Chapter 12: Magnetic Vector Potential and the Biot-Savart Law
Chapter Learning Objectives: After completing this chapter the student will be able to: Calculate the magnetic vector potential due to a region of current density. Use the magnetic vector potential to calculate the magnetic flux density near an electrical current. Use the Biot-Savart Law to calculate magnetic flux density.
You can watch the video associated with this chapter at the following link:
Historical Perspective: Jean-Baptiste Biot (1774-1862) and Felix Savart (1791-1841) were French physicists who discovered the Biot-Savart Law in 1820. They were both professors at the Collège de France in Paris.
Photo credits: https://commons.wikimedia.org/wiki/File:Jean_baptiste_biot.jpg, [Public domain], via Wikimedia Commons. http://fameengineers.blogspot.com/2011/09/felix-savart.html [Public domain].
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12.1 Review: Electrostatics and Magnetostatics
We have now introduced all the fundamental laws that are necessary to understand and study both electrostatics (the electric fields caused by immobile charges) and magnetostatics (the magnetic fields caused by charges moving at a constant velocity). Recall the following two equations, which describe the fundamentals of magnetostatics:
(Copy of Equation 11.8)
(Copy of Equation 11.12)
The first of these equations is one of Maxwell’s equations, and the second is an incomplete form of Ampere’s Law, which will become another of Maxwell’s equations once it is complete. It turns out that we can prove that specifying the divergence and the curl of a field fully specifies the field at all points. This is known as Helmholtz’s theorem, and it proves that based on these two equations, we have specified the magnetic field at all points.
You will also hopefully remember Gauss’s Law, which in its point form specifies the divergence of the electric field:
(Copy of Equation 5.6)
For now, we are ignoring the impact of dielectric materials, so we can use E rather than D. Including the effect of dielectric materials would give the same result.
We have not yet mentioned the curl of the electric field, but it turns out to be zero. If you think about the electric field, it always tends to move in straight lines away from charges or regions of charge, and even when the electric field lines curve, they don’t tend to exhibit rotation. Thus:
(Equation 12.1)
Vector fields that have zero curl at all points are referred to as irrotational. Thus, the electric field is irrotational. Irrotational fields can be proven to be conservative, so equation 12.1 can be used to prove the continuous form of Kirchhoff’s Voltage Law:
(Copy of Equation 7.26)
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So, we now have enough information to calculate the electric field and the magnetic field within a region of space, given the charge density and the current density within that region. We could stop here, but we’re going to build on the magnetic field equations to obtain two additional tools that can be used to help us calculate B from a specified J. Just as we can use Coulomb’s Law, Gauss’s Law, or the definitions of voltage and electric field to calculate E, we will have three different methods for calculating B.
12.2 Magnetic Vector Potential
Recall that we performed a series of mathematical operations and abstractions to move from Coulomb’s Law, a very specific rule that is somewhat difficult to apply, to the principle of voltage (potential), which is quite a bit easier to apply. We will now do the same for magnetic fields.
We will define the magnetic vector potential A to be the vector that, when the curl of A is taken, we obtain B.
(Equation 12.2)
This definition seems useless now, but we will soon see that it is actually quite helpful. We can now substitute Equation 12.2 into Ampere’s Law (Equation 11.12) and obtain:
(Equation 12.3)
Now, we will rely on our dear friends the mathematicians to prove (outside the scope of this course) the following theorem:
(Equation 12.4)
We will further rely on expert mathematicians to tell us that we can choose the value of the divergence of A (∙A) to take on any value we want. After all, we defined A to be a function whose curl is B, but to fully specify A, we would also have to specify its divergence. Since there are an infinite number of functions that would satisfy Equation 12.2, we will select the one of them for which the divergence of A is also equal to zero. This is known as “selecting the Coulomb gauge,” and it is entirely analogous to specifying the reference voltage to be zero volts or the elevation at sea level to be zero feet. In this course, we will always select the Coulomb gauge, which means that:
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(Equation 12.5)
Applying this new information to Equation 12.4, we obtain the following result:
(Equation 12.6)
Notice that this equation allows us to calculate the magnetic vector potential A from the current density J, just as Poisson’s Equation allowed us to calculate the electric potential V from the charge density .
(Copy of Equation 7.34)
This similarity (or complementarity) between the equations controlling electric fields and magnetic fields will be an ongoing theme throughout the remainder of this book.
