Metallic carbon nanotubes were first discovered about 15 years ago. They are sma

Question

Metallic carbon nanotubes were first discovered about 15 years ago. They are small conducting cylinders made of a wrapped sheet of carbon atoms (one atom-thick). Consider a double-wall carbon nanotubes (two concentric cylinders), were the smallest tube has a radius of 1 nm (10^-9m) and the larger tube has a radius of 2 nm. Assume that the two shells are infinitely thin and are separated by air. If the tube is 2 µm long, what is the capacitance of this double-wall nanotube? (hint: You do not need to re-derive the expression for the electric field, take it from the book.

** You do need to derive the expression for the capacitance.)

Explanation / Answer

Let a and b be the inner and outer radius of the clinder. Let the charge density per unit length of the cylinder is and the cylinder length is l.

Let us draw a cylinder or radius r such that a<r<b,

So applying Gauss's law

Integraton E.ds = q(enclosed)/0 (integration over the surface of the drawn cylinder)

E.2rl=l/0

E=/20r

Now let us do a line integration from a to be, which will give the potential difference

int. E.dr(from a to b)=V

/20(log e   r)(a to b)=V

/20(log e   b/a)=V=q/C=l/C

C=20l/ln (b/a)

Just substitute the value of 0, l, b,a.

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