$[3/6$ Points$]$

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SERPSE$1025.$C$.$P$\mathrm{.}049.$

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A capacitor is constructed from two square, metallic plates of sides $l$ and separation $d.$ Charges $+Q$ and $-Q$ are placed on the plates, and the power supply is then removed. A material of dielectric constant $x$ is inserted a distance $x$ into the capacitor as shown in the figure below. Assume $d$ is much smaller than $x.$ Suggestion: The system can be considered as two capacitors connected in parallel. $($Use the following as necessary: ${}_0,x,l,Q,d,$ and $x.)$

$($a$)$ Find the equivalent capacitance of the device.

$c_{eq}=$

$($b$)$ Calculate the energy stored in the capacitor.

$u$

$($c$)$ Find the direction and magnitude of the force exerted by the plates on the dielectric.

magnitude $\frac{Q^{2}d(k-1)}{2{}_{0}l((k-1)x+1{)}^2}$

direction

$($d$)$ Obtain a numerical value for the force when $x=\frac{l}{2},$ assuming $l=5\mathrm{.}40cm,d=1\mathrm{.}90mm,$ the dielectric is glass $(x=4\mathrm{.}50),$ and the capacitor was charged to $1\mathrm{.}80{10}^{3}vv$ before the dielectric was inserted.

magnitude $$