329lect10 - 10 Capacitance and conductance Parallel-plate...

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10 Capacitance and conductance Parallel-plate capacitor: Consider a pair of conducting plates with surface areas A separated by some distance d in free space (see margin). z x y - Q d A Q A The plates are initially charge neutral, but then some amount of electrons are transferred from one plate to the other so that the plates acquire equal and opposite charges Q and - Q , distributed with surface densities of ± Q A on plate surfaces facing one another (as shown in the margin). That way, in steady state and for d ² A , a Feld conFguration con- Fned mainly to the region between the plates is acquired, satisfying the condition that static Feld inside a conductor should be zero. A weak “fringing Feld” can be ignored if d ² A and thus the geometry well approximates the case with inFnite plates. A constant displacement Feld D x Q A satisFes the normal boundary condition at the left plate boundary as well as Gauss’s law ∇· D =0 in the region between the plates. The corresponding electrostatic Feld is E = D ± o x Q ± o A , and the voltage drop from (positive charged) left plate to (negative 1
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V = ± ( d, 0 , 0) (0 , 0 , 0) E · d l = ± d x =0 Q ± o A dx = d ± o A Q. The last result can be expressed as a linear charge-voltage relation Q = CV with C ± o A d representing the capacitance of the parallel conducting plate ar- rangement that we call parallel plate capacitor . As we know from our circuit courses such a capacitor can be used in diverse ways in time-varying Flter circuits as well as for energy and charge storage. A capacitor connected to an external circuit will con- duct a current I = dQ dt ±owing externally in the direction of voltage drop V across capacitor plates, obeying a linear current-voltage relation z x y - Q d A Q A I = C dV dt V ( t ) + - I = C dV dt
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329lect10 - 10 Capacitance and conductance Parallel-plate...

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