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3. Magnetism and Currents

Moving Charges Create Magnetic Fields

In the last section we learned that a magnetic field affects moving charges. By Newton’s third law, the moving charges must exert an equal and opposite force on whatever produced the \(\mathbf{B}\) field. In other words, the moving charge must create its own \(\mathbf{B}\) field! By using Newton’s third law we can complete the description of the indirect model we started earlier:

\[\text{Moving charges } \xrightarrow{\text{creates field}} \mathbf{ B } \xrightarrow{\text{exerts force on}} \text{Moving charges}\]

As we learned in Physics 7B moving charges constitute an electric current; a concept that is particular useful if we have a steady flow of charge. Considering a separate charge \(q\), the indirect model becomes: 

\[\text{Current } \xrightarrow{\text{creates field}} \mathbf{ B } \xrightarrow{\text{exerts force on}} \text{Moving charge }q\]

Electromagnetism: A History

While we have worked this out, it is far from clear what currents and moving charges have to do with anything related to the fridge magnets or bar magnets that make magnetism familiar to us. In essence we have cheated: the ideas of how a magnetic field affects moving charges were not known until the mid-1800s. Before that, the only thing known about magnetism was that some materials can produce magnetic fields and these attract (or repel) certain kinds of other similar materials, and that the Earth had its own magnetic field which aligns these magnetic materials. These facts were known to the Greeks as early as 600 BC. The question of why certain materials where magnetic while others did not appear to be, and what phenomenon created these magnetic fields was not addressed until 1820.

In 1820, Dutch physicist Hans Christian Ørsted had set up an experiment to show that large electric currents could be used to heat a wire. While demonstrating this to a group of students in his house, he noticed that a compass on his bookshelf changed direction whenever his “kettle” was switched on. After months of investigation, Ørsted concluded that an electric current could create a magnetic field. This was big news at the time, because prior to this only magnetic fields were known to affect other magnetic materials. This was a watershed moment in the history of science, as it was the first link between electric and magnetic phenomena. Originally we experience these as two distinct forces, two distinct fields. Ørsted’s finding was the first step on the road that led humankind to find that these apparently dissimilar phenomena were in fact linked. This unification of seemingly disconnected ideas is still at the core of fundamental research: much hope is placed on possibly unifying all forces in nature.

What is a Magnet?

The finding that electricity and magnetism are linked caused a huge revolution in science, but we now want to return to our question of what makes a magnet a magnet. Ørsted showed us that electric currents created a magnetic field, but where are the currents in a magnetized piece of Iron? People could not answer this question until the late 1800s, and even then they were met with skepticism. The answer relied on the existence of atoms: in a nutshell, the origins of magnetism are found to be the electric currents produced by the electrons orbiting (i.e. making a current loop) atomic nuclei.

The magnetism of certain materials also depends on spin orientation. Spin endows the electrons with an intrinsic angular momentum. This “intrinsic spin” can only have two possible values (you might have heard that an electron can be “spin up” or “spin down” in your physics or chemistry classes). Spin is a purely quantum mechanical phenomenon, but for the purposes of thinking about magnetism in our current discussion, consider spin an additional way to produce a loop of current (the smallest one you can imagine!)

The fact that atoms exist is something we now take for granted, but most scientists thought of it as ludicrous until 1905 (due in large part to a separate contribution by Einstein)!  They reasoned that all things in they saw in motion lost energy due to friction, and the idea of electrons perpetually moving around in atoms seemed absurd to them.

Only Some Materials Are Magnetic

To summarize, all our experiments point to the following finding: to get a magnetic field, we need a net motion of charge. How does this idea explain magnetism in materials? Imagine helium, an atom with two electrons. Now if these electrons go around in opposite directions, the currents they produce will be opposite to each other. The magnetic field of one current loop will cancel the magnetic field of the other, leading to no net magnetic field. The spin also affects the magnetic field, and if the spins are pointed in the same direction (both up or both down) the field gets stronger, while if they are aligned in opposite directions the field gets weaker. In helium, the spins of the two electrons are paired up-down, so helium would not be very magnetic.  As it turns out, helium is an “anti-magnet” and tries to stop any magnetic field going through it. This effect is called diamagnetism and is a manifestation of Lenz’s law which we have not covered yet.

