needle, he was not dealing with extremely large currents. How does the shape of wires carrying current affect the shape of the magnetic field created? We noted before that a electric current loop created a magnetic field similar to that of a bar magnet, but what almost a straight wire or a toroid (doughnut)? How is the direction of a current-created field related to the direction of the electric current? Answers to these questions are explored in this section, together with a brief word of the law governing the fields created by currents.

Magnetic Field Created past a Long Straight Current-Conveying Wire: Right Hand Rule two

Magnetic fields take both direction and magnitude. As noted before, one mode to explore the direction of a magnetic field is with compasses, as shown for a long direct current-conveying wire in Figure 1. Hall probes tin can decide the magnitude of the field. The field around a long straight wire is establish to be in circular loops. The right manus dominion 2 (RHR-2) emerges from this exploration and is valid for any current segment—indicate the pollex in the direction of the current, and the fingers roll in the management of the magnetic field loops created by it.

Effigy ane. (a) Compasses placed virtually a long straight electric current-conveying wire point that field lines form circular loops centered on the wire. (b) Right hand dominion 2 states that, if the correct hand pollex points in the direction of the current, the fingers curl in the direction of the field. This rule is consistent with the field mapped for the long directly wire and is valid for any current segment.

The magnetic field force (magnitude) produced by a long straight electric current-carrying wire is constitute by experiment to exist

[latex]B=\frac{{\mu}_{0}I}{two\pi r}\left(\text{long straight wire}\right)\\[/latex],

where I is the current, r is the shortest distance to the wire, and the constant [latex]{\mu }_{0}=four\pi \times ten^{-7}\text{T}\cdot\text{ m/A}\\[/latex] is the permeability of free space. (μ ₀ is i of the basic constants in nature. We will see later that μ ₀ is related to the speed of low-cal.) Since the wire is very long, the magnitude of the field depends only on distance from the wire r, not on position along the wire.

Example 1. Calculating Current that Produces a Magnetic Field

Discover the current in a long straight wire that would produce a magnetic field twice the strength of the Earth'south at a distance of 5.0 cm from the wire.

Strategy

The Earth's field is about 5.0 × ten⁻⁵T, then here B due to the wire is taken to be 1 . 0 × ten ^{− 4} T . The equation [latex]B=\frac{\mu_{0}I}{two\pi r}\\[/latex] tin can be used to discover I, since all other quantities are known.

Solution

Solving for I and entering known values gives

[latex]\begin{array}{lll}I& =& \frac{2\pi rB}{\mu _{0}}=\frac{2\pi\left(5.0\times 10^{-2}\text{ m}\right)\left(ane.0\times 10^{-4}\text{ T}\right)}{iv\pi \times x^{-7}\text{ T}\cdot\text{m/A}}\\ & =& 25\text{ A}\stop{array}\\[/latex]

Discussion

So a moderately large current produces a pregnant magnetic field at a distance of 5.0 cm from a long direct wire. Note that the answer is stated to only 2 digits, since the Earth's field is specified to only two digits in this example.

Ampere's Law and Others

The magnetic field of a long straight wire has more than implications than you might at first suspect. Each segment of current produces a magnetic field like that of a long direct wire, and the total field of whatsoever 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 chosen the Biot-Savart law. Integral calculus is needed to sum the field for an arbitrary shape current. This results in a more complete law, called Ampere'south law, which relates magnetic field and current in a general way. Ampere'southward police in turn is a office of Maxwell's equations, which give a complete theory of all electromagnetic phenomena. Considerations of how Maxwell's equations appear to dissimilar observers led to the mod theory of relativity, and the realization that electric and magnetic fields are unlike manifestations of the same thing. Most of this is beyond the scope of this text in both mathematical level, requiring calculus, and in the amount of space that can be devoted to information technology. But for the interested student, and particularly for those who continue in physics, technology, or similar pursuits, delving into these matters further will reveal descriptions of nature that are elegant besides equally profound. In this text, nosotros shall go along the general features in mind, such as RHR-2 and the rules for magnetic field lines listed in Magnetic Fields and Magnetic Field Lines, while concentrating on the fields created in certain important situations.

