Intermediate Physics for Medicine and Biology - Russell K. Hobbie & Bradley J. Roth
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Problems |
249 |
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Symbol |
Use |
Units |
First |
0 |
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Electrical permittivity |
C2 |
N−1 |
229 |
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used on |
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of vacuum |
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m−2 |
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page |
κ |
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Dielectric constant |
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229 |
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p, pc, po |
Probability |
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240 |
η |
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Coe cient of viscocity |
Pa s |
248 |
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q |
Charge |
C |
233 |
λD |
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Debye length |
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m |
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230 |
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r, r |
Position |
m |
229 |
λ |
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Characteristic length |
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m |
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235 |
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r |
Radius in cylindrical co- |
m |
236 |
ν |
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Frequency |
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Hz or s−1 |
244 |
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ordinates |
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ρ, ρext |
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Charge density |
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C m−3 |
229 |
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r |
Radius in spherical coor- |
m |
232 |
ρ |
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Resistivity |
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Ω m |
235 |
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dinates |
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σq , σq |
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Charge per unit area |
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C m−2 |
233 |
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t |
Time |
s |
242 |
σ |
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Conductivity |
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S m−1 |
235 |
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u |
rv(r) |
V m |
232 |
σi |
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Standard |
deviation |
of |
A |
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242 |
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u, uo |
Energy (normalized to |
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235 |
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current |
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v, v |
kB T ) |
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σn |
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Standard |
deviation |
of |
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242 |
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Potential |
V |
227 |
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number of ions |
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vNernst |
Nernst potential |
V |
236 |
σq |
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Standard deviation of |
C |
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242 |
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w |
Energy |
J |
240 |
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charge |
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x |
Position |
m |
229 |
σq |
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Charge per unit area |
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C m−2 |
246 |
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x |
Distance along |
m |
236 |
σv |
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Standard deviation of |
V |
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242 |
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cylindrical axis |
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voltage |
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z |
Valence |
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227 |
τ |
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Time constant |
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s |
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235 |
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A, B, A , B |
Constants |
V |
231 |
τ |
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Torque |
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N m |
248 |
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Bearth |
Earth’s magnetic field |
T |
247 |
τt |
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Tissue time constant |
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s |
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246 |
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B0 |
Amplitude of applied os- |
T |
248 |
θ |
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Angle |
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248 |
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cillating magnetic field |
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φ |
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Angle in cylindrical |
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236 |
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C, C |
Concentration |
m−3 |
227 |
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coordinates |
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Ci |
Concentration of species |
m−3 |
230 |
χ |
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Susceptibility |
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N−1 s−1 |
234 |
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[Cl] , [Cl ] |
i |
m−3 |
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ω, ωs, ω0 |
Solute permeability |
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237 |
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Chloride concentration |
228 |
ω |
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Angular frequency |
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s−1 |
246 |
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C |
Capacitance |
F |
242 |
ωt |
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Characteristic angular |
s−1 |
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D, De , D0 |
Di usion constant |
m2 s−1 |
234 |
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frequency of tissue |
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E, Ex, E0, E1 |
Electric field |
V m−1 |
229 |
ξ |
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Energy in units of kB T |
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230 |
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Eext |
External electric field |
V m−1 |
234 |
Γ |
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Radial concentration |
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236 |
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Epol |
Polarization electric |
V m−1 |
233 |
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factor |
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field |
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E |
Photon energy |
J |
244 |
Problems |
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F |
Faraday constant |
C mol−1 |
227 |
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F, F |
Force |
N |
234 |
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G |
Conductance |
S |
235 |
Section 9.1 |
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J |
Current per unit area of |
A m−2 |
237 |
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[K] , [K ] |
membrane |
m−3 |
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Problem 1 The chloride ratio between plasma and in- |
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Potassium |
228 |
terstitial fluid is 0.95. Plasma protein has a valence of |
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concentration |
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L |
m |
235 |
about |
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18. In the interstitial fluid, Na = Cl = 155 |
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Separation |
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M+ , M+ |
Concentration of imper- |
m−3 |
228 |
mmol l−1. Find the sodium, chloride and protein concen- |
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M− , M− |
meant cations |
m−3 |
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trations in the plasma and the potential di erence across |
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Concentration of imper- |
228 |
the capillary wall, assuming Donnan equilibrium. |
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meant anions |
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[M] , [M ] |
Net concentration of im- |
m−3 |
228 |
Problem 2 Suppose that there are |
two |
compartments |
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permeant ions |
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with equal volume V = 1 l, separated by a membrane that |
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N |
Number per unit volume |
m−3 |
233 |
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NA |
Avogadro’s number |
mol−1 |
234 |
is permeable to K and Cl ions. Impermeant positive ions |
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[Na] , [Na ] |
Sodium concentration |
m−3 |
228 |
have a concentration 0 on the left and M = M+ =10 |
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P |
Polarization |
C m−2 |
234 |
mmol l−1 on the right. The initial concentration of potas- |
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R |
Gas constant |
J K−1 |
227 |
sium is [K0] = 30 mmol l−1 on the left. T = 310 K. |
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mol−1 |
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(a) Find |
the initial concentrations |
of potassium |
and |
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R |
Resistance |
Ω |
235 |
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chloride on both sides and the potential di erence. |
