The Exterior Potential and the Electrocardiogram

Abstract

This chapter analyzes the small electrical potential produced outside a nerve or muscle fiber. Far from the fiber, we often describe this potential using a multipole approximation. We then review the electrical behavior of the heart, leading to a model for the electrocardiogram based on a current-dipole source. Different features of the electrocardiogram are related to various heart pathologies. The bidomain model is a more advanced description of the anisotropic electrical properties of cardiac tissue. We analyze the electrical stimulation of nerves and muscle, including the strength-duration curve, and we apply these ideas to the implantable cardiac pacemaker. The chapter ends by discussing the electroencephalogram: the electrical potential produced by the brain.

Appendices

Symbols Used in Chapter 7

Table 2

Problems

7.2.1 Section  7.1

Problem 1.

A single nerve or muscle cell is stretched along the $x$ axis and embedded in an infinite homogeneous medium of conductivity $\sigma _{o}$. Current $i_{0}$ leaves the cell at $x=b$ and enters the cell again at $x=-b$. Find the current density $\mathbf {j}$ at distance $r$ from the axis in the $x=0$ plane.

Problem 2.

An axon is stretched along the $x$ axis. At one instant of time an impulse traveling along the axon has the form shown in the graph. The electrical conductivity inside the axon of radius $a$ is $\sigma _{i}$. In the infinite external medium it is $\sigma _{o}$. Find an expression for the potential at point $(x_{0},y_{0})$.

Problem 3.

The interior potential of a cylindrical cell is plotted at one instant of time. Distances along the cell are given in terms of length $b$. The cell has radius $a$ and electrical conductivity $\sigma _{i}$. The resting potential is $0$ and the depolarized potential is $v_{0}$. The conductivity of the external medium is $\sigma _{o}$.

• (a) Find expressions for, and plot, the current along the cell in the four regions $(x<0,0<x<2b,2b<x<3b,3b<x)$.

• (b) Find the potential at a point $(x,y)$ outside the cell in terms of the parameters given in the problem. The point is not necessarily far from the cell.

7.2.2 Section  7.2

Problem 4.

Modify the closing argument of Sect. 7.2 by considering electrodes that are disks rather than spheres. (Hint: The capacitance you will need is given in Sect. 6.19.)

Problem 5.

Suppose an axon is surrounded by a thin layer of extracellular fluid of thickness $d$. Use arguments based on the intracellular and extracellular resistances to estimate the ratio $\Delta v_{o}/\Delta v_{i}$ in this case.

7.2.3 Section  7.3

Problem 6.

Starting with Eq. 7.4, make the Taylor’s series expansions described in the text, and use them to derive Eq. 7.16.

Problem 7.

What would be the current-dipole moment of a nerve cell of radius 2μm when it depolarizes? Would myelination make any difference? Does the result depend on the rise time of the depolarization? If the impulse lasts 1 ms and the conduction speed is 5 m s⁻¹, how far apart are the rising and falling edges of the pulse?

Problem 8.

An axon or muscle cell is stretched along the $x$ axis on either side of the origin. As it depolarizes, a constant current dipole $\mathbf {p}$ pointing to the right sweeps along the axis with velocity $u$. An electrode at $(x=0,y=a)$ measures the potential with respect to $v=0$ at infinity. Ignore repolarization. Find an expression for $V$ at the electrode as a function of time and sketch it. Assume that at $t=0$, $\mathbf {p}$ is directly under the electrode at $x=0$.

Problem 9.

An electrode at $(x=0,y=a)$ measures the potential outside an axon with respect to $v=0$ at infinity. A nerve impulse is at point $x$ along the axon, measured from the perpendicular from the electrode to the axon. At $x+b$ a current dipole points to the right, representing the depolarization wave front. At $x-b$ a vector of the same magnitude points to the left, representing repolarization. Obtain an expression for $v$ as a function of $x$, $b$, $p$, and $a$. Plot it in the case $a=1$, $b=0.05$.

