Chapter 26: Q. 74 (page 741)

A vacuum-insulated parallel-plate capacitor with plate separation $d$ has capacitance $C_{0}$ . What is the capacitance if an insulator with dielectric constant $κ$ and thickness $d / 2$ is slipped between the electrodes without changing the plate separation?

Short Answer

Expert verified

If an insulator with dielectric constant $κ$ and thickness $d / 2$ is slipped between the electrodes without changing the plate separation then the capacitance is $C_{s} = \frac{2 κ C_{0}}{κ + 1}$

Step by step solution

Given Information

Parallel-plate capacitor with plate separation $= d$

Capacitance $= C_{0}$

Dielectric constant $= k$

Thickness $= d / 2$

Explanation

If we insert an insulator into such a capacitor with separation $d$ and $k = 1$ , then we can treat it as two capacitors connected in series, each with thickness $d / 2$ and $κ = 1$

It is clear that the capacitance of the air half will be

$C_{1} = 2 C_{0}$

Since the capacitance is given as

$C = ε_{0} κ \frac{A}{d} .$

The capacitance of the other half will then clearly be

$C_{2} = 2 κ C_{0} .$

Resulting capacitance of the considered system will be

$C_{s} = {(\frac{1}{2 C_{0}} + \frac{1}{2 κ C_{0}})}^{- 1} = {(\frac{κ + 1}{2 κ C_{0}})}^{- 1}$

The resulting capacitance will be

role="math" localid="1648714430037" $C_{s} = (\frac{2 κ C_{0}}{κ + 1})$

Final Answer

Hence, the capacitance is $C_{s} = \frac{2 κ C_{0}}{κ + 1} .$

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A vacuum-insulated parallel-plate capacitor with plate separation $d$ has capacitance $C_{0}$ . What is the capacitance if an insulator with dielectric constant $κ$ and thickness $d / 2$ is slipped between the electrodes without changing the plate separation?

Short Answer

Step by step solution

Given Information

Explanation

Final Answer

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A vacuum-insulated parallel-plate capacitor with plate separation dhas capacitance C0. What is the capacitance if an insulator with dielectric constant κand thickness d/2is slipped between the electrodes without changing the plate separation?

Short Answer

Step by step solution

Given Information

Explanation

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

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A vacuum-insulated parallel-plate capacitor with plate separation $d$ has capacitance $C_{0}$ . What is the capacitance if an insulator with dielectric constant $κ$ and thickness $d / 2$ is slipped between the electrodes without changing the plate separation?