P26_031

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

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

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