Problem 10.

A dipole $\mathbf {p}$ located at the origin $(0,0,0)$ is oriented in the $x$ direction. The potential $v_{o}(x)$ produced by this dipole is measured along the line $y=0$, $z=d$.

• (a) Find an equation for $v_{o}(x)$ in terms of $x$, $d$, $\sigma _{o}$ (the conductivity of the medium) and the dipole strength, $p$.

• (b) Find an expression for the depth $d$ of the dipole in terms of the distance $\Delta x$, defined as the distance between the minimum and maximum of $v_{o}(x,y)$. This is an example of an “inverse problem,” in which you try to learn about the source (in this case, the depth of $\mathbf {p}$) from measurements of $v_{o}$.

Problem 11.

The solid angle theorem is often used to interpret electrocardiograms. The relationship between the exterior potential and the solid angle in Eq. 7.15 is a general result: the potential is proportional to the solid angle subtended by the wave front. Use this result to explain (a) why a closed wave front produces no exterior potential, and (b) why an open wave front produces a potential that depends only on the geometry of its opening or rim.

7.2.4 Section  7.4

Problem 12.

Run the program of Fig. 7.11 and plot the potential for different distances from the axon.

Problem 13.

Modify the program of Fig. 7.11 to calculate the potential from a single Gaussian action potential and plot the potential.

Problem 14.

Let the intracellular potential be zero except in the range $-a<x<a$, where it is given by

$$ {v_i} = \left\{ {\begin{array}{*{20}{c}} {{{\left( {\frac{{a + x}}{a}} \right)}^2},}&{ - a < x < - a/2} \\ {1 - 2{{\left( {\frac{x}{a}} \right)}^2},}&{ - a/2 < x < a/2} \\ {2{{\left( {\frac{{a - x}}{a}} \right)}^2},}&{a/2 < x < a.} \end{array}} \right. $$

Plot $v_{i}$ vs $x$. Use Eq. 7.21 to calculate the exterior potential at $(x_{0},y_{0}).$ You may need the integral

$$ {\displaystyle\int} \frac{dx}{\sqrt{x^{2}+b^{2}}}=\sinh^{-1}\left( \frac{x}{b}\right) . $$

7.2.5 Section  7.5

Problem 15.

Suppose a wave front propagates at a speed of 0.25 m s$^{-1}$ and its refractory period lasts 250 ms. Calculate the minimum path length of its reentrant circuit. Most reentrant wave fronts are somewhat slower and briefer than this, so their paths may be shorter.

7.2.6 Section  7.7

Problem 16.

Two electrodes are placed in a uniform conducting medium 10 cm from a cell of radius 5 μm and 10 cm from each other, so that the two electrodes and the cell form an equilateral triangle. When the cell depolarizes the potential rises 90 mV. What will be the potential difference between the two electrodes when the cell orientation is optimum? How many cells would be needed to give a potential difference of 1 mV between the electrodes? Assume $\sigma_{i}/\sigma_{o}=10$.

Problem 17.

Guess whatever parameters you need to predict the voltage at the peak of the QRS wave in lead II. Compare your results to the electrocardiogram of Fig. 7.23.

Problem 18.

At a particular instant of the cardiac cycle, $\mathbf {p}$ is located at the midpoint of a line connecting two electrodes that are 50 cm apart, and $\mathbf {p}$ is parallel to that line. At that instant the magnitude of the potential difference between the electrodes is 1.5 mV. Upon depolarization, the potential change within the cells has magnitude 90 mV.

• (a) What is the magnitude of $\mathbf {p}$?

• (b) If $\sigma _{i}/\sigma _{o}=10$, what is the cross-sectional area of the advancing region of depolarization?

Problem 19.

