Respuesta :
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
In this exercise we have to use our knowledge of thermodynamics to calculate the time, so we have:
[tex]t = 2.87*10^{-3}[/tex]
First, organizing the information given in the statement, we have that:
- Diameter of spherical cell = 0.040 mm
- thickness = L = 9 nm
- Resistivity = ρ = 3.6×10⁷ Ω⋅m
- Dielectric constant = k = 9.0
Using the constant time formula we find that:
[tex]T = RC[/tex]
Then it is necessary to start calculating the area to use the formula above, like this:
[tex]A = 4\pir^2\\A = 4\pi(d/2)^2\\A = 4\pi(0.040*10^{-3}/2)^2\\A = 5.026*10^{-9} m^2[/tex]
Using the resistance formula we find that:
[tex]R = \rho L/A\\R = (3.6*10^7*9*10{-9})/5.026*10^{-9}\\R = 6.45*10^7[/tex]
Using the capacitance formula we have:
[tex]C = (9*8.854*10^{-12}*5.026*10^{-9})/9*10^{-9}\\C = 44.5 pF\\C = 44.5*10^{-12} F[/tex]
Returning to the time formula given earlier, we find that:
[tex]T= 6.45*10^{7}*44.5*10^{-12}\\T= 2.87*10^{-3} seconds[/tex]
See more about time at brainly.com/question/2570752
