a. Suppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are Q and −Q, respectively. What is the thickness d of the membrane?
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:
## Strain in the Membrane 2
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The wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant k and an equilibrium thickness d0. Assume that the membrane has a very large area A and negligible curvature.
The neuron has "ion pumps" that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is ϵ.