i. Assume that in the steady state, conditions have been established so that a nonzero, steady current flows, and the electric potential again satisfies V(x)∝xb′, where b′ is another exponent you must find. Derive an expression for the current in terms of V0 and the other given parameters.
Given constants:g=9.8N/kgk=1/4πϵ0=8.99×109N⋅m2/C2c=3.00×108m/sNA=6.02×1023(mol)−1σ=5.67×10−8J/(s⋅m2⋅K4)1eV=1.602×10−19Jme=9.109×10−31kg=0.511MeV/c2sinθ≈θ−θ3/6 for ∣θ∣≪1G=6.67×10−11N⋅m2/kg2km=μ0/4π=10−7T⋅m/AkB=1.38×10−23J/KR=NAkB=8.31J/(mol⋅K)e=1.602×10−19Ch=6.63×10−34J⋅s=4.14×10−15eV⋅s(1+x)n≈1+nx for ∣x∣≪1cosθ≈1−θ2/2 for ∣θ∣≪1
Extra Supplementary Reading Materials:
Electric Slide
Two large parallel plates of area A are placed at x=0 and x=d≪A in a semiconductor medium. The plate at x=0 is grounded, and the plate at x=d is at a fixed potential −V0, where V0>0. Particles of positive charge q flow between the two plates. You may neglect any dielectric effects of the medium.
Context question:
a. For large V0, the velocity of the positive charges is determined by a strong drag force, so that
v=μE
where E is the local electric field and μ is the charge mobility.
i. In the steady state, there is a nonzero but time-independent density of charges between the two plates. Let the charge density at position x be ρ(x). Use charge conservation to find a relationship between ρ(x),v(x), and their derivatives.
Context answer:
vdxdρ+ρdxdv=0
Extra Supplementary Reading Materials:
a. For large V0, the velocity of the positive charges is determined by a strong drag force, so that
v=μE
where E is the local electric field and μ is the charge mobility.
Context question:
ii. Let V(x) be the electric potential at x. Derive an expression relating ρ(x),V(x), and their derivatives. (Hint: start by using Gauss's law to relate the charge density ρ(x) to the derivative of the electric field E(x).)
Context answer:
dx2d2V=−ϵ0ρ
Context question:
iii. Suppose that in the steady state, conditions have been established so that V(x) is proportional to xb, where b is an exponent you must find, and the current is nonzero. Derive an expression for the current in terms of V0 and the other given parameters.
Context answer:
I=8d39ϵ0μAV02
Extra Supplementary Reading Materials:
b. For small V0, the positive charges move by diffusion. The current due to diffusion is given by Fick's Law,
I=−ADdxdρ
Here, D is the diffusion constant, which you can assume to be described by the Einstein relation
D=qμkBT
where T is the temperature of the system.