P26_031 - 31. We first need to find an expression for the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 31. We first need to find an expression for the energy stored in a cylinder of radius R and length L, whose surface lies between the inner and outer cylinders of the capacitor (a < R < b). The energy density at any point is given by u = 1 ε0 E 2 , where E is the magnitude of the electric field at that point. If q is the 2 charge on the surface of the inner cylinder, then the magnitude of the electric field at a point a distance r from the cylinder axis is given by q E= 2πε0 Lr (see Eq. 26-12), and the energy density at that point is given by u= 1 q2 . ε0 E 2 = 2 ε L2 r 2 2 8π 0 The energy in the cylinder is the volume integral UR = u dV . Now, dV = 2πrL dr , so R UR = a q2 q2 2πrL dr = 2 ε L2 r 2 8π 0 4πε0 L R a q2 R dr = ln . r 4πε0 L a To find an expression for the total energy stored in the capacitor, we replace R with b: Ub = q2 b ln . 4πε0 L a We want the ratio UR /Ub to be 1/2, so ln or, since 1 2 R 1b = ln a 2a ln(b/a) = ln( b/a), ln(R/a) = ln( b/a). This means R/a = b/a or R = √ ab. ...
View Full Document

This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

Ask a homework question - tutors are online