However, there are many materials whose atoms have an odd number of electrons but who don't exhibit magnetism, how can that be? So far we have discussed the effects of the magnetic field on individual atoms; we need to consider the possibilities that these atoms are also interacting strongly with each other. Since a material is made up of \(\sim 10^{23}\) atoms, tiny effects, like the ones between atoms, can become large if each atom contributes to it.  Highly magnetic materials are generally metals, because metals have many outer electrons that act as if they're "free," and can interact strongly with external fields and with each other.

Ampère’s Law

We see that magnetism boils down to moving charges affecting other moving charges. It is no surprise that an explanation of the phenomena of bar magnets took so long; it required a serendipitous observation of how two superficially unlike phenomena (electricity and magnetism) affected one another and the atomic model. We have already studied how a magnetic field affects a moving charge.  Now we turn to the quantitative question of how, exactly, a current produces a magnetic field.

Field Produced by a Long, Straight Wire

We will first study a simple test case: a long straight wire carrying a current. We want to understand the magnetic field produced by this wire, i.e. how strong it is in magnitude, where it points (recall it is a vector), and how does it vary with position. In other words, we want to map the \(\mathbf{B}\) field.

We will retrace some of Ørsted’s steps. He showed that a current-carrying conductor produces a magnetic field. A simple way to demonstrate this is to place several compass needles in a horizontal plane (for example, the surface of a table) near a long wire placed vertically (running up and down through the table surface). Let’s assume the current direction to be coming from the bottom and going toward the top of the table. When there is no current in the wire, all needles point in the same direction (magnetic north). As soon as  current starts flowing in the wire, all needles will deflect. We have produced a magnetic field!

The first thing that one notices when doing this experiment, is that the needles orient themselves in a recognizable pattern; if we draw a circle on the table with the wire at the center and we place the compasses along the circle, we will notice that all the compass needles will orient themselves tangential to the circle. In other words, the field lines for \(\mathbf{B}\) from a long straight wire at a distance \(r\) from the wire will have the shape of concentric circles of radius \(r\). For our experiment with the current coming out of the table, we find that the \(\mathbf{B}\) field direction is counterclockwise along the circles. If we flip the direction of the current, the compass needles will still point tangent to the circle, but now their north poles point in a clockwise direction.

Right-Hand Rule 

This observation for the direction of the \(\mathbf{B}\) field can be summarized with the following convenient rule, right-hand-rule #1 (RHR #1):  Point the thumb of your right hand along a wire in the direction of the conventional (positive) current. Your fingers will now curve naturally in the direction of the magnetic field.

What can we infer about the magnitude of the magnetic field? By symmetry, the magnetic field should have the same magnitude everywhere along the circle. Why is this? Every point along the circle is equidistant from the wire, and have the same distance \(r\) from the wire. Likewise, if the wire is infinitely long, we could chose to place a horizontal plane (with compasses on it) anywhere along the wire. Moving this horizontal plane up or down along the wire, should also have no effect in our results.  To be mathematically precise, we can set the plane of the wire to be the x-y plane, and the direction along the wire to be the z-axis.  We see that, by symmetry, the magnetic field does not depend on the z coordinate.

Mapping the Field

Let's mathematically relate every quantity that we've talked about.  We expect that the magnitude of the field at any point will depend only on the perpendicular distance \(r\) between the wire and that point. All points at the same distance \(r\) will have the same magnitude (that is why the compass needles around a wire arranged in a circle with radius \(r\)). It is also not unexpected that the magnitude of the field will be larger if we have a larger current. It turns out that the magnitude is proportional to the current. Likewise, you probably expect that if we start moving away from the wire, the magnetic field will get weaker the farther we move. The equation that relates all these quantities to the magnetic field magnitude at a point \(\overrightarrow{r}\) is:

\[| \mathbf{B} (\overrightarrow{r})| = \dfrac{\mu_0 I}{2 \pi r}\]

where \(I\) is the current in the wire and \(r\) is the perpendicular distance from the wire to the point we are interested in. In our experiment with a wire along the z-axis, the distance \(r\) would be the length of a vector perpendicular to the wire that points directly to our position. The constant \(\mu_0\) is a proportionality constant called the magnetic permeability of the vacuum, which has the value \(\mu_0 = 4 \times 10^{−7} \text{T }· \text{ m/A}\)

Example #1

a) A long straight wire carrying a current \(I\) produces a magnetic field of \(\mathbf{B} = 1.0 \times 10^{−4} \text{ T}\) at a distance of \(2 \text{ cm}\) away from the wire. Find the current \(I\) carried by the wire. How close to the wire is the field a magnitude of \(1 \text{ T}\)?

b) A proton is moving at \(\mathbf{v} = 1.5 \times 10^3 \text{ m/s}\) parallel to the wire, in the same direction as the current, a distance of \(1.0 \text{ cm}\) away from the wire. Find the magnitude and direction of the magnetic force exerted on the proton by the magnetic field produced by the wire.

