If the capacitance per unit length of a cylindrical capacitor is 20 picofarads per meter, what is the ratio of the radii of the two cylinders?
We can call the capacitance per unit length, 20 picofarads per meter, 𝐶 per 𝐿. We want to solve for the ratio of the radii of the two cylinders of the cylindrical capacitor. If our cylindrical capacitor looks like this, with an inner radius we call 𝑟 sub 𝑖 and an outer radius we call 𝑟 sub 𝑜, then we want to solve for the ratio 𝑟 sub 𝑖 to 𝑟 sub 𝑜.
To begin, let’s recall the equation for the capacitance of a cylindrical capacitor. That capacitance 𝐶 equals two 𝜋 times ε naught, the permittivity of free space, times the length of the cylindrical capacitor divided by the natural log of the outer radius over the inner radius. The permittivity of free space, we’ll treat as exactly 8.85 times 10 to the negative 12th farads per meter. Regarding our cylindrical capacitor, we don’t know the length 𝑙 of 𝑛. But we do know that the capacitance per unit length is given information.
Let’s rearrange this equation to isolate the radii and solve for the ratio. The natural log of 𝑟 sub 𝑜 over 𝑟 sub 𝑖 equals two 𝜋ε naught 𝐿 over 𝐶. If we raise both sides of this equation to 𝑒, then on the left-hand side, this function cancels out with the natural logarithm, leaving us with 𝑟 sub 𝑜 over 𝑟 sub 𝑖. That fraction equals 𝑒 to the two 𝜋ε naught 𝐿 over 𝐶. We’re given the ratio of 𝐶 over 𝐿 in the problem statement. And ε naught is a constant we know. So we can plug in for the values in this expression. When we do, we’re careful to use units of farads per meter in our expression for capacitance per length.
Entering this term at our calculator, to two significant figures, 𝑟 sub 𝑜 over 𝑟 sub 𝑖 is 16. That means we can write the ratio 𝑟 sub 𝑖 to 𝑟 sub 𝑜 as one to 16. This is the ratio of the inner radius to the outer radius.