A semi-infinite slab of myocardium occupies the region $z>0$. A hemispherical wave of depolarization moves radially away from the origin through the slab. At some instant of time the radius of the hemispherical depolarizing wavefront is $R$. Assume that $\mathbf {p}=\int d\mathbf {p}$, that $d\mathbf {p}$ is everywhere perpendicular to the advancing wavefront, and that the magnitude of $d\mathbf {p}$ is proportional to the local area of the wavefront. Find $\mathbf {p}$. Assume that the observation point is very far away compared to $R$.

Problem 20.

Make measurements on yourself and construct Fig. 7.19.

Problem 21.

Experiments have been done in which a dog heart was stimulated by an electrode deep within the myocardium. No exterior potential difference was detected until the spherical wave of depolarization grew large enough so that part of it intercepted one wall of the heart. Why?

Problem 22.

Prove directly from Eq. 7.32 that I${}-{}$II${}+{}$III${}=0$. (It is sometimes said that the equilateral nature of Einthoven’s triangle is necessary to prove this.)

Problem 23.

Derive Eqs. 7.32.

7.2.7 Section  7.8

Problem 24.

Estimate the lower limit for the duration of the QRS complex by calculating the time required for a wave front to propagate across the heart wall. Assume the wall thickness is 10 mm and the propagation speed is 0.2 m s$^{-1}.$

Problem 25.

In an ECG recording, the width of one large square corresponds to 200 ms. A normal heart rate is between 60 and 100 beats min$^{-1}$. The heart rate is usually measured by counting the number of large squares between adjacent QRS complexes.

• (a) How many large squares are there for a normal heart rate?

• (b) In Fig. 7.30 determine the rate of the atria and of the ventricles.

Problem 26.

Consider Lead II of the normal ECG in Fig. 7.23. The QRS wave and the T wave are both positive. Use a 1-dimensional model to convince yourself that the QRS complex and the T wave should have opposite polarities. Why then is the T wave inverted? Find a way to explain the inverted T wave by letting the action potential duration vary between epicardium (outside) and endocardium (inside). On which surface should the duration be longest?

7.2.8 Section  7.9

Problem 27.

Ohm’s law says that $\mathbf {j}=\sigma \mathbf {E}$. Draw what $\mathbf {j}$ and $\mathbf {E}$ look like (a) in a circuit consisting of a battery and a resistor; (b) for the current flowing when a nerve cell depolarizes.

Problem 28.

Obtain the values for $\beta$ for a cube of length $a$ on a side, for a cylinder of radius $a$ and length $h$, and for a sphere of radius $a$.

Problem 29.

Show that Eq. 7.36a is the same as Eq. 6.51 by considering the interior of a single cell stretched along the $x$ axis as in Fig. 6.28. Consider the charge in a small cylindrical region of axoplasm of length $h$ and radius $A$, the cylindrical surface of which is surrounded by cell membrane. Show that the total charge $Q$ within the axoplasm changes according to

$$ \begin{aligned} \frac{\partial Q}{\partial t} & =\pi a^{2}h\frac{\partial\rho_{i}}{\partial t}=C\frac{\partial v_{m}}{\partial t}+i_{m}\\ & =2\pi ah\left( c_{m}\frac{\partial v_{m}}{\partial t}+j_{m}\right) , \end{aligned} $$

and that this can be combined with Eq. 7.36a to give

$$ \begin{aligned} c_{m}\frac{\partial v_{m}}{\partial t}+j_{m} & =\frac{\pi a^{2}h}{2\pi ah}\sigma_{i}\dfrac{\partial^{2}v_{i}}{\partial x^{2}}\\ & =\frac{\sigma_{i}a}{2}\dfrac{\partial^{2}v_{i}}{\partial x^{2}}, \end{aligned} $$

which is the same as Eq. 6.51, except that it is written in terms of $\sigma _{i}$, $A$, and $h$ instead of $A$ and $r_{i}$.

Problem 30.

