Electromagnetism without B: dipoles and loops

Dipoles and Loops

Daniel M. Dobkin / Nicholas E. Dobkin
October 2007

All Currents Great and Small

Consider just about the simplest electrical system: a short piece of wire with an electric current flowing on it. Because the wire is of finite length, the current has to go to 0 at the wire ends: there's nowhere for the electrons to go (at least when the wire is cool enough for thermionic emission to be negligible). If there's a finite current in the middle of the wire and no current at the end, electrons must be accumulating somewhere: that is, there's a net charge on the wire. If the wire is short enough, it looks something like this:

Now, according to the source equations, the charge and current create radiating vector and scalar potentials. The vector potential is everywhere in the direction of the current, and falls as the inverse of the distance. The scalar potential cancels in the plane perpendicular to the wire, because the charges on top and bottom of the wire are equal and opposite. The scalar potential does NOT cancel along the axis of the wire, because the top and bottom of the wire are at different distances from the observation point (even when that point is infinitely distant), and so are seen at differing times:

Therefore in the plane perpendicular to the wire, only the vector potential is present, and it propagates away from the wire at the speed of light, giving rise to an electric field. The field can induce a voltage in another wire (more about that below), and thus the wire radiates in this plane.

It is not unduly difficult to show that along the axis, the gradient of the scalar potential is of the same magnitude as the time derivative of the vector potential, but oppositely oriented. The two contributions cancel: there is no electric field along the axis of the wire.

By making a provision for injecting current into the wire (e.g. splitting it in the middle to form a dipole), we obtain an antenna: a device that converts electric currents into radiating waves. We have been able to do this because we did not provide a return path for the current, so there is no equal and opposite current to cancel the contribution on the first wire. The price is that we must account for the accumulation of charge and consequent scalar potential. As we will see in a moment, this charge accumulation represents a capacitive impedance, making it difficult to get much current to flow in the wire if it is very short compared to a wavelength, and thus establishing a tradeoff between antenna size and ease of use.

Bring Your Own Popcorn:

Time for a movie...

The currents and charges on the surface of a wire antenna adjust themselves to ensure zero electric field inside the antenna. For example, currents flow on a wire to cancel an incident electric field; the resulting induced voltage can be detected by an amplifier and used to receive the signal due to the incident wave. Short antennas behave like capacitors, long antennas like inductors, and at a specific length -- the resonant length, just less than half a wavelength -- the capacitance and inductance cancel, and a large current flows (limited mainly by re-radiation due to the current flow). We've provided a short Flash animation that illustrates these concepts, to provide a starting point for the math.

Note that for simplicity we've imagined that positive charges move on the wire, though of course it is normally electrons that do the moving; it's just easier to minimize the number of negative signs that one has to keep track of when trying to explain a concept.

Click here to watch the animation. Hit the BACK button on your browser when you want to return to this page.

Dipole Details: Capacitance, Inductance, and Resonance

To dig a little deeper, let's return to our wire suspended in space, with an impinging vector potential A and consequent electric field –iωA. For reasonable wire thicknesses, we can assume that currents and charges are only present in a thin layer on the surface of the wire, and adjust themselves to ensure zero field well within the wire. What we'd like to do is to use this knowledge to estimate how much current flows for a given incident field (or equivalently, voltage): that is, we'd like to be able to create a rough equivalent circuit for the antenna:

The electric field in the wire arises from the potentials due both to the incident wave and the local charges and currents on the wire. The latter portion is often called the scattered field, though the terminology is perhaps a bit more appealing when our observation point is farther away. To get 0 total field, the scattered field must be equal in magnitude and opposite in direction to the incident field. To see how this comes about it is very helpful to partition the scattered field into three pieces:

