You are watching: The magnetic field of a solenoid depends upon n, which represents

How much current is needed to develop a far-reaching magnetic field, perhaps as strong as the earth field? Surveyors will tell you the overhead electrical power lines produce magnetic areas that interfere v their compass readings. Indeed, as soon as Oersted uncovered in 1820 the a current in a wire impacted a compass needle, he to be not handling extremely large currents. Just how does the shape of wires delivering current affect the form of the magnetic ar created? We provided earlier that a current loop created a magnetic field similar to that of a bar magnet, but what about a straight wire or a toroid (doughnut)? how is the direction the a current-created field related come the direction of the current? Answers come these questions are explored in this section, along with a quick discussion that the legislation governing the fields created by currents.

Magnetic areas have both direction and magnitude. As provided before, one way to discover the direction of a magnetic ar is with compasses, as presented for a lengthy straight current-carrying cable in number 1. Room probes deserve to determine the size of the field. The field about a long straight wire is found to it is in in circular loops. The * ideal hand ascendancy 2* (RHR-2) increase from this exploration and is precious for any kind of current segment—*point the ignorance in the direction that the current, and also the fingers curl in the direction of the magnetic field loops* produced by it.

Figure 1. (a) Compasses placed near a long straight current-carrying wire indicate that field lines form circular loops centered on the wire. (b) best hand preeminence 2 states that, if the ideal hand thumb points in the direction the the current, the fingers curly in the direction the the field. This dominion is continual with the field mapped because that the long straight wire and also is precious for any current segment.

The * magnetic field strength (magnitude) developed by a lengthy straight current-carrying wire* is uncovered by experiment come be

where *I* is the current, *r* is the shortest street to the wire, and also the constant * permeability of cost-free space*. (*μ*0 is one of the straightforward constants in nature. We will certainly see later on that *μ*0 is regarded the speed of light.) due to the fact that the cable is an extremely long, the size of the field depends only on distance from the cable *r*, not on position along the wire.

Find the present in a lengthy straight wire that would develop a magnetic ar twice the toughness of the earth at a street of 5.0 centimeter from the wire.

**Strategy**

The Earth’s field is about 5.0 × 10−5 T, and also so here *B* as result of the cable is taken to be 1.0 × 10−4 T. The equation *I*, because all other quantities room known.

**Solution**

Solving because that *I* and also entering well-known values gives

**Discussion**

So a moderately large current produce a far-reaching magnetic ar at a distance of 5.0 centimeter from a long straight wire. Keep in mind that the prize is proclaimed to only two digits, due to the fact that the Earth’s ar is specified to only two digits in this example.

The magnetic field of a long straight wire has much more implications 보다 you might at first suspect. *Each segment of existing produces a magnetic field like the of a lengthy straight wire, and also the total field of any type of shape current is the vector amount of the fields because of each segment.* The officially statement that the direction and magnitude of the field as result of each segment is referred to as the * Biot-Savart law*. Integral calculus is required to sum the ar for one arbitrary shape current. This outcomes in a more complete law, called * Ampere’s law*, i m sorry relates magnetic field and also current in a basic way. Ampere’s law consequently is a component of * Maxwell’s equations*, which give a complete theory of all electromagnetic phenomena. Considerations of just how Maxwell’s equations show up to various observers resulted in the modern-day theory the relativity, and the realization the electric and magnetic areas are various manifestations the the very same thing. Many of this is past the scope of this text in both mathematical level, inquiry calculus, and also in the lot of room that can be devoted to it. Yet for the interested student, and particularly for those who proceed in physics, engineering, or similar pursuits, delving into these matters more will expose descriptions that nature that space elegant as well as profound. In this text, us shall keep the general features in mind, such as RHR-2 and also the rules because that magnetic field lines noted in Magnetic Fields and also Magnetic field Lines, when concentrating ~ above the fields produced in specific important situations.

Hearing all we do around Einstein, we sometimes gain the impression that he invented relativity out of nothing. ~ above the contrary, one of Einstein’s motivations to be to solve obstacles in knowing how various observers view magnetic and also electric fields.

The magnetic field near a current-carrying loop of cable is displayed in figure 2. Both the direction and the size of the magnetic field developed by a current-carrying loop are complex. RHR-2 can be provided to provide the direction the the field near the loop, but mapping with compasses and the rules around field lines offered in Magnetic Fields and Magnetic field Lines are needed for more detail. There is a straightforward formula because that the * magnetic ar strength in ~ the facility of a one loop*. The is

where *R* is the radius of the loop. This equation is very comparable to that for a directly wire, but it is valid *only* at the center of a one loop of wire. The similarity of the equations does suggest that similar field strength deserve to be obtained at the facility of a loop. One method to get a larger field is to have *N* loops; then, the field is *B *= *Nμ*0*I*/(2*R*). Note that the bigger the loop, the smaller sized the field at that is center, because the present is aside from that away.

Figure 2. (a) RHR-2 offers the direction the the magnetic field inside and outside a current-carrying loop. (b) much more detailed mapping with compasses or v a hall probe completes the picture. The field is comparable to that of a bar magnet.

