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5 September, 18:20

What is the time constant of a 9.0-nm-thick membrane surrounding a 0.040-mm-diameter spherical cell? Assume the resistivity of the cell membrane as 3.6*107 Ω⋅m and the dielectric constant is approximately 9.0. Express your answer with the appropriate units.

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  1. 5 September, 18:34
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    Given Information:

    Diameter of spherical cell = 0.040 mm

    thickness = L = 9 nm

    Resistivity = ρ = 3.6*10⁷ Ω⋅m

    Dielectric constant = k = 9.0

    Required Information:

    time constant = τ = ?

    Answer:

    time constant = 2.87*10⁻³ seconds

    Explanation:

    The time constant is given by

    τ = RC

    Where R is the resistance and C is the capacitance.

    We know that resistivity of of any material is given by

    ρ = RA/L

    R = ρL/A

    Where area of spherical cell is given by

    A = 4πr²

    A = 4π (d/2) ²

    A = 4π (0.040*10⁻³/2) ²

    A = 5.026*10⁻⁹ m²

    The resistance becomes

    R = (3.6*10⁷*9*10⁻⁹) / 5.026*10⁻⁹

    R = 6.45*10⁷ Ω

    The capacitance of the cell membrane is given by

    C = kεoA/L

    Where k = 9 is the dielectric constant and εo = 8.854*10⁻¹² F/m

    C = (9*8.854*10⁻¹²*5.026*10⁻⁹) / 9*10⁻⁹

    C = 44.5 pF

    C = 44.5*10⁻¹² F

    Therefore, the time constant is

    τ = RC

    τ = 6.45*10⁷*44.5*10⁻¹²

    τ = 2.87*10⁻³ seconds
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