Ohm’s Law: V = IR

Power dissipated (Joule heating) =

P = I2R = IV

Kirchoff’s Loop equation:

Kirchoff’s junction rule:

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 + …

0iclosed loopVΣ =

/0ientering leaving junction

IΣ =

Group Problems1. R combo – all 100 Ω

a) Find I from the batteryb) Find I through each R2. a) Find equivalent R (in terms of R)

b) Find the batterycurrent if R = 100 Ω

R4

R2R1

R5

R325V

R

R

10V

R

R

Simple RC Series circuit

• Discharging: Q(t) = Qoexp(-t/RC)and I(t) = Ioexp(-t/RC)

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60

time (ms)

Norm

aliz

ed Q

or

I

When t = RC = 1 time constant, then Q = Qoexp(-1) = 0.37Qo

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 mean-free 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/σto write R = ρL/A

IC IN a IK IL

outside

inside

C

V N a V K V L

GN a GK GL

Cell body

axon

Myelin sheath

dendrites

Terminal endings

retinal “starburst” cell (red) found in visual processing network, Courtesy of Thomas Euler, Max Planck Institute for Medical Research, Heidelberg

x x + Δ x x + 2Δ x

I n s id e

O u ts id e

cΔ x cΔ x cΔ xgΔ x gΔ x gΔ x

r iΔ x r iΔ x

r oΔ x r oΔ x

I i(x ) I i(x + Δ x ) I i(x + 2Δ x )

Io(x ) Io(x + Δ x ) Io(x + 2Δ x )

Im (x ) Im (x + Δ x ) Im (x + 2Δ x ) V M (x ,t) V M (x + Δ x ,t)

V V V

I (pA)

Time (ms)

0

1

Time (ms)

I (pA)

V (mV)