Average Energy Density of Capacitor

AI Thread Summary
The discussion revolves around calculating the average energy density of an air-filled capacitor formed by two coaxial cylindrical shells with given radii and an electric potential. The user successfully determined the charge per length but struggled to find the correct length to use in energy density equations. Initially, they misapplied the surface area equation instead of the volume equation for energy density calculations. After realizing the mistake and reorganizing their work, they were able to solve the problem correctly. The thread highlights the importance of using the appropriate formulas in capacitor energy density calculations.
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Homework Statement


An air-filled capacitor is formed from two long conducting cylindrical shells that are coaxial and have radii of 48 mm and 72 mm. The electric potential of the inner conductor with respect to the outer conductor is -536 V (k = 1/4πε0 = 8.99 × 109 N · m2/C2) The average energy density of the capacitor is closest to ?

Homework Equations


V = 2k(lambda) ln(b/a)
Energy stored = energy density * volume

The Attempt at a Solution


Using the first equation, I was able to determine the charge/length to be 7.3*10^-8 C/m.
So I know the two radii, the voltage, and the charge per length.
However, I can't seem to figure out how to get the length to be able to get either charge or capacitance to be able to plug it into

U = 1/2 qv or U = 1/2 cv^2.
 
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I realized that the L's cancel. So if lambda * L is q, using .5*λL / (2π(.0072-.0048)) = .0013 but that's not the correct answer. What am I doing wrong?
 
EDIT: I was using the surface area equation and not the volume one.. whoops. I guess I just needed to organize my work more and I was able to figure it out.
 
I multiplied the values first without the error limit. Got 19.38. rounded it off to 2 significant figures since the given data has 2 significant figures. So = 19. For error I used the above formula. It comes out about 1.48. Now my question is. Should I write the answer as 19±1.5 (rounding 1.48 to 2 significant figures) OR should I write it as 19±1. So in short, should the error have same number of significant figures as the mean value or should it have the same number of decimal places as...
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