# 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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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
5,448
784
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
52
0
c = Aeo/(d1/er1 + d2/er2 + d3/er3)

Is that correct?

4. ### steveb Senior Member

Jul 3, 2008
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469
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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321

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
5,448
784
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
2,433
469
TNK,

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

8. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
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Your are quite right. My error. Apologies to 'smarch'.

9. ### smarch Thread Starter Active Member

Mar 14, 2009
52
0
No worries TNK, I appreciate your help!