Solution

a) We use our equation \(| \mathbf{B} (\overrightarrow{r})| = \frac{\mu_0 I}{2 \pi r}\) to solve for the current \(I\). Once we find the current, we set the field equal to \(1 \text{ T}\) and solve for the distance \(r\) from the wire.

Solve for \(I\): Rearranging our equation we find \[I = 2 \pi r |\mathbf{B}|/ \mu_0\] \[I = 2 \pi (0.02\text{ m})(1.0 \times 10^{-4} \text{ T})/ 4 \pi \times 10^{-7} \text{ T m/A}\] \[I = 10 \text{ A}\]

Solve for \(r\) with new \(\mathbf{B}\): \[r = \mu_0 I/ 2 \pi |\mathbf{B}| = 2 \mathrm{\mu m}\]

b) To find the force on the proton, we must know the direction of \(\mathbf{B}\) at the proton's location and from there we can determine the direction of the force and its magnitude.

Direction of \(\mathbf{B}\):  From the Right-Hand Rule above,  we find that the magnetic field produced by the wire at the proton’s location is going into the page.

Direction of \(\mathbf{F}\): By  Right-Hand Rule #2,  the magnetic force on the proton is directed towards the wire.

Magnitude of \(\mathbf{F}\): We know that \(|\mathbf{B}| = 0.1 \text{ T}\) at 2 cm from the wire.  At 1 cm from the wire, because \(|\mathbf{B}| \propto 1/r\), we must have \(|\mathbf{B}| = 0.2 \text{ T}\).  We now insert \(\mathbf{B}\) into the equation for the magnetic force \[\mathbf{F} = q|\mathbf{v}||\mathbf{B}| \sin \theta\] \[\mathbf{F} = (1.6 \times 10^{−19}\text{ C})(1.5 \times 10^3 \text{ m/s})(0.2 \text{ T})(\sin 90°)\] \[\mathbf{F} = 4.8 \times 10^{-17} \text{ N}\]

Field Produced by a Circular Current

We would now like to describe the magnetic field from another simple configuration of electrical current. Consider a coil of radius \(r_0\), made up of \(N\) loops of wire, all carrying a current \(I\).  For ease, we adopt a convention where the axis of the coil is the z-axis.  This configuration is displayed below from two views.

Suppose we are interested in the magnetic field along the coil’s axis, at point \(P\), a distance \(z\) along the axis from the coil. We can treat the coil as \(N\) copies of a single loop of current \(I\), or as one loop with current \(NI\).  If we apply the right-hand-rule to a small piece of these loops, we find that no matter which part of the loop we choose, the resulting \(\mathbf{B}\)-field points upward at our position \(P\), or indeed any point on the axis of the coil (the z-axis).

It makes sense that the magnitude of \(\mathbf{B}\) will decrease as we get further away from the coil.  It also makes sense that the magnitude will be directly proportional to the number of wire loops and to the current flowing through the loop.  Indeed, it turns out the magnetic field at distance \(z\) along the axis (as long as we are relatively far from the coil) is given by:

\[|\mathbf{B}| = \dfrac{\mu_0 NI {r_0}^2}{2 z^3}\]

Now, what if we’re instead interested in the magnetic field at point \(Q\), which resides in the plane of the coil? Again we turn to the right-hand-rule for currents. Notice that this time not all parts of the loop contribute to the \(\mathbf{B}\)-field in the same direction. The side of the loop closest to \(Q\) gives a contribution to the field in the down direction, while the side farthest from \(Q\) contributes in the up direction. Since the field strength drops with distance, the closest side has the larger contribution, and the overall field is in the down direction. But one consequence is that, if \(Q\) and \(P\) were at similar distances away from the coil, the magnitude of \(\mathbf{B}\) at \(Q\) (in the coil plane) is significantly less than the magnitude of \(\mathbf{B}\) at \(P\) (along the axis).

The overall magnetic field lines are shown in the picture below, with points \(P\) and \(Q\) labeled.

Note the similarity between the field lines here and the field lines of a bar magnet. The similarity demonstrates that the magnetic field of a permanent magnet really can be described as the result of many small current loops lined up with each other.