Inserting a dielectric between the plates of a parallel plate capacitor

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  • #1
archaic
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Homework Statement:
We have a capacitor made of two square plates of length ##\ell##, and separated by a distance ##d##. Charges ##+Q## and ##-Q## are placed on the plates.
A material of dielectric constant ##\kappa=4.50## is inserted a distance ##x## into the capacitor as shown in the figure below.
The system can be considered as two capacitors connected in parallel.
a) Find the equivalent capacitance.
b) Calculate the energy stored in the capacitor.
c) Find the direction and magnitude of the force exerted by the plates on the dielectric.
b) If the capacitor was charge to ##2.20\times10^3## V before the dielectric was inserted, then give a numerical value of the force in micro newtons if ##x=\ell/2##, ##\ell=5.10## cm, and ##d=1.80## mm.
Relevant Equations:
$$C=\frac{Q}{\Delta V}$$
26-p-071.gif

a)$$C=\frac{\kappa\epsilon_0\ell x}{d}+\frac{\epsilon_0\ell(\ell-x)}{d}=\frac{\epsilon _0ℓ\left(\kappa x+ℓ-x\right)}{d}$$
b)$$U=\frac{1}{2C}Q^2=\frac{dQ^2}{2\epsilon _0ℓ\left(\kappa x+l-x\right)}$$
c)$$F=-\frac{dU}{dx}=\frac{dQ^2\left(\kappa -1\right)}{2\epsilon _0ℓ\left(\kappa x+ℓ-x\right)^2}$$and it is directed to the right, since it is positive.
b) I first find the charge using ##Q=C_0\Delta V=\frac{\epsilon_0\ell^2\Delta V}{d}##$$\begin{align*}
F&=\frac{d\left(\kappa -1\right)}{2\epsilon _0ℓ\left(\kappa\fracℓ2+ℓ-\fracℓ2\right)^2}\frac{\epsilon_0^2\ell^4(\Delta V)^2}{d^2}\\
&=\frac{d\left(\kappa -1\right)}{2\epsilon _0\frac{ℓ^3}{4}\left(\kappa+1\right)^2}\frac{\epsilon_0^2\ell^4(\Delta V)^2}{d^2}\\
&=\frac{2\epsilon_0\ell(\Delta V)^2(\kappa-1)}{d(\kappa+1)^2}\\
&=\frac{2(5.10\times10^{-2})(2.20\times10^3)^2(4.50-1.00)}{(4\pi\times8.99\times10^9)(1.80\times10^{-3})(4.50+1)^2}\\
&=0.000281\,N=281\,\mu N
\end{align*}$$
 

Answers and Replies

  • #2
archaic
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I think that I have a mistake in my force calculation, but I can't see it..
 
  • #3
haruspex
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I think that I have a mistake in my force calculation, but I can't see it..
What makes you think it is wrong?
In exactly what form did you present your answer?
 
  • #4
archaic
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What makes you think it is wrong?
In exactly what form did you present your answer?
I have found the same exercise in my textbook, but with different values, and when I plug their values I get a different answer.
It might be that they are wrong..
Do you see no mistakes?
 
  • #5
haruspex
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I have found the same exercise in my textbook, but with different values, and when I plug their values I get a different answer.
It might be that they are wrong..
Do you see no mistakes?
I can't see any. Didn't check the numerical result precisely, but it is in the ballpark.
 
  • #6
archaic
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I can't see any. Didn't check the numerical result precisely, but it is in the ballpark.
It was the correct answer after all.. Thanks!
 

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