Ohm’s Law: V = IR
Power dissipated (Joule heating) =
P = I2R = IV
Kirchoff’s Loop equation:
Kirchoff’s junction rule:
Σ
closed loop
Σ
Vi = 0
entering / leaving junction
Ii = 0
Resistors in series: Req = R1 + R2 + …
Resistors in parallel: Req-1 = R1-1 + R2-1 +…
Capacitors in series: Ceq-1 = C1-1 + C2-1 + …
Capacitors in parallel: Ceq = C1 + C2 + …
Group Problems
1. R combo – all 100 Ω
R1
25V
R2
R3
a) Find I from the battery
R5
R4
b) Find I through each R
2. a) Find equivalent R (in terms of R)
R
b) Find the battery
current if R = 100 Ω
R
R
10V
R
Simple RC Series circuit
Norm alized Q or I
• Discharging: Q(t) = Qoexp(-t/RC)
and
I(t) = Ioexp(-t/RC)
1
0.8
0.6
0.4
0.2
0
0
10
20
30
tim e (m s)
When t = RC = 1 time constant, then
Q = Qoexp(-1) = 0.37Qo
40
50
60
Microscopic Picture of Electric
Current
• E field in wire produces forces on free electrons
leading to a net drift velocity: vdrift = aτ, where a is
the acceleration (a = F/m = eE/m = e(V/L)/m where V is
the applied voltage across the wire of length L) and τ is the meanfree time between collisions
• Solving for the charge drifting by allows us to
compute the current I: I=[(ne2τ/m)(A/L)]V =
[1/R]V = GV, where n is the free charge density, e is the
electron charge, A is the wire cross-sectional area, R is the wire’s
resistance and G is its conductance
• We can also introduce intrinsic parameters:
conductivity, σ = ne2τ/m and resistivity, ρ = 1/σ
to write R = ρL/A
outside
IC
IN a
IK
GNa
IL
GK
GL
C
V Na
VK
inside
VL
dendrites
axon
Myelin sheath
Cell body
Terminal endings
retinal “starburst” cell (red) found in visual processing network, Courtesy of Thomas
Euler, Max Planck Institute for Medical Research, Heidelberg
I n s id e
I i( x )
r iΔ x
I i( x + Δ x )
Im (x )
gΔ x
cΔ x
V M ( x ,t)
cΔ x
V
Io(x )
roΔ x
x
r iΔ x
I i( x + 2 Δ x )
Im (x + Δ x )
V M ( x + Δ x ,t)
gΔ x
cΔ x
V
Io(x + Δ x )
x+Δx
O u t s id e
roΔ x
Im (x + 2 Δ x )
gΔ x
V
Io(x + 2 Δ x )
x+2Δx
I (pA)
0
1
Time (ms)
V (mV)
I (pA)
Time (ms)