Magnetism and Inverse Cube versus Inverse Square
Date: Spring 2013
I learned a few months ago that magnetic flux intensity abides by the inverse cube law. I was surprised to learn this, because I already know that light, sound, and electric fields follow the inverse square law. Why does magnetism follow the inverse cube law?
You are correct in thinking that the magnetic (B) field behaves in the same way as an electric (E) field and falls off as the inverse square of the distance. The difference comes into play when considering how a B-field is created compared to an E-field. The E-field can be created from a single charged particle. The field for a particle is then characterized in 3-dimensional space by Coulomb's law. Magnetic structures are different. As far as we know presently, we have not found evidence for magnetic monopoles. That is, unlike positive or negative charged particles, a North pole or South pole does not exist by itself - they have to come in pairs. This is shown in Gauss's law for magnetism where the net B-field flux through a closed surface is zero - whatever comes out of a closed surface surrounding a magnet had to have gone into that closed surface. Therefore, you can model a magnetic structure as a dipole, that is, isolated North and South poles close together. When this is done, and the distance to the point of interest is large compared to the space between N and S, the field falls off as the inverse cube. This is the same behavior that can be shown with positive and negative charges of the same magnitude constructed as a dipole.
Hope this helps!
Electric field from a single charge follows the inverse square law. Electric field from an electric dipole follows an inverse cube law. A simple electric dipole is made from two opposite charges of equal magnitude fairly close together. The simplest magnetic field pattern is a dipole pattern. This is because a magnetic monopole does not exist. A north pole or south pole magnet has never been seen. A magnet always has both.
Dr. Ken Mellendorf
Illinois Central College
Thanks for the question. A very simple (and not 100% accurate) way of explaining this inverse cube is that it comes from the Biot-Savart Law for magnetism. The Biot-Savart Law is (equation omitted -search web). You'll notice the r^3 in the denominator--this gives the inverse cube dependence for magnetism in simple scenarios. Please note that for more complicated scenarios such as curved wires, the inverse cube dependence no longer holds.
I hope this helps. Please let me know if you have more questions.
First, light is an "electro-magnetic" (EM) wave composed of both an electric field and a magnetic field. Both follow the same law in that both the magnetic and electric field intensities drops as 1/r, where r is the distance away from the source. What drops as 1/r-squared is the power of the light. Electromagnetic fields have analogies in Ohm's law where the power within the circuit is related to the voltage x current. The power of the light is related to the electric field intensity times the magnetic field intensity. Because these drop as 1/r, we have (1/r) x (1/r) = 1/r-squared. We usually measure the intensity of light as its power or the luminous intensity so this is where this rule comes from.
In order to understand the cubic behavior of fields, we need to understand the source of the field. Every field will have a source. A radio wave (which is an EM wave just like light) usually gets launched by an antenna. Far away from the antenna the fields behaves as expected with the power of the radio wave dropping as 1/r-squared. This behavior is called the "far field" behavior. Near the antenna, the fields behave in a much more complex manner (called the "near field" behavior). Some of the fields will "cling" close to the antenna and not radiate. We do not see these in the far field, and thus it is reasonable to use the inverse square law here. Near the antenna, the field behavior, both electric and magnetic, can have inverse cubic behavior.
Now consider a magnet. A bar magnet does not radiate energy because its field is static. What is important here is the near field because the field drops off so rapidly away from the magnet. Note, however, that we can create an oscillating magnetic field which can radiate energy like an antenna by using an electromagnet and driving it with an alternating current. The north and south poles will flip in response to the alternating current, and under the right conditions the electromagnet will radiate electromagnetic energy. In the far field, we will see the same field behavior with both a magnetic and an electric field and a resulting inverse-square power law.Yes, the oscillating magnetic field will create an oscillating electric field as the two are needed for radiation (thus the term "electro-magnetic").
The behavior of electric and magnetic fields can be quite complex mathematically, but they follow the equations of Maxwell. In many specific situations, the mathematics simplifies so that we can use rules like the "inverse square law" which holds under many conditions but does not apply in all conditions. You might call this a "rule of thumb" in this case. Such rules are common in science in order to simplify the mathematics, but it is important to also know their limits in order to avoid confusion.
Kyle Bunch, PhD, PE
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