A cylindrical layer of a homogeneous dielectric with the dielectric constant ε is introduced into a cylindrical capacitor so that the layer fills the gap of width d between the plates. The mean radius of the plates is R such that R >> d. The capacitor is connected to a source of a permanent voltage U. Find the force pulling the dielectric inside the capacitor.
Text Solution
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Sol. Using the formula W = q 2 /2C for the energy of a capacitor, we find that, in accordance with, the required force is
F x = – 
= 
= 
. ….. (1)
Since d << R the capacitance of the given capacitor can be calculated by the formula for a parallel-plate capacitor. Therefore, if the dielectric is introduced to a depth x and the capacitor length is l, we have
C = 
+ 
= 
(εx + l – x)... (2)
Substituting (2) into (1), we obtain
F x = ε 0 (ε – 1)πRU 2 /d.
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