I will refer to Equation 12.6 as “Poisson’s Equation for Magnetic Fields,” although this name is not widely recognized. Just as with all other vector equations in this subject, this expression comes in two forms: the point form, as shown in Equation 12.6, and the integral form, which is shown below:
(Equation 12.7)
where
(Equation 12.8)
The derivation between Equation 12.6 and Equation 12.8 is beyond the scope of this course. It requires the use of a method called Green’s functions, which is a very useful method for solving advanced differential equations like these.
The geometry of the problem associated with Equation 12.7 can be seen in Figure 12.1, which shows that the vector r’ is pointing from the origin to the location of the current density element, r is pointing from the origin to the point where we want to calculate A, and R is pointing from the current density to the location where we want to find A.
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Also, we can see from this figure that the magnetic vector potential caused by a differential element of current density will always point in the same direction as the current density. z
J
y R
x
A
Figure 12.1. The geometry associated with Equation 12.7.
Example 12.1: Determine the magnetic vector potential A caused by a current I along a straight wire of length a. The point to be studied lies a distance along the perpendicular bisector of the wire.
I
a az I af
a
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Example 12.2: Given the result of the previous example, determine the magnetic flux density, B, at the same point.
Example 12.3: Given the result to Example 12.2, determine the magnetic flux density a distance away from an infinitely long wire carrying current I.
12.3 Biot-Savart Law
As we have just observed, the magnetic vector potential A allows us to calculate the magnetic flux density B due to any current density J or any current I. However, the problem must always be solved in two steps: First, calculate A, and then calculate B=xA. Is it possible to do some extra work up front in order to find an equation that will directly yield B as a function of J? It turns out that it is, and the result of this effort is the Biot-Savart Law.
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We will begin with the relationship between B and A (Equation 12.2), and we can then substitute equation 12.7 for A:
(Equation 12.9)
Bringing the cross product into the integral and performing a little bit of vector calculus yields:
(Equation 12.10)
This equation is designed for problems in which three-dimensional regions of current density are flowing. Typically, this is not the case we will encounter, so we can modify this equation further in order to customize it for a closed loop of one-dimensional current:
(Equation 12.11)
This is known as the Biot-Savart Law, named after the two physicists who discovered it in 1820. This is the first equation we have that directly relates current flow (I) in arbitrary directions (dl’) to the magnetic flux density it creates. The definition of r, r’, and R as defined in Figure 12.1 still hold here.
Example 12.4: Repeat the calculation of the magnetic flux density caused by a current I along a straight wire of length a, this time using the Biot-Savart Law.
I
a az I af
a
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Example 12.5: Using the Biot-Savart Law, determine the magnetic flux density caused by a circular loop of current I with a radius a. Consider a point at a distance z above the center of the loop.
az af z a
a
I
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We will encounter loops of current like the one in Example 12.5 so often that it is worthwhile to introduce an extra mathematical tool to help work with them. We will refer to such a loop of current as a magnetic dipole, since it will have a north pole (where the magnetic flux lines flow out) and a south pole (where the magnetic flux lines flow in). There are two magnetic poles, so we call it a magnetic dipole. This is illustrated in Figure 12.2. B
North Pole
a
I
South Pole
Figure 12.2. A magnetic dipole, with the north and south poles labeled.
As you would expect (and as we have seen in Example 12.5), the magnetic flux density will increase with the current I, and it also increases with the surface area of the loop (a2). Thus, we will define a quantity called the Magnetic Dipole Moment m, which is defined as:
(Equation 12.12)
A “moment” is a quantity that incorporates a great deal of information about the geometry of a problem into a pre-calculated value, which makes future calculations easier to perform. Both mechanical and civil engineers tend to use them more frequently than electrical and computer engineers, but here we will use it.
Example 12.6: What is the magnetic dipole moment of a loop of wire in the xy plane with radius 2.5 cm with a current of 100mA?
Reconsidering the result of Example 12.5, and recognizing that 푅 = √푎2 + 푧2, we see that the calculation can be substantially simplified by the use of the magnetic dipole moment:
(Equation 12.13)
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12.5 Summary
We have now seen the two equations from which all of electrostatics and magnetostatics can be derived:
The magnetic vector potential A is defined as the vector field that, when you take the curl of it, you get the magnetic flux density. We will also choose the Coulomb gauge, so the divergence of A is zero.
Poisson’s Equation for Magnetic Fields (in both point form and integral form) allows us to calculate A given the current density:
We can combine these two steps into one using the Biot-Savart Law:
From this equation, we can calculate the magnetic flux density caused by any arbitrary shape and quantity of current. For example, we can calculate the magnetic flux density caused by a current I passing through a wire of length a or around a loop of radius a:
(A distance from a wire of length a) (A distance z above a loop of radius a)
The magnetic dipole moment is defined as:
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