Capacitance of a parallel plate capacitor?

Discussion in 'Homework Help' started by smarch, Jun 30, 2010.

  1. smarch

    Thread Starter Active Member

    Mar 14, 2009
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    Could someone please help me with this question:

    A parallel plate capacitor has a surface area of 1 cm2 and its plates are
    separated by 3 mm. In between the plates are three layers of dielectric,
    each 1 mm thick, with relative permittivities of 3, 5 and 11 respectively. What
    is the capacitance of the whole capacitor?

    I know I use the formula c=eoerA/d.
    Do I add the dielectrics together so er=19? and just apply the formula as usual?
     
  2. t_n_k

    AAC Fanatic!

    Mar 6, 2009
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    Treat this as three capacitors (C1, C2 & C3) in series, each with the same physical dimensions (1mm plate separation) but with different relative permittivity [er1=3, er2=5, er3=11].

    1/Ctot=1/C1+1/C2+1/C3
     
    Last edited: Jun 30, 2010
  3. smarch

    Thread Starter Active Member

    Mar 14, 2009
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    Thanks for your reply, so I use the formula :
    c = Aeo/(d1/er1 + d2/er2 + d3/er3)

    Is that correct?
     
  4. steveb

    Senior Member

    Jul 3, 2008
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    That looks correct to me from memory (which gets less reliable as I age).

    Based on the other problems you are working on, I recommend that you make sure you can derive this formula from Maxwell's equations. TNK's approach is a good one, but the difficulty you are having with the other problems may go away if you work out this simple case from first principles.
     
  5. KL7AJ

    AAC Fanatic!

    Nov 4, 2008
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    Does this really work? I probably ran across this ages ago, but I don't think I've even encountered the question in recent history. :) Very cool!

    Eric
     
  6. t_n_k

    AAC Fanatic!

    Mar 6, 2009
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    No ... your equation is incorrect

    C_1=\epsilon_0\epsilon_{r1}\frac{A}{d}

    C_2=\epsilon_0\epsilon_{r2}\frac{A}{d}

    C_3=\epsilon_0\epsilon_{r3}\frac{A}{d}

    \frac{1}{C_{tot}}=\frac{1}{C_1}+ \frac{1}{C_2}+ \frac{1}{C_3}

    or

    C_{tot}=\frac{C_1C_2C_3}{C_1C_2+C_2C_3+C_1C_3}

    or

    C_{tot}=\epsilon_0 \frac{A}{d}\[\frac{\epsilon_{r1} \epsilon_{r2} \epsilon_{r3}}{\epsilon_{r1} \epsilon_{r2}+\epsilon_{r2} \epsilon_{r3} + \epsilon_{r1} \epsilon_{r3}}\]
     
    Last edited: Jun 30, 2010
  7. steveb

    Senior Member

    Jul 3, 2008
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    TNK,

    His equation appears to me to just be a different form of your equation.
     
  8. t_n_k

    AAC Fanatic!

    Mar 6, 2009
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    Your are quite right. My error. Apologies to 'smarch'.
     
  9. smarch

    Thread Starter Active Member

    Mar 14, 2009
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    No worries TNK, I appreciate your help!
     
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