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Rp |
Pore radius |
m |
236 |
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S |
Area |
m2 |
229 |
(b) A fixed amount of potassium chloride (10 mmol) is |
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T |
Temperature |
K |
227 |
added on the left. After things have come to equilibrium, |
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U, W |
Energy |
J |
242 |
find the new concentrations and potential di erence. |
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V |
Particle velocity |
m s−1 |
234 |
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α |
Proportionality |
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240 |
Problem 3 The extracellular space in cartilage contains |
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constant |
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large, immobile, negatively charged molecules called gly- |
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β |
Linear viscous drag coef- |
N s m−1 |
234 |
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ficient |
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coaminoglycans (GAGs). An early sign of osteoarthritis |
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β |
Rotational viscous drag |
N s m |
248 |
is the loss of GAGs. The concentration of the GAGs is |
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coe cient |
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di cult to |
measure directly, but |
Shapiro |
et al. (2002) |
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References |
253 |
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Problem 32 Obtain Eq. 9.79 from the expression U = |
Adair, R. K. (2000). Static and low-frequency magnetic |
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−mB cos θ that was derived in Problem 8.29, by making |
field e ects: Health risks and therapies. Rep. Prog. Phys. |
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a suitable expansion for small angles. |
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63: 415–454. |
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Problem 33 Electric fields in the body caused by expo- |
Adair, R. K., R. D. Astumian, and J. C. Weaver (1998). |
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Detection of weak electric fields by sharks, rays |
and |
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sure to power lines are produced by two mechanisms: di- |
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skates. Chaos 8: 576–587. |
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rect coupling to the power line electric field, and Fara- |
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Astumian, R. D., J. C. Weaver, and R. K. Adair (1995). |
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day induction from the power line magnetic field. Con- |
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Rectification and signal averaging of weak electric fields |
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sider a high-voltage power line that produces an electric |
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by biological cells. Proc. Nat. Acad. Sci. USA 92: 3740– |
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field of 10 kV/m and a magnetic field of 50 mT [Barnes, |
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3743. |
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(1995)]. Estimate the electric field induced in the human |
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Barnes, F. S. (1995). Typical electric and magnetic field |
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body by these two mechanisms. Which is larger? Com- |
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exposures at power-line frequencies and their coupling |
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pare the strength of these powerline-induced electric fields |
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to biological systems. In M. Blank, ed. Electromagnetic |
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to the strength of naturally occurring electric field pro- |
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Fields: Biological Interactions and Mechanisms. Wash- |
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duced in the body by the heart (estimate the strength of |
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ington, American Chemical Society, pp. 37–55. |
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this endogenous field using the data in Fig. 7.23). |
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Bastian, J. (1994). Electrosensory organisms. Physics |
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Problem 34 Derive the equations for the electric field |
Today 47(2): 30–37. |
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Bockris, J. O’M., and A. K. N. Reddy (1970). Modern |
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shown |
in Fig. |
9.19. Use |
the following |
method. |
Let |
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Electrochemistry. New York, Plenum, Vol. 1. |
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the potentials be voutside |
= |
A cos θ/r |
2 |
− E1r cos θ and |
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Bren, S. P. A. (1995). 60 Hz EMF health e ects—a |
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vintracellular = Br cos θ, where A and B are unknown con- |
scientific uncertainty. IEEE Eng. Med. Biol. 14: 370–374. |
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stants. At the cell surface, the following boundary condi- |
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Carstensen, E. L. (1995). Magnetic fields and cancer. |
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tion applies when the cell membrane is thin and obeys |
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IEEE Eng. Med. Biol. 14: 362–369. |
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Ohm’s law: |
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Chandler, W. K., A. L. Hodgkin, and H. Meves (1965). |
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∂voutside |
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σ |
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The e ect of changing the internal solution on sodium |
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outside |
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∂r |
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inactivation and related phenomena in giant axons. J. |
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r=a |
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= σ |
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∂vintracellular |
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Physiol. 180: 821–836. |
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Crouzy, S. C., and F. J. Sigworth (1993). Fluctuations |
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intracellular |
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∂r |
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r=a |
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in ion channel gating currents: Analysis of nonstationary |
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σmembrane |
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= (voutside |
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vintracellular) |
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shot noise. Biophys. J. 64: 68–76. |
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b |
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r=a |
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DeFelice, L. J. (1981). Introduction to Membrane |
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(a) |
Verify |
that |
the |
expressions |
for |
voutside |
and |
Noise. New York, Plenum. |
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vintracellular obey Laplace’s equation and behave properly |
Denk, W., and W. W. Webb (1989). Thermal-noise- |
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at r = 0 and r = ∞. |
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limited transduction observed in mechanosensory recep- |
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(b) Use the boundary condition to determine A and B. |
tors of the inner ear. Phys. Rev. Lett. 63(2): 207– |
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(c) Use your expressions for the potential to determine |
210. |
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the electric fields given in Fig. 9.19. |
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Doyle, D. A., J. M. Cabral, R. A. Pfuetzner, A. Kuo, J. |
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M. Gulbis, S. L. Cohen, B. T. Chait, and R. MacKinnon |
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(1998). The structure of the potassium channel: Molecu- |
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References |
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lar basis of K+ conduction and selectivity. Science 280: |
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69–77. |
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Abramowitz, M., and I. A. Stegun (1972). Handbook |
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Adair, R. (1991). Constraints on |
biological e ects |
of Biological E ects of Electromagnetic Fields. Boca Ra- |
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of weak extremely-low-frequency electromagnetic fields. |
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Hafemeister, D. (1996). Resource Letter BELFEF-1: |
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on biological e ects of weak extremely-low-frequency |
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Adair, R. (1993). E ects of ELF magnetic fields on |
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biological magnetite. Bioelectromagnetics 14: 1–4. |
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high-resolution current recording from cells and cell-free |
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Adair, R. (1994). Constraints of thermal noise on the |
membrane patches. Pfl¨ugers Arch. 391: 85–100. |
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e ects of weak 60-Hz magnetic fields acting on biological |
Hille, B. (2001). Ion Channels of Excitable Membranes, |
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magnetite. Proc. Nat. Acad. Sci. USA 91: 2925–2929. |
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