Question

6. (12 points) The simplest (homogeneous, isotropic) 'Ohmic' materials are characterized by J=oE, where o is the constant) electrical conductivity (do not confuse it with the surface charge density!), and E is the electric field within the conductor. (a) (4 points Using the time-dependent form of the continuity equation, along with Ohm's law from above, derive a differential equation that describes the time and spa- tial dependence of the volume charge density par, t) within some conductor. Charac- terize (mathematically) the differential equation (partial/ordinary, linear/nonlinear, nth-order, homogeneous/nonhomogeneous, separable/non-separable). (b) (4 points) Yes, you guessed it, solve the differential equation obtained in part (a). Assume an initial condition is specified as part = 0) = po(r). (c) (2 points) What does your solution tell you about the charge density in the conductor? Is the charge moving? If so, where is it going? If not, why not? It might be helpful to imagine the conductor as finite in size. (d) (2 points) What is the typical timescale for the motion of charges in copper (see Griffiths)? Explain the physical significance of this timescale.

Answer 1

If T is the cutrent par urit area (ie.lume current denaity) vand o is the vodume ehotrge vdensily then the continuity Equation can lie wmitten vas J- oP JÁ given where r is the conductivs ty of the matereal vand Ē is wthe electric field. Putting it into O ne obaen Now ने ক . है) Since ( prarn Mam nellib Euation) Le is the fomittivity of the matouiat] maliual de This is the dtiffirlential equation that deseriles the time and Apatial dependence sf the vodume charge dimity withim Aome conductor

Aitam the difb: em: of O (de a mp - - + me Et + plnp-me plř, 0) =P Te (7,t)- Su Any initial free change El Po disiputes sin a sharadtóristic tina t- t/r. yes the chorge. is moving. they ore iy flowing out to the edges of the condiritor.

The Ly pical timescale for coppert s z=10^198, Borically the time constant ? menaures how sord a conductor is. FOn l e te borfeel conductor eis much less then T=0. Foll va good conduotor vthe othor tele vand teimes in that ptroblem ( suppuse in a seillatrny syslemi, rce do copper * ( Vw) otso facoa is a good eondurtor (4 210-135¢ which is A shont).