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?

 

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

 

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

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

Is that correct?

 

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

Is that correct?

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.

 

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

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

 

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

Is that correct?

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}\[\frac{\epsilon_{r1} \epsilon_{r2} \epsilon_{r3}}{\epsilon_{r1} \epsilon_{r2}+\epsilon_{r2} \epsilon_{r3} + \epsilon_{r1} \epsilon_{r3}}\]\)

 

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}\[\frac{\epsilon_{r1} \epsilon_{r2} \epsilon_{r3}}{\epsilon_{r1} \epsilon_{r2}+\epsilon_{r2} \epsilon_{r3} + \epsilon_{r1} \epsilon_{r3}}\]\)

TNK,

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

 

TNK,

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

Your are quite right. My error. Apologies to 'smarch'.

 

No worries TNK, I appreciate your help!