Capacitance of a spherical cell

[fatsack]

Homework Statement

A cell membrane is composed of lipid molecules and approximately 10.29 nm thick. If the dielectric constant of the lipid is kappa= 5 , what is the approximate capacitance of a spherical cell that has a diameter of 10.29 micro meters?

Homework Equations

E=KQ/r^2
C=Q/deltaV=1/K(1/r1-1/r2)
r2=D/2

The Attempt at a Solution

So I've come up with the final solution for to get the answer is Kappa*C=>Kappa(1/K(1/r1-1/r2). I'm not really sure what I am doing wrong, but I keep coming up with answers like 5.72e-18F, which is incorrect. Any help would be greatly appreciated.

 

[kuruman]

You need to show exactly how you came up with that number. Otherwise, we can't figure out why it is incorrect. Do you know the correct answer?

 

[fatsack]

I don't know the correct answer.
so what I did was used the aforementioned formula and insterted all the values(after converting them). My guess is I went wrong somewhere with the (1/r1-1/r2). I used the thickness for r1 and the diameter of the sphere/2 for r2. anyway, here is the what I got:
5(1(8.99e9((1/1.029e-8m)-(1/1.029e-5/2))))=5.73e-18

 

[kuruman]

You cannot use the thickness for r1. In the expression for the capacitance r1 is the inner radius and r2 is the outer radius. The thickness is d = r2 - r1. Now note that this thickness is one-thousandth of the diameter, very thin. You need an approximation. What do you get for the difference

$$\frac{1}{r_1}-\frac{1}{r_2}$$

after subtracting the fractions? Do it with symbols - don't put in numbers yet.

 

[fatsack]

so r1=(d/2)-thickness
that makes a lot more sense and yielded a correct answer!
thanks a bunch!

 

[kuruman]

Not what I had in mind, but with today's calculators, I guess it is possible. For whatever it's worth, this is where I was going

$$\frac{1}{r_1}-\frac{1}{r_2}=\frac{r_{2}-r_{1}}{r_{2}r_{1}}\approx\frac{d}{r^{2}}$$

The last approximation is valid because the radii are so close to each other. It is a useful approximation to know.