2210. Magnetic Fields Produced by Currents: Ampere’s LawLearning Objectives
How much current is needed to produce a significant magnetic field, perhaps as strong as 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 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 of a currentcreated field related to the direction of the current? Answers to these questions are explored in this section, together with a brief discussion of the law governing the fields created by currents. Magnetic Field Created by a Long Straight CurrentCarrying Wire: Right Hand Rule 2Magnetic fields have both direction and magnitude. As noted before, one way to explore the direction of a magnetic field is with compasses, as shown for a long straight currentcarrying wire in Figure 1. Hall probes can determine the magnitude of the field. The field around a long straight wire is found to be in circular loops. The The where $I$ is the current, $r$ is the shortest distance to the wire, and the constant ${\mu}_{0}=\mathrm{4\pi}\phantom{\rule{0.25em}{0ex}}\times \phantom{\rule{0.25em}{0ex}}{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\mathrm{T}\cdot \text{m/A}$ is the Example 1: Calculating Current that Produces a Magnetic FieldFind the current in a long straight wire that would produce a magnetic field twice the strength of the Earth’s at a distance of 5.0 cm from the wire. Strategy The Earth’s field is about $5\text{.}0\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\mathrm{T}$, and so here $B$ due to the wire is taken to be $1\text{.}0\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\mathrm{T}$. The equation $B=\frac{{\mu}_{0}I}{2\mathrm{\pi r}}$ can be used to find $I$, since all other quantities are known. Solution Solving for $I$ and entering known values gives $$\begin{array}{lll}I& =& \frac{2\pi \text{rB}}{{\mu}_{0}}=\frac{2\pi \left(5.0\times {\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\mathrm{m}\right)\left(1.0\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\mathrm{T}\right)}{4\pi \times {\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\mathrm{T}\cdot \text{m/A}}\\ & =& \text{25 A.}\end{array}$$Discussion So a moderately large current produces a significant magnetic field at a distance of 5.0 cm from a long straight wire. Note that the answer is stated to only two digits, since the Earth’s field is specified to only two digits in this example. Ampere’s Law and OthersThe magnetic field of a long straight wire has more implications than you might at first suspect. Each segment of current produces a magnetic field like that of a long straight wire, and the total 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 Making Connections: RelativityHearing all we do about Einstein, we sometimes get the impression that he invented relativity out of nothing. On the contrary, one of Einstein’s motivations was to solve difficulties in knowing how different observers see magnetic and electric fields. Magnetic Field Produced by a CurrentCarrying Circular LoopThe magnetic field near a currentcarrying loop of wire is shown in Figure 2. Both the direction and the magnitude of the magnetic field produced by a currentcarrying loop are complex. RHR2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in Magnetic Fields and Magnetic Field Lines are needed for more detail. There is a simple formula for the where $R$ is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid only at the center of a circular loop of wire. The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. One way to get a larger field is to have $N$ loops; then, the field is $B={\mathrm{N\mu}}_{0}I/(2R)$. Note that the larger the loop, the smaller the field at its center, because the current is farther away. Magnetic Field Produced by a CurrentCarrying SolenoidA The magnetic field inside of a currentcarrying solenoid is very uniform in direction and magnitude. Only near the ends does it begin to weaken and change direction. The field outside has similar complexities to flat loops and bar magnets, but the where $n$ is the number of loops per unit length of the solenoid $(n=N/l$, with $N$ being the number of loops and $l$ the length). Note that $B$ is the field strength anywhere in the uniform region of the interior and not just at the center. Large uniform fields spread over a large volume are possible with solenoids, as Example 2 implies. Example 2: Calculating Field Strength inside a SolenoidWhat is the field inside a 2.00mlong solenoid that has 2000 loops and carries a 1600A current? Strategy To find the field strength inside a solenoid, we use $B={\mu}_{0}\text{nI}$. First, we note the number of loops per unit length is $${n}^{1}=\frac{N}{l}=\frac{\text{2000}}{\mathrm{2.00\; m}}=\text{1000}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{1}=\text{10}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{1}\text{.}$$Solution Substituting known values gives $$\begin{array}{lll}B& =& {\mu}_{0}\text{nI}=\left(\mathrm{4\pi}\times {\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\mathrm{T}\cdot \text{m/A}\right)\left(\text{1000}\phantom{\rule{0.25em}{0ex}}{\mathrm{m}}^{1}\right)\left(\text{1600 A}\right)\\ & =& 2\text{.01 T.}\end{array}$$Discussion This is a large field strength that could be established over a largediameter solenoid, such as in medical uses of magnetic resonance imaging (MRI). The very large current is an indication that the fields of this strength are not easily achieved, however. Such a large current through 1000 loops squeezed into a meter’s length would produce significant heating. Higher currents can be achieved by using superconducting wires, although this is expensive. There is an upper limit to the current, since the superconducting state is disrupted by 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 one another, perhaps inducing fusion. But the charged particles do not cross field lines and escape the toroid. A whole range of coil shapes are used to produce all sorts of magnetic field shapes. Adding ferromagnetic materials produces greater field strengths and can have a significant effect on the shape of the field. Ferromagnetic materials tend to trap magnetic fields (the field lines bend into the ferromagnetic material, leaving weaker fields outside it) and are used as shields for devices that are adversely affected by magnetic fields, including the Earth’s magnetic field. PhET Explorations: GeneratorGenerate electricity with a bar magnet! Discover the physics behind the phenomena by exploring magnets and how you can use them to make a bulb light. Section Summary
Conceptual QuestionsExercise 1Make a drawing and use RHR2 to find the direction of the magnetic field of a current loop in a motor (such as in Section 229). Then show that the direction of the torque on the loop is the same as produced by like poles repelling and unlike poles attracting.
