The Leyden jar, invented by chance in 1746, was one of the earliest types of cap

Question

The Leyden jar, invented by chance in 1746, was one of the earliest types of capacitors (see the figure below), made by covering the inner and outer surfaces of a glass jar with conducting layers (for example, aluminum foil). It is a good approximation to treat the Leyden jar as a parallel-plate capacitor. (a) Find the capacitance for a cylindrical jar of height 19 cm and radius 6 cm, made of 3 mm thick Pyrex glass. (The dielectric strength of the Pyrex glass is 1.00 107 V/m.) Incorrect: Your answer is incorrect. (b) What is the maximum charge that could be stored on it?

Explanation / Answer

Hi,

a) If we have a cylindrical capacitor, then we know that its capacitance can be calculated as:

C = L/[ 2ke Ln(R/(R-e)) ] ; where L is the height of the cylinder, ke is the electric constant (9*109 Nm2/C2 ), R is the radius of the cylinder and e is the thickness of the cylinder.

However, the previous equation was deduced considering that the gap between the two surfaces was filled with air. As this is not true and we have a dielectric, the real capacitance should be:

C = k Co ; where k is the dielectric constant of the material (pyrex glass in this case).

If we investigate a little, we can find that the value of k for the Pyrex glass is about 5.6.

So, the capacitance of the Leyden jar would be:

Co = (19*10-2 m) / [ 2*9*109 Nm2/C2 Ln (60 mm / (60 - 3) mm) ] = 2.06*10-10 F

C = (5.6)(2.06*10-10 F) = 1.15*10-9 F

(b) The relation between the charge and the voltage was:

Q = CV ; therefore, if we want the maximum charge, we need the maximum voltage (since the capacitance is a constant)

The maximum voltage can be found if we consider that the electric field in this capacitor is uniform. If that is true, then:

V = eE ; where e is the thickness of the cylinder and E is the dielectric strength.

So, the value of the maximun charge that can be stored is:

Vmax = eEmax = (3*10-3m)(1*107 V/m) = 30000 V

Qmax = CVmax = (1.15*10-9 F)(30000 V) = 3.45*10-5 C

I hope it helps.

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