Lecture 16 Wednesday February 23, Chapter 23:9 The nervous system
1. Take: Collect HW
2. Demos—none?
3. Review of RC circuit
a. Charging: I (t )  I 0 e

t
RC
and
t



VC (t )   1  e RC 
i.


  RC
b.
c.
I (t )  I 0 e
Discharging
i.
d.
e.

t
RC
V (t )  V0 e
Voltage across C: C

t
RC
Find Q(t) from Q(t)=CVC(t)
X
4. 23:9 Electricity in the nervous system
a. Cell membrane about 7 nm thick
i. Transport through membrane
1. Sodium-potassium exchange pumps
a. Na+ out, K+ in
b. Resting state concentration of Na+ outside 10x inside
c. K+ more concentrated on inside
d. Net is that cell is at a lower potential than surrounding fluid
2. Sodium and potassium channels
a. Normally closed, when open Na+ flows in, K+ out
3. Since more positive charge outside than in, electric field across cell
membrane. Voltage difference is about 70 mV
a. What is the electric field in the membrane? E= 1. X 107 inward
4. For the membrane, ρ = 36 x 106 Ω•m
a. What is the resistance of the membrane? Section of length L and
cell is a sphere of radius 50 μm.?
b. A=πr2 = π (50x10-6 m)2 = 7.85 x 10-9 m2, L=7nm
c. RMEMBRANE = ρL/A= 36 x 106 Ω•m*7x10-9/7.85 x 10-9 =32MΩ
5. What is capacitance of the membrane, κ = 9.0?
a. C=Kε0A/d=9*8.85x10-12*7.85 x 10-9/7x10-9 = 8.9x10-11
6. What is time constant τ for the membrane? Τ=RC=32x106 * 8.9x10-11 =
3 ms
5.
6.
7.
8.
9.
10.
2. Model of conduction…
3.
4.
If it took 3ms to charge each node and the length of a node is about 1mm.
a. V=LNODE/τ = .001m/.003s = .3 m/s
b. So what has mother Nature done?
i. Wrap nerve cell in a myelin sheath which makes the capacitance smaller by
increasing d so C ≈1.6 pF per segment (compared to 90 pF above)
ii. Resistance between one node and the next is ≈ 25MΩ, so now
1. Ττ ≈ 25x106Ω*1.6x10-12F= 40 μs
2. v = .001/(40x10-6) = 25 m/s
c. MS and ALS
5. Assignments that were due Friday:
a.
b.
c.
d.
e.
f.
g.
*CQ8: Two bulbs in series, shunt one
*CQ9: 3 R in series compared to one, V applied same
*CQ10: same in parallel
MC33: given V across 3 in parallel and total current, find 3rd R given 2
18: series parallel equivalent
23: Resistor reduction, most difficulty in reducing diagram to simple case
WB 7-12: R=ρL/A
6. Assignments due today:
a. Problems: 23: 25, 26, 29, 30, 31, 35, 36, 40, WB: 13-17, 19-21
i. 25:given P and V, find I and R
ii. 26: loss in wiring
iii. 29: Given P,V, find energy
iv. 30: 3 C in parallel, find eq
v. 31: ditto in series
vi. 35: 2C in parallel in series with 3rd
vii. 36: tau for C and 2R in series
viii. 40: I(t) for discharging capacitor
7. Assignments due Wednesday
a. 23: CQ27, MC36,42,74,75,76, 77
i. CQ27: Halve the distance between nodes of Ranier,
1. How are R and C affected? R and C each halved
2. Time constant? ¼ τ = 10μs
3. Vv = L/τ = (1/2) / (1/4) = 12 x speed
ii. MC36: double cell d, R and C?
iii. 42: speed of propagation given t and d
iv. 74: RC for defibrillator: given R and C and V which graph
v. 75: same find τ
vi. 76: increase R, what happens to I and τ
vii. 77: change V what happens to τ
11.