Making Connections: Relativity

Hearing all we practise about Einstein, we sometimes get the impression that he invented relativity out of nothing. On the contrary, one of Einstein'south motivations was to solve difficulties in knowing how different observers see magnetic and electric fields.

Magnetic Field Produced by a Electric current-Conveying Circular Loop

The magnetic field near a current-carrying loop of wire is shown in Effigy 2. Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex. RHR-ii tin be used to give the direction of the field most the loop, but mapping with compasses and the rules nearly field lines given in Magnetic Fields and Magnetic Field Lines are needed for more detail. There is a simple formula for the magnetic field strength at the center of a round loop. It is

[latex]B=\frac{\mu_{0}I}{2R}\left(\text{at center of loop}\right)\\[/latex],

where R is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid just at the centre of a circular loop of wire. The similarity of the equations does signal that similar field forcefulness can be obtained at the eye of a loop. One way to get a larger field is to accept N loops; then, the field is B=Nμ ₀ I/(iiR). Note that the larger the loop, the smaller the field at its center, because the current is farther away.

Figure 2. (a) RHR-two gives the direction of the magnetic field inside and outside a current-carrying loop. (b) More detailed mapping with compasses or with a Hall probe completes the picture. The field is similar to that of a bar magnet.

Magnetic Field Produced past a Electric current-Conveying Solenoid

A solenoid is a long coil of wire (with many turns or loops, every bit opposed to a apartment loop). Because of its shape, the field inside a solenoid tin can be very compatible, and also very strong. The field but outside the coils is well-nigh zero. Figure iii shows how the field looks and how its direction is given by RHR-2.

Effigy iii. (a) Considering of its shape, the field inside a solenoid of length l is remarkably uniform in magnitude and direction, as indicated past the straight and uniformly spaced field lines. The field outside the coils is well-nigh naught. (b) This cutaway shows the magnetic field generated by the electric current in the solenoid.

The magnetic field inside of a current-carrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and modify management. The field outside has similar complexities to apartment loops and bar magnets, but the magnetic field strength inside a solenoid is only

[latex]B={\mu }_{0}nI\left(\text{inside a solenoid}\right)\\[/latex],

where n is the number of loops per unit of measurement length of the solenoid (northward=N/50, with Northward being the number of loops and l the length). Annotation that B is the field strength anywhere in the uniform region of the interior and not only at the eye. Large uniform fields spread over a large volume are possible with solenoids, as Example 2 implies.

Example 2. Calculating Field Force inside a Solenoid

What is the field within a 2.00-yard-long solenoid that has 2000 loops and carries a 1600-A current?

Strategy

To detect the field strength inside a solenoid, we use [latex]B={\mu }_{0}nI\\[/latex]. Beginning, we note the number of loops per unit length is

[latex]n=\frac{North}{fifty}=\frac{2000}{2.00\text{ m}}=m\text{ m}^{-i}=10{\text{ cm}}^{-1}\\[/latex].

Solution Substituting known values gives

[latex]\begin{array}{lll}B & =& {\mu}_{0}nI=\left(four\pi \times x^{-7}\text{ T}\cdot\text{chiliad/A}\right)\left(thou\text{ m}^{-1}\right)\left(1600\text{ A}\correct)\\ & =& 2.01\text{ T}\cease{array}\\[/latex]

Discussion

This is a large field force that could exist established over a large-diameter solenoid, such equally in medical uses of magnetic resonance imaging (MRI). The very large electric current is an indication that the fields of this strength are not easily achieved, however. Such a large current through grand loops squeezed into a meter's length would produce significant heating. Higher currents tin can exist achieved by using superconducting wires, although this is expensive. There is an upper limit to the current, since the superconducting state is disrupted past very large magnetic fields.

