Capacitance of a Spherical Capacitor?

[Destroxia]

Homework Statement

The plates of a spherical capacitor have radii 51.8 mm and 55.0 mm. (a) Calculate the capacitance. (b) What must be the plate area of a parallel-plate capacitor with the same plate separation and capacitance?

(givens)

a = 51.8 mm
b = 55.0 mm
d = 3.2 mm
ε₀ = 8.85E-12

(unknown)

Part a:
C_sphere = ?

Part b:
A (plate area of parallel-plate capacitor) = ?

Homework Equations

(Spherical Capacitor) C_sphere = 4πε₀ * [(a*b)/(b-a)]

(Parallel Capacitor) C_parallel = (ε₀*A)/d

C_sphere = C_parallel

The Attempt at a Solution

C_sphere = 4π(8.85E-12[F/m]) * ((51.8[mm]*55.0[mm]*E-6)/((55.0[mm] - 51.8[mm])*E-3))

= 9.896E-11 F

= 98.86 pF

I could easily solve the second part just by plugging in this capacitance for the parallel-plate capacitor, but the program I am plugging the answer into tells me my answer is wrong. I have double checked the answer, even triple checked the answer, it seems to be coming down to some calculation error?

 

[AlexCdeP]

I don't think you've done the calculation wrong it looks correct the way you have written it, except the units inside your equation are in meters since you have already converted to that unit. Are you out by a factor of tens/tens. If that's not the issue then I don't know what the problem is.
Just to clarify you have written for example 51.8[mm] when it should be 51.8[m].

 

[BvU]

Hello Ryan, welcome to PF !

Clear posting ! But difficult to help: I followed your working and:

Funny. I get the 99.0 pF too. And A = 358 cm². Not much we can do then, eh ?

 

[gneill]

Your capacitance value looks okay. Is your program expecting some particular units or number of significant figures?

 

[mfb]

I could easily solve the second part just by plugging in this capacitance for the parallel-plate capacitor, but the program I am plugging the answer into tells me my answer is wrong. I have double checked the answer, even triple checked the answer, it seems to be coming down to some calculation error?

It would help to know the area you got, to check that.
It might require some specific units as input?

 

[Destroxia]

Hello Ryan, welcome to PF !

Clear posting ! But difficult to help: I followed your working and:

Funny. I get the 99.0 pF too. And A = 358 cm². Not much we can do then, eh ?

I tried your answer of 99.0 pF and it worked! I looked into it, and apparently even though the program usually takes +/- 1 digit in the 2nd significant digit, this one was to the 3rd significant digit... It appears it was my problem for not looking into the programs syntax for that particular problem.

I don't think you've done the calculation wrong it looks correct the way you have written it, except the units inside your equation are in meters since you have already converted to that unit. Are you out by a factor of tens/tens. If that's not the issue then I don't know what the problem is.
Just to clarify you have written for example 51.8[mm] when it should be 51.8[m].

Your capacitance value looks okay. Is your program expecting some particular units or number of significant figures?

It would help to know the area you got, to check that.
It might require some specific units as input?

Thank you all for your help! Stupid program mistake on my end!

 

[BvU]

I did notice a small change in your post #1 going from = 9.896E-11 F to = 98.86 pF

(and subtly typed 99.0 to leave you room to discover the typo )