A * solenoid* is a lengthy coil of wire (with numerous turns or loops, as opposed come a flat loop). Due to the fact that of that shape, the field inside a solenoid can be an extremely uniform, and also really strong. The field just exterior the coils is nearly zero. Number 3 shows how the ar looks and how that direction is given by RHR-2.

Figure 3. (a) since of that shape, the ar inside a solenoid of size l is substantial uniform in magnitude and direction, as indicated by the straight and also uniformly spaced ar lines. The field external the coils is virtually zero. (b) This cutaway shows the magnetic ar generated by the present in the solenoid.

The magnetic field inside the a current-carrying solenoid is an extremely uniform in direction and also magnitude. Only close to the end does it start to threaten and readjust direction. The field exterior has similar complexities to level loops and bar magnets, yet the * magnetic ar strength within a solenoid* is simply

where *n* is the number of loops every unit length of the solenoid (*n *= *N*/*l*, through *N* gift the number of loops and also *l* the length). Note that *B* is the field strength anywhere in the uniform an ar of the interior and also not just at the center. Big uniform areas spread over a large volume are possible with solenoids, as example 2 implies.

What is the field inside a 2.00-m-long solenoid that has actually 2000 loops and also carries a 1600-A current?

**Strategy**

To uncover the field strength within a solenoid, we use

**Solution** Substituting recognized values gives

**Discussion**

This is a large field strength that could be created over a large-diameter solenoid, such as in clinical uses the magnetic resonance imaging (MRI). The very large current is one indication that the areas of this strength space not quickly achieved, however. Such a large current through 1000 loops squeezed right into a meter’s length would produce far-reaching heating. Higher currents deserve to be accomplished by using superconducting wires, back this is expensive. Over there is an top limit to the current, since the superconducting state is disrupted by very large magnetic fields.

There are interesting variations of the flat coil and also solenoid. Because that example, the toroidal coil offered to confine the reactive corpuscle in tokamaks is lot like a solenoid bent into a circle. The field inside a toroid is very solid but circular. Fee particles travel in circles, following the field lines, and also collide through one another, probably inducing fusion. Yet the fee particles execute not cross field lines and escape the toroid. A whole variety of coil shapes are provided to develop all sorts of magnetic ar shapes. Adding ferromagnetic materials produces greater field strengths and also can have a far-ranging effect on the shape of the field. Ferromagnetic materials tend to catch magnetic fields (the field lines bend right into the ferromagnetic material, leaving weaker fields external it) and are used as shields for gadgets that space adversely impacted by magnetic fields, consisting of the earth magnetic field.

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## Section Summary

The strength of the magnetic field produced by present in a lengthy straight cable is offered bywhere *I* is the current, *r* is the shortest distance to the wire, and the constant*Point the ignorance of the appropriate hand in the direction the current, and also the fingers curl in the direction that the magnetic ar loops* produced by it.The magnetic field developed by current following any kind of path is the amount (or integral) that the fields as result of segments follow me the path (magnitude and direction as for a straight wire), causing a basic relationship in between current and also field well-known as Ampere’s law.The magnetic ar strength at the center of a one loop is given by

where *R* is the radius the the loop. This equation becomes B =* **μ*0*nI*/(2*R*) for a flat coil of *N* loops. RHR-2 provides the direction of the field about the loop. A long coil is called a solenoid.The magnetic ar strength within a solenoid is

where *n* is the number of loops per unit size of the solenoid. The ar inside is an extremely uniform in magnitude and also direction.

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1. Do a drawing and use RHR-2 to discover the direction the the magnetic ar of a present loop in a engine (such as in figure 1 from Torque top top a current Loop). Then present that the direction the the torque on the loop is the same as produced by favor poles repelling and also unlike poles attracting.

## Glossary

right hand ascendancy 2 (RHR-2):a rule to identify the direction of the magnetic ar induced by a current-carrying wire: allude the thumb of the right hand in the direction of current, and also the fingers curly in the direction of the magnetic field loopsmagnetic field strength (magnitude) produced by a lengthy straight current-carrying wire:defined as*I*is the current,

*r*is the shortest street to the wire, and

*μ*0 is the permeability of totally free spacepermeability of cost-free space:the measure of the ability of a material, in this case complimentary space, to assistance a magnetic field; the consistent

*R*is the radius that the loopsolenoid:a thin wire wound into a coil that produces a magnetic ar when an electric present is passed with itmagnetic field strength inside a solenoid:defined together

*n*is the number of loops per unit size of the solenoid

*n = N/l*, through

*N*gift the number of loops and

*l*the length)Biot-Savart law:a physical law that describes the magnetic ar generated by one electric current in regards to a details equationAmpere’s law:the physical law that states that the magnetic field roughly an electric existing is proportional come the current; every segment of existing produces a magnetic ar like that of a long straight wire, and also the complete field of any type of shape present is the vector amount of the fields due to each segmentMaxwell’s equations:a set of four equations that describe electromagnetic phenomena