17_Oct_08 - Electricity and Magnetism I (P331) M. R....

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Electricity and Magnetism I (P331) M. R. Shepherd October 17, 2008 A capacitor with a non-uniform dielectric Consider a parallel-plate capacitor filled with non-uniform dielectric. The dielectric is still linear, in other words, P = ± 0 χ e E , but this time χ e varies as a function of position in the dielectric. Let’s say that χ e varies such that ± r = ± r 0 + ax. Here ± r 0 is a dimensionless constant, x is the distance from one plate, and a is constant with units 1 / length. Let’s assume the area of the plates is A and their separation is d . What is the capacitance? x=0 x=d x Remember the strategy for computing capacitance: put a charge Q on the capacitor and then compute the potential difference between the plates in terms of this charge. The capacitance can then be identified by comparing this expression to V = Q/C . In order to compute V we first need to find the electric field. Let’s charge up the plates so that we have a free surface charge density of + σ f on the left plate and -
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