Clark and Plonsey (1968) solved Eq. 7.34 for a cylindrical axon of radius $a$ using the following method. Assume that the potentials all vary in the $z$ direction sinusoidally, for instance $v_{m}(z)=V\sin (kz),$ where $V$ is a constant.

• (a) Show that the intracellular and extracellular potentials can be written as

  $$ \begin{aligned} v_{i} & = AI_{0}(kr)\sin(kz)\\ v_{o} & =BK_{0}(kr)\sin(kz), \end{aligned} $$

  (7.53)

  where $I_{n}$ and $K_{n}$ are modified Bessel functions obeying the equation

  $$ \frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial v}{\partial r}\right) -\left( k^{2}+\frac{n^{2}}{r^{2}}\right) v=0. $$

• (b) Determine the constants $A$ and $B$ in terms of $V$, using the following two boundary conditions: $v_{m}=v_{i}-v_{o},$ and $\sigma _{i}(\partial v_{i}/\partial r)=\sigma _{o}(\partial v_{o}/\partial r),$ both evaluated at $r=a$. You will need to use the Bessel function identities $dI_{0}(kr)/dr=kI_{1}(kr)$ and $dK_{0}(kr)/dr=-kK_{1}(kr)$. Clark and Plonsey used this result and Fourier analysis (Chap. 11) to determine $v_{i}$ and $v_{o}$ when they are not sinusoidal in $z.$

Problem 31.

Starting with the bidomain equations, divide Eq. 7.44a by $\sigma _{ix}$ and Eq. 7.44b by $\sigma _{ox}$. Now subtract one equation from the other. Under what conditions do the equations contain $v_{m}=v_{i}-v_{o}$ but not $v_{i}$ and $v_{o}$ individually?

7.2.9 Section  7.10

Problem 32.

Verify Eq. 7.47.

Problem 33.

Verify the values given for rheobase and chronaxie in Table 7.1 that are based on Table 6.1.

Problem 34.

An approximation to the error function is given by Abramowitz and Stegun (1972)

$$ \begin{aligned} \operatorname{erf}(x)\approx 1-\left( 1+0.278393x+0.230389x^{2}\right. \\ \left. +0.000972x^{3}+0.078108x^{4}\right) ^{-4},\,x>0. \end{aligned} $$

Calculate $\operatorname {erf}(x)$ using this approximation for $x=0,0.5,$ 1.0, 2.0 and $\infty .$ Using trial and error, determine the value of $x$ for which $\operatorname {erf}(x)=0.5.$ (See Eq. 7.50.)

Problem 35.

Find the equivalent of Eq. 7.45 in terms of the charge required for the stimulation.

Problem 36.

If the medium has a constant resistance, find the energy required for stimulation as a function of pulse duration.

Problem 37.

A typical pacemaker electrode has a surface area of 10 mm$^{2}$. What is its resistance into an infinite medium if it is modeled as a sphere? If it is modeled as a disk? (You will have to use results from Chap. 6 and assign a value for $\sigma _{o}$.)

Problem 38.

Equation 6.51 is the cable equation for a nerve axon. Assume that the axon membrane is passive ($j_{m}=g_{m}(v_{i}-v_{o})$, where $g_{m}$ is a constant).

• (a) Express the equation in terms of $v_{m}$ and $v_{o}$ instead of $v_{i}$ and $v_{o}$, where $v_{m}=v_{i}-v_{o}$.

• (b) Divide the resulting equation by $g_{m}$, and then write the cable equation in terms of the time constant $c_{m}/g_{m}$ and the length constant $1/\sqrt{2\pi a{{r}_{i}}{{g}_{m}}}$.

• (c) Put all the terms containing $v_{m}$ on the left side, and terms containing $v_{o}$ on the right side. The resulting equation should look like Eq. 6.55, except for a new source term on the right side equal to $-\lambda ^{2}\partial ^{2}v_{o}/\partial x^{2}.$ (Measure $v_{m}$ with respect to resting potential so $v_{r}=0$ in Eq. 6.55). The negative of this new term has been called the activating function (Rattay 1987). It is useful when studying electrical stimulation of nerves.