An instantaneous electric potential φ_scresults from the accumulation of charge, primarily near the ends of the wire, where charge must build up since current cannot flow past the ends. For short antennas we can ignore the time delay between the charge location r’ and the axis of the wire r. The electric field tracks the accumulation of charge in the wire. This is the result of the dipole capacitance.
An instantaneous magnetic component A_sc,in, in phase with the local current, whose value at each location is mainly determined by the current in that vicinity. The electric field lags the current by 90 degrees and so opposes changes in the current: this field is the result of the dipole inductance.
A delayed magnetic component A_sc,d, dependent on the integral of the total current along the wire. For harmonic time dependence, this component lags the instantaneous component by 90 degrees – that is, it is along the negative imaginary axis. Note that the consequent electric field is in phase but directly opposed to the current flow, suspiciously like a conventional resistor. We will find that this is the contribution that leads to the radiation resistance of the antenna.

(There is also a delayed contribution to the scattered electric (scalar) potential. It is smaller than that due to the vector potential because the contributions of the positive and negative charge cancel to first order, so we can neglect it in a qualitative examination; however, it must be included to obtain accurate estimates of the radiation resistance.)

Corresponding to each scattered potential and field we can define an induced voltage. Working in terms of voltages may be more familiar for circuit-type people, and lends itself to approximate calculations rather more gracefully than focusing on fields. We shall somewhat arbitrarily choose to measure this voltage from left to right, in the same direction in which we define the current to be positive. (This is an electrical engineer's view: the voltage across a resistor is measured from the current source to the current sink.) The electrostatic voltage is thus the difference in scalar potential between the left and right sides of the wire, and the magnetic components are the line integrals of the corresponding electric fields along the wire. These three contributions are combined to obtain the net scattered voltage; we then arrange the phase and amplitude of the current so that this scattered voltage approximately cancels the incident voltage V_inc, ensuring that the interior of the wire is field-free as it must be.

In the movie we asserted that the different contributions scale in different ways as the wire length changes. Let's put some meat on those bones by getting some very rough ideas of how the voltages depend on current and length. The first contribution is the instantaneous scattered potential due to the charges. The charge is just the accumulation of the current; if the current were constant in time we'd have:

In the general case the charge is the integral of the current; for a harmonic time dependence we get:

If we treat this charge as a sort of lump at the end of the wire, the rough magnitude of the potential it creates can be gotten from Coulomb's law:

where in the interests of simplicity we're blithely ignoring details like exactly where the charge is and where we're measuring the potential. If this bothers you, have a beer and try again. (If it still bothers you -- you'll need to read the detailed treatment of the problem in my article on short wire antennas. Drinking more beer probably won't help.) We plug in the harmonic expression for the charge and find:

That looks a bit obscure but if we remember that the frequency is related to the wavelength (ω = 2 π c/ λ) we can write this expression in a very appealing fashion:

Here we've defined a sort of characteristic voltage V₀ = IZ₀, as the product of the current and the characteristic impedance of free space, Z₀ =μ₀c= 377 ohms. So the antenna voltage due to the charge accumulation scales as the ratio of wavelength to the antenna size, multiplied by this characteristic voltage associated with the current but not with the antenna geometry.

The diagram below schematically depicts the charge and scalar potential at the moment when the current falls to near zero and the charge is a maximum value. (That is, this is 90 degrees behind the maximum value of the current.) Since the voltage is defined from left to right, the voltage at this moment is negative. The scaling behavior arises simply from the fact that the shorter the wire is, the closer we have to pack the charges for the same current. More charge means more mutual repulsion and a higher voltage.