There are interesting variations of the flat coil and solenoid. For example, the toroidal coil used to confine the reactive particles in tokamaks is much like a solenoid bent into a circle. The field inside a toroid is very strong but circular. Charged particles travel in circles, following the field lines, and collide with i some other, perhaps inducing fusion. But the charged particles do not cantankerous field lines and escape the toroid. A whole range of scroll shapes are used to produce all sorts of magnetic field shapes. Adding ferromagnetic materials produces greater field strengths and can take a meaning effect on the shape of the field. Ferromagnetic materials tend to trap magnetic fields (the field lines bend into the ferromagnetic textile, leaving weaker fields exterior it) and are used as shields for devices that are adversely affected by magnetic fields, including the Globe's magnetic field.

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Department Summary

• The strength of the magnetic field created by current in a long directly wire is given by

  [latex]B=\frac{{\mu}_{0}I}{2\pi r}\left(\text{long straight wire}\right)\\[/latex]

  where I is the current, r is the shortest distance to the wire, and the constant[latex]{\mu}_{0}=4\pi \times 10^{-7}\text{ T}\cdot\text{ 1000/A}\\[/latex] is the permeability of free space.

• The management of the magnetic field created past a long direct wire is given by correct hand rule 2 (RHR-2): Betoken the pollex of the correct hand in the direction of current, and the fingers curl in the direction of the magnetic field loops created by it.
• The magnetic field created by current following any path is the sum (or integral) of the fields due to segments forth the path (magnitude and direction every bit for a direct wire), resulting in a general relationship between current and field known as Ampere's law.
• The magnetic field strength at the center of a circular loop is given past

  [latex]B=\frac{\mu_{0}I}{2R}\left(\text{at heart of loop}\right)\\[/latex]

  where R is the radius of the loop. This equation becomes B = μ ₀ nI / ( 2 R ) for a flat gyre of N loops. RHR-2 gives the direction of the field near the loop. A long gyre is called a solenoid.

• The magnetic field force within a solenoid is

  [latex]B={\mu }_{0}\text{nI}\left(\text{inside a solenoid}\right)\\[/latex]

  where n is the number of loops per unit length of the solenoid. The field inside is very compatible in magnitude and direction.

Conceptual Questions

1. Make a drawing and use RHR-2 to find the direction of the magnetic field of a current loop in a motor (such as in Figure 1 from Torque on a Electric current Loop). Then show that the direction of the torque on the loop is the aforementioned as produced by like poles repelling and dissimilar poles attracting.

Glossary

right manus rule 2 (RHR-2):

a rule to determine the direction of the magnetic field induced by a current-conveying wire: Point the thumb of the correct paw in the direction of current, and the fingers curl in the direction of the magnetic field loops

magnetic field forcefulness (magnitude) produced by a long direct current-carrying wire:

divers as [latex]B=\frac{\mu_{0}I}{2\pi r}\\[/latex], where Iis the current, r is the shortest distance to the wire, and μ ₀ is the permeability of free space

permeability of gratis space:

the measure out of the ability of a material, in this case free space, to support a magnetic field; the constant [latex]\mu_{0}=four\pi \times 10^{-7}T\cdot \text{m/A}\\[/latex]

magnetic field strength at the center of a circular loop:

defined every bit [latex]B=\frac{{\mu }_{0}I}{2R}\\[/latex] where R is the radius of the loop

solenoid:

a thin wire wound into a coil that produces a magnetic field when an electric electric current is passed through it

magnetic field strength inside a solenoid:

defined as [latex]B={\mu }_{0}\text{nI}\\[/latex] where n is the number of loops per unit length of the solenoid n = N / l , with N being the number of loops and lthe length)

Biot-Savart law:

a physical law that describes the magnetic field generated by an current in terms of a specific equation

Ampere's police:

the physical law that states that the magnetic field effectually an electric current is proportional to the current; each segment of 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

Maxwell's equations:

a fix of four equations that describe electromagnetic phenomena

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Source: https://courses.lumenlearning.com/physics/chapter/22-9-magnetic-fields-produced-by-currents-amperes-law/