Problem 39.

For this problem, use the activating function derived in Problem 38. Assume that $\lambda$ and $\tau$ are negligibly small, so that $v_{m}$ simply equals the activating function. Consider a point electrode in an infinite, homogeneous volume conductor at distance $d$ from the axon. The extracellular potential is $v_o = \left ( 1/4\pi \sigma_o \right) I/r.$

• (a) Calculate $v_{m}$ as a function of position $x$ along the axon ($x=0$ is the closest position to the electrode).

• (b) Assume that the axon will fire an action potential if $v_{m}$ somewhere along the axon is greater than $V_{\text {threshold}}$. Calculate the ratio of the stimulation current $i$ needed to excite the axon for a cathode (negative electrode) and an anode (positive electrode).

Problem 40.

For this problem, use the activating function derived in Problem 38. An action potential can be excited if a stimulus depolarizes an axon to a value greater than $V_{\text {threshold}}$, and a propagating action potential can be blocked if a stimulus hyperpolarizes to a value of $v_{m}$ less than $-V_{\text {block}}$ ($V_{\text {block}}>V_{\text {threshold}}$).

• (a) For a cathodal electrode [$v_o = \left ( 1/4\pi \sigma_o \right) I/r$] calculate the ratio of the threshold current to the current needed to block propagation.

• (b) Use two electrodes (one cathodal and one anodal ) to design a stimulator that will result in one-way propagation along the axon (say, propagation only in the positive $x$ direction, but blocked in the negative $x$ direction). For an application of such electrodes during functional electrical stimulation, see Ungar et al. (1986).

Problem 41.

For this problem, use the activating function derived in Problem 38, and block by hyperpolarization derived in Problem 40. The factor of $\lambda ^{2}$ in the activating function implies that larger diameter axons are easier to stimulate than smaller diameter axons. Sometimes you want to excite the smaller fibers without the larger fibers (physiological recruitment) . Describe qualitatively how you can use a single electrode and block in the hyperpolarized region to obtain physiological recruitment. For a more complete discussion, see Tai and Jiang (1994).

Problem 42.

In second degree heart block, the wave front sometimes passes through the conduction system and sometimes does not. Qualitatively sketch the ECG for a heart with second degree block for at least five beats. Specifically include the case where every third wave is blocked. Include the P wave, the QRS wave, and the T wave.

Problem 43.

During sinus exit block the SA node functions normally but the wave front fails to propagate from the SA node to the atria. Sketch five beats of an ECG with all beats normal except the third, which undergoes sinus exit block.

Problem 44.

In sick sinus syndrome the SA node has a slow and erratic rate. The AV node and conduction system function properly. You plan to implant a pacemaker in the patient. Should it stimulate the atria or the ventricles? Why?

Problem 45.

A patient with intermittent heart block has an AV node that functions normally most of the time with occasional episodes of block, lasting perhaps several hours. Design a pacemaker to treat the patient. Ideally, your design will not stimulate the heart when it is functioning normally. Describe

• (a) whether you will stimulate the atria or ventricles

• (b) which chambers you will monitor with a recording electrode

• (c) what logic your pacemaker will use to determine when to stimulate. Your design may be similar to a demand pacemaker described in Jeffrey (2001, p. 132).

Problem 46.

The Lapicque strength-duration (SD) curve is

$$ \frac{i}{i_{R}}=1+\frac{t_{C}}{t}, $$

the SD curve in terms of the error function is

$$ \frac{i}{i_{R}}=\frac{1}{\operatorname{erf}\left( \sqrt{0.228t/t_{C}}\right) }, $$

and the SD curve derived in Chap. 6 Problem 37 is

$$ \frac{i}{i_{R}}=\frac{1}{1-e^{-0.693t/t_{C}}}. $$

• (a) Plot all three curves for $0<t/t_{C}<5.$ Use the equation in Problem 34 to evaluate the error function.