What about the comparable voltage resulting from the current (the antenna inductance)? The source integral for the wire is:

Most of the contribution to the integral arises from the nearby currents (where nearby is defined relative to the diameter of the wire):

So very roughly we can consider the integral to be just I*a/a = I. (The actual dependence on wire length is logarithmic, as we might expect from our consideration of wire inductance.) Multiplying by angular frequency and re-expressing the result in terms of wavelength we get the electric field due to inductive effects:

Again we see that the characteristic voltage V₀ = IZ₀ pops up, so we can write the voltage due to this term as:

The inductive voltage is of the opposite sign to the capacitive voltage above, and scales the opposite way with antenna length, just as claimed in the animation. The two influences oppose one another, and dominate in different regimes of antenna length; it is reasonable to infer that at some frequency for any given antenna length they are equal and opposite, so that the inductive and capacitive voltages cancel one another -- the antenna becomes resonant. Our very simple calculations would suggest that this takes place when the length of the wire equals one wavelength, but of course we have made no effort to account for the actual current distribution. More accurate estimates show that resonance occurs at a bit less than half a wavelength in length. An example of this behavior is shown below for a model of a 10-cm antenna with a quadratic current distribution (which slightly underestimates the actual resonance); the resonant frequency is about 1250GHz, with a wavelength of 24 cm.

inductive and capacitive voltages vs frequency

So far we've provided a bit more substantial justification for the assertions made in the animation. Antennas scale relative to a wavelength. Short antennas are charge-dominated (capacitive); a little current generates a lot of charge and thus field to oppose the incident field. That is, the antenna presents a large capacitive impedance opposing current flow. Long antennas are current-dominated (inductive); again a little current creates a lot of voltage, but this time leading instead of lagging the current. At some resonant length the two effects will (at least approximately) cancel: that is to say, no scattered electric field results from current flow. Since the scattered electric field must cancel the incident field inside the wire, it would seem that arbitrarily large currents must flow at resonance. In fact, large currents do flow, but they are limited by more subtle effects related to the finite value of the speed of light. In the next section we'll look at the delayed magnetic potential and see how it gives rise to a resistive current.

Delayed Potentials and Radiation Resistance

The delayed component is the result of considering the finite speed of light. The potential due to currents from far away is increasingly delayed – for harmonic time dependence, more of that potential is along the delayed (negative imaginary) axis. To first order, this effect cancels the (1/r) decrease in the contribution of more distant currents, so that the contribution of each current element to the delayed potential is independent of position (assuming, of course, that the antenna is short compared to a wavelength). Mathematically, for a harmonic time dependence, the contribution to the potential of any current element must be multiplied by an exponential term e^-ikr. When we expand the exponential to first order, (1-ikr), we find that we have already accounted for the first term, which is the instantaneous potential. The first-order delayed term, -ikr, is linearly dependent on the distance between the current element and the point of measurement; this linear dependence compensates for the inverse weighting of the more distant currents to give a potential integral with no dependence on the distance between the measurement and the currents:

The delayed component is thus linearly proportional to the average current and the wavevector k = 2π/λ. If the current doesn’t vary too much over the wire, the integral is roughly just the product of the current and the wire length. We obtain:

The electric field is again derived from the vector potential; using ω = 2 π c/ λ we obtain:

Once again, the product of the current and the impedance of free space -- that is, the characteristic voltage V₀ -- appears. The voltage is as usual roughly the product of the length and the field, which is in this approximation independent of position:

Note that the electric field is real and opposed to the current. If this is the only contribution to the field, in order to cancel the incident field the current will be in phase with the incident field so that the scattered and incident fields are of opposite sign: that is, the current is flowing in phase with the incident field. The wire is acting like a resistor even though we have completely ignored the finite conductivity of the wire. We can write I = V_inc/R_rad, where R_rad is known as the radiation resistance of the wire, and is proportional to the impedance of free space and the square of the normalized wire length. The energy dissipated by this apparent resistance is radiated away to the distant world, though the value of the resistance is obtained through a purely local calculation. (However, in making the calculation we included only the retarded field -- that is, we assumed time-asymmetric propagation. This is equivalent to making an assumption that the distant universe is ergodic; thus we really didn't quite do everything locally.)

In this approximation the radiation resistance arises from the sum of the current over the antenna. A current distribution whose average is zero -- such as that in a small loop -- produces no delayed voltage and thus no radiation.

