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Unformatted text preview: 1 ' ( # . In the second dielectric the electric field is reduced to K1 K1!0 A ) 2 * K1 2!0 A E0 Q Qd Qd Qd Qd % 1 1& %d & # 0 # and Vbc # E2 ' ( # . Thus Vac # Vab 0 Vbc # '0 (. K 2 K 2!0 A K1 2!0 A K 2 2!0 A 2!0 A ) K1 K 2 * ) 2 * K 2 2!0 A % 2! A & % K K & 2! A % K K & Qd % K1 0 K 2 & Q # Q ' 0 ( ' 1 2 ( # 0 ' 1 2 (. ' ( . This gives C # Vac d ) K1 0 K 2 * 2!0 A ) K1K 2 * ) Qd * ) K1 0 K 2 * EVALUATE:! An equivalent way to calculate C is to consider the capacitor to be two in series, one with dielectric constant K1 and the other with dielectric constant K 2 and both with plate separation d/2. (Can imagine inserting a thin conducting plate between the dielectric slabs.) !A !A C1 # K1 0 # 2 K1 0 d /2 d !0 A !0 A C2 # K 2 # 2K2 d /2 d 11 1 CC 2! A % K K & Since they are in series the total capacitance C is given by so C # 1 2 # 0 ' 1 2 ( #0 C C1 C2 C1 0 C2 d ) K1 0 K 2 * Vac # 24.72. IDENTIFY:! This situation is analogous to having two capacitors in parallel, each with an area A/2. SET UP:! For capacitors in parallel, Ceq # C1 0 C2 . For a parallel-plate capacitor with plates of area A/ 2, C # !0 A / 2 !0 A / 2 !0 A 0 # ( K1 0 K 2 ) d d 2d !A EVALUATE:! If K1 # K 2 , Ceq # K 0 , which is Eq.(24.19). d EXECUTE:! Ceq # C1 0 C2 # !0 ( A/2) . d Capacitance and Dielectrics 24.73. 24-23 IDENTIFY and SET UP:! Show the transformation from one circuit to the other: Figure 24.73a EXECUTE:! (a) Consider the two networks shown in Figure 24.73a. From Circuit 1: Vac # q1 ! q3 q 0 q3 and Vbc # 2 . Cy Cx %q CxC yCz % q1 q2 & q3 q1 ! q3 q2 ! q3 q& ! T K' 1 ! 2 (. # # . This gives q3 # ' ' C y Cx ( ( ' C y Cx ( Cz Cy Cx Cx 0 C y 0 Cz ) * ) * %1 %1 q1 q1 0 q2 1& 1 q2 q1 0 q2 1 1& # q1 ' 0 ( 0 q2 and Vbc # 0 # q1 0 q2 ' 0 ( . Setting the From Circuit 2: Vac # 0 C1 C3 C3 C2 C3 C3 ) C1 C3 * ) C2 C3 * q3 is derived from Vab : Vab # coefficients of the charges equal to each other in matching potential equations from the two circuits results in three...
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This document was uploaded on 03/11/2014 for the course PHYSICS 240 at University of Michigan.

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