• (b) Find approximations for each curve for $t/t_{C}\ll 1$. You may need the Taylor’s series expansions $e^{x}\approx 1+x$ and $\operatorname (x)\approx 2x/\sqrt \pi .$

• (c) Discuss the physical assumptions that were used to derive each curve.

Problem 47.

Consider a pacemaker delivering a $2-\operatorname {mA}$, $1-\operatorname {V}$, $1-\operatorname {ms}$ pulse every second. Pacemakers are often powered by a lithium-iodide battery that can deliver a total charge of $2$ ampere hours.

• (a) What is the energy per pulse?

• (b) What is the average power?

• (c) How long will the battery last?

• (d) Your answer to (c) is an overestimate of battery lifetime, in part because the battery voltage begins to decline before all its charge has been delivered, and in part because the pacemaker circuitry requires a small, constant current. For this pacemaker, add a constant current drain of 5 μA and assume that the useful lifetime of the battery is over when 75 % of the total charge has been delivered. How long will the battery last in this case?

Problem 48.

During stimulation of cardiac tissue through a small anode, the tissue under the electrode and in the direction perpendicular to the myocardial fibers is hyperpolarized, and adjacent tissue on each side of the anode parallel to the fiber direction is depolarized. Imagine that just before this stimulus pulse is turned on the tissue is refractory. The hyperpolarization during the stimulus causes the tissue to become excitable. Following the end of the stimulus pulse, the depolarization along the fiber direction interacts electrotonically with the excitable tissue, initiating an action potential (break excitation). (This type of break excitation is very different than the break excitation analyzed on page 181.)

• (a) Sketch pictures of the transmembrane potential distribution during the stimulus. Be sure to indicate the fiber direction, the location of the anode, the regions that are depolarized and hyperpolarized by the stimulus, and the direction of propagation of the resulting action potential.

• (b) Repeat the analysis for break excitation caused by a cathode instead of an anode. For a hint, see Wikswo and Roth (2009).

Problem 49.

The signal measured during optical mapping, $V$, is a weighted average of the transmembrane potential, $V_m(z)$, as a function of depth,

$$ V = \int_0^\infty {{V_m}(z)w(z)dz}, $$

where $w(z)$ is a normalized weighting function. Suppose the incident light that produces the fluorescence decays with depth exponentially, with an optical length constant $\delta$. Then $w(z) = \textrm {exp}(-z/\delta )/\delta$. Often a shock will cause $V_m(z)$ to fall off exponentially with depth, $V_{m}(z)=V_0\,\textrm {exp}(-z/\lambda )$, where $V_0$ is the transmembrane potential at the tissue surface and $\lambda$ is the electrical length constant (see Sect. 6.6.12).

• (a) Perform the required integration to find an analytical expression for the optical signal, $V$, as a function of $V_0$, $\delta$ and $\lambda$.

• (b) What is $V$ in the case $\delta \ll \lambda$? Explain this result physically.

• (c) What is $V$ in the case $\delta \gg \lambda$? Explain this result physically.

• (d) For which limit do you obtain an accurate measurement of the transmembrane potential at the surface, $V=V_0$?

For additional analysis, see Janks and Roth (2002).

Problem 50.

Consider a two-dimensional sheet of cardiac tissue represented as a bidomain having unequal anisotropy ratios: $\sigma _{ix}=\sigma _{ex}=0.2,\sigma _{iy}=0.02,$ and $\sigma _{ey}=0.08\,$S m$^{-1}$. Assume an insulated obstacle that current must go around is at the center of the sheet. At any point in the tissue, current will divide between the intracellular and extracellular spaces according to their conductivities, with a larger fraction of the current in the space with greater conductivity.