To obtain quantitative estimates for the values of capacitance, inductance, and resistance in our equivalent circuit we would need to perform the integrals, which means we must make some assertion about the current distribution. It turns out that for short antennas we can do a remarkably good job using a quadratic current distribution, maximized in the middle of the antenna and going to zero at the ends (since there is nowhere for the electrons to go). Sinusoidal currents produce more accurate results at the expense of introducing new special functions (the sine and cosine integrals); the true current distributions can be obtained numerically.

Our wire is not really an antenna -- there's nowhere to get the voltage out. To access the signal we could cut the wire in half and attach wires to each end to form a dipole antenna:

The current induced on the wire flows through the load impedance (for example, the input impedance of a radio receiver), creating a signal voltage. The current distribution is now dependent on the load impedance. If the load is a short circuit, the current peaks in the center of the antenna and it looks like a wire. If the load is an open circuit, the current must goes to zero at the center, and the antenna looks rather like two wires stuck end to end. Intermediate loads are a combination of the open- and short-circuit current distributions, since the system is linear.

With the current and consequent charge distributions we can obtain the scalar potential at the center of the wire segments, and thus the voltage for a given current and incident field. The ratio of voltage to current is the source impedance; typical behavior vs. frequency for the same 10-cm dipole is depicted below, comparing the impedance found using only the potentials and a quadratic current distribution to that for a similar structure using the conventional method of moments (from Stutzman & Thiele's Antenna Theory and Design).

The details of these calculations can be found in the article on short wire antennas. All the computations are done without the need to compute the magnetic field B. Everything we need to know is obtained from the two source integrals and the effect of the vector and scalar potential on the electrons in the wire.

Loop Antennas Made Simple(r)

We've already considered the inductance of a loop when propagation delays are completely negligible. Just as in the case of a dipole, as the loop gets bigger, we have to take into account the finite time required for the effects of currents and charges to propagate across the loop. To lowest order there's no charge on the wire (though we'll have more to say about that shortly). Furthermore, the first imaginary term in the expansion of the exponential (the -ikr term) integrates to zero for loop, because the the average current in a loop is always zero: each current element is cancelled by the diametrically opposed element.

We have to go to the third-order term in the exponential expansion to get a finite contribution:

The quadratic term in distance means that distant currents contribute more strongly to the imaginary part of A than nearby currents, so the loop currents no longer cancel. To obtain a semi-quantitative estimate we can once again approximate the loop as an equivalent square. If we choose a test point in the middle of one of the sides, the integral around the loop is mainly just due to the current on the opposite side:

To a good approximation all the current elements along this side are the same distance (L_eff ) from the test point, so the integration becomes just multiplication of the current by the length of the wire:

The electric field due to this potential is:

Note that the electric field is real (in phase with the current), and that furthermore the scattered field from the current flows points in the opposite direction to the current flow (recall that our test point at the moment is at the left side, where the current is directed in the negative z direction). The total electric field in the wire must be 0, so the voltage must have a real part pointing in the direction of current flow -- a real resistance. Simplistically assuming that the field is the same along the perimeter of the loop, the voltage is just the field multiplied by the perimeter:

Now, what should we use for the size of the coil? At this level of approximation the choice is sort of arbitrary, but the quadratic dependence on position hints that setting the effective square side for the same perimeter as a circle is not a good choice. Instead let's set the area of the square and circle equal:

We can then express the radiation resistance as a function of the ratio of radius (or perimeter) over wavelength:

A comparison of this result with the conventional estimate for a small loop, taken from Balanis, is shown below.

Naturally the nearly perfect agreement is somewhat fortuitous given the sloppy approximations we've made, but the fundamental point is that we've reproduced the correct dependence of the radiation resistance on the fourth power of the loop size. Once again there is no need to compute a magnetic field or integrate radiated power at infinite distance.

COMING SOON: We'll show that the qualitative properties of larger loops can also be obtained from very simple arguments about keeping track of the charge responsible for neutralizing the influence of the vector potential.