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Energy of a Capacitor

A capacitor h...

Question

A capacitor has rectangular plates of length a and width b. The top plate is inclined at a small angle as shown in the figure. The plate separation varies from $d = y_{0}$ at the left to d $= 2 y_{0}$ at the right where $y_{0}$ is much less than a or b. What is capacitance of the system?

\frac{ε_{0} a b}{2 y_{0}} l n 2

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\frac{2 ε_{0} a b}{2 y_{0}} l n 2

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\frac{ε_{0} a b}{y_{0}} l n 2

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\frac{ε_{0} a b}{2 y_{0}} l n 2

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Solution

The correct option is C $\frac{ε_{0} a b}{y_{0}} l n 2$

We consider a differential strip of width dx and length b to approximate a differential capacitor of area bdx and separation d $= y_{0} + (\frac{y_{0}}{a}) x .$
thus, dC = $\frac{ε_{0} b d x}{y_{0} + \frac{y_{0}}{a} x}$
$C = \int_{0}^{3} \frac{ε_{0} b d x}{y_{0} + \frac{y_{0}}{a} x} ∴ C = \frac{ε_{0} a b}{y_{0}} l n 2$

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A capacitor has rectangular plates of length a and width b. The top plate is inclined at a small angle as shown in the figure. The plate separation varies from d=y0 at the left to d =2y0 at the right where y0 is much less than a or b. What is capacitance of the system?

The correct option is C ε0aby0ln2 We consider a differential strip of width dx and length b to approximate a differential capacitor of area bdx and separation d =y0+(y0a)x. thus, dC = ε0bdxy0+y0ax C=∫30ε0bdxy0+y0ax∴C=ε0 aby0ln 2

A capacitor has rectangular plates of length a and width b. The top plate is inclined at a small angle as shown in the figure. The plate separation varies from $d = y_{0}$ at the left to d $= 2 y_{0}$ at the right where $y_{0}$ is much less than a or b. What is capacitance of the system?