• (a) If current is passed through the tissue in the $x$-direction, determine qualitatively where the tissue is depolarized and where it is hyperpolarized in the region surrounding the insulator. Recall, depolarization occurs where current passes from the intracellular into the extracellular space, and hyperpolarization where current passes from the extracellular into the intracellular space.

• (b) Repeat this analysis if current is passed in the $y$-direction.

• (c) What would be the transmembrane potential if the tissue had equal anisotropy ratios?

  For additional analysis, see Langrill and Roth (2001).

7.2.10 Section  7.11

Problem 51.

When measuring the EEG with electrodes distributed according to the 10–20 system, you obtain measurements of the potential difference between the $i$th electrode ($i=1,\dots ,20$) and the reference electrode ($i=21$). Show that by computing the average reference $v_{i}^{\ast }=\left ( v_{i}-v_{21}\right ) -\left ( 1/20\right ) \sum _{j=1}^{20}\left ( v_{j}-v_{21}\right ) ,$ the resulting values of $v_{i}^{\ast }$ are independent of the reference potential $v_{21}$.

Problem 52.

Consider a very simple model of the EEG: a dipole $\mathbf {p}$ pointing in the $z$ direction at the center of a spherical conductor of radius $R$ and conductivity $\sigma _{o}$. The potential $v_{o}$ can be written as the sum of two terms: the potential of a dipole in an unbounded medium plus a potential that obeys Laplace’s equation

$$ v_{o}=\frac{p\cos\theta}{4\pi\sigma_{o}r^{2}}+Ar\cos\theta $$

where $r$ and $\theta$ are in spherical coordinates, and $A$ is an unknown constant.

• (a) Use Appendix L to show that the second term in the expression for $v_{o}$ obeys Laplace’s equation.

• (b) If the region outside the spherical conductor is air (an insulator), determine the value of $A$ by using the boundary condition that the radial current at the surface of the sphere is zero.

• (c) Calculate $v_{o}$ as measured at the sphere surface $(r=R)$, and determine by what factor $v_{o}$ differs from what it would be in the case of an unbounded volume conductor.

Problem 53.

Suppose you measure the EEG potential $v_{j}$ at $N$ locations $\mathbf {r}_{j}=(x_{j},y_{j},z_{j}),\ j=1,\!\cdots \!,N$. Assume $v_{j}$ is produced by a dipole $\mathbf {p}=(p_{x},p_{y},p_{z})$ located at the origin. Define

$$ R={\displaystyle\sum\limits_{j=1}^{N}} \left[ \frac{p_{x}x_{j}+p_{y}y_{j}+p_{z}z_{j}}{4\pi\sigma\left( x_{j}^{2}+y_{j}^{2}+z_{j}^{2}\right) ^{3/2}}-v_{j}\right] ^{2}, $$

which measures the least-squares difference between the data and the potential predicted by a single-dipole model. (Chap. 11 explores the least-squares method in greater detail.) The goal is to find the dipole components $p_{x},p_{y},p_{z}$ that fit the data best (minimize $R$).

• (a) Minimize $R$ with respect to $p_{x}$ (set $dR/dp_{x}=0$) and find an equation relating $p_{x},p_{y},$ and $p_{z}.$

• (b) Repeat for $p_{y}$ and $p_{z}.$

• (c) Write the three equations in the form $\mathbf {Ap=b}$, where $\mathbf {A}$ is a $3\times 3$ matrix and $\mathbf {b}$ is a $3\times 1$ vector. Find expressions for the components of $\mathbf {A}$ and $\mathbf {b}.$

• (d) If we had not assumed that we knew the location of the dipole, the problem would be much more difficult. Assume the dipole is at location $\mathbf {r}_{p}=(x_{p},y_{p},z_{p})$. Modify $R$ and then try to minimize it with respect to $\mathbf {r}_{p}.$ Carry the calculation far enough to convince yourself that you must now solve nonlinear equations to determine $\mathbf {r}_{p}.$ Press et al. (1992) discuss methods for making nonlinear least squares fits.