Conductor capacitance

1.Explain the concept of "conductor capacitance"!

Solution:

Capacitance of a conductor CC is the ability of a conductor to bind and distribute electric charge.

C=QUC = \frac{Q}{U} [C]=C⋅V−1=1F=farad[C] = C \cdot V^{-1} = 1F = farad

Q=Q = charge located in the electric field of the conductor
U=U = voltage of the conductor

Capacitance of a parallel-plate capacitor:

C=ε0⋅εr⋅SlC = \frac{\varepsilon_0 \cdot \varepsilon_r \cdot S}{l}

l=l = distance between the plates, S=S = effective area of the capacitor plates.
ε0=8.854⋅10−12 C2⋅N−1⋅m−2 (F⋅m−1)\varepsilon_0 = 8.854 \cdot 10^{-12} \, C^2 \cdot N^{-1} \cdot m^{-2} \, (F \cdot m^{-1}), permittivity of vacuum.
εr=\varepsilon_r = relative permittivity of the medium between the capacitor plates.

Capacitance of a sphere:

C=4πε0εrRC = 4 \pi \varepsilon_0 \varepsilon_r R

Energy of the electric field of a charged capacitor:

Ee=12CU2=12Q2CE_e = \frac{1}{2} C U^2 = \frac{1}{2} \frac{Q^2}{C}

Capacitors connected in series:

1C=1C1+1C2+1C3+…+1Cn\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots + \frac{1}{C_n}

Capacitors connected in parallel:

C=C1+C2+C3+…+CnC = C_1 + C_2 + C_3 + \ldots + C_n

 

2.Derive the dimension of the physical unit farad = F. Show that it holds: C2.N-1.m-2 = F.m-1

Solution:

 fyzika-kapacita-vodica-2.gif

Thus we have shown that 1 F = A2.kg-1.m-2.s4 and that C2.N-1.m-2 = F.m-1

3.Calculate the capacitance of a plate capacitor composed of 11 plates with dimensions 3 cm and 2 cm, if the distance between the plates is 0.2 mm. The dielectric between the plates is mica with εr = 6. There are 10 gaps between the 11 plates.

Solution:

Analysis:

S = a.b = 3cm·2cm = 6 cm2 = 6·10-4 m2,  l = 2·10-4 m,  εr = 6, ε0 = 8,854·10-12 C2.N-1.m2

fyzika-kapacita-vodica-3.gif 

The capacitance of the capacitor is C' = 1594 pF.

4.What is the voltage between the plates of an air capacitor with two square plates of side 10 cm, separated by 2 cm, if its charge is 8,854·10–3 μC?

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5.What is the capacitance of our Earth?

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6.To what potential will a conductor with a capacitance of 20 picofarads be charged by a charge of 1 microcoulomb?

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7. Determine the relative permittivity of the dielectric in a parallel-plate capacitor whose plates with an area of 1000 cm2 are separated by 0.1 mm and the capacitor is charged by a charge of 17.7·10-6C to 100 V.

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8. Capacitors with capacitances 6·10-6F and 4·10-6F are connected in series, and in parallel with them a capacitor with capacitance 2·10-6F is connected. What is their resulting capacitance?

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9.Two capacitors of the same capacitance are connected a) in series b) in parallel. The difference between the resulting capacitances of both connections is 3·10-6F. Determine the capacitance of each capacitor.

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10. What energy will be stored in a capacitor with plates of area 0.3 m2 and plate separation 1 mm, if we charge it to a voltage of 1000 V? The relative permittivity is 20.

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11. Two capacitors with capacitances 12·10–6F and 24·10–6F are connected in series to a DC voltage source of 30 V. Find:

  • a.)    the resulting capacitance
  • b.)    the charges on the plates of the capacitors
  • c.)    the ratio of the voltages across the individual capacitors.
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12. The plates of a capacitor are 5 mm apart and have an area of 2 m2. The plates are in vacuum. Calculate:

  • a.) the charge on each plate
  • b.) the surface charge density. The voltage across the capacitor is 10 000 V.
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13. Capacitors with capacitances C1 = 2·10-6F and C2 = 3·10-6F are connected in parallel. On the capacitor with capacitance C1 there is a charge Q1 = 6·10-6C. What is the voltage and what is the charge on the second capacitor?

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14. Three capacitors with capacitances C1 = 2·10-6F,  C2 = 3·10-6F,  C3 = 6·10-6F are connected in series. What is the resulting capacitance of this connection? What will be the voltage across each capacitor if the whole battery is connected to a voltage of 200 V?

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15.How will a) the capacitance of the capacitor b) the electric field intensity change if we insert paper between the plates of the capacitor, completely filling the space between them?

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16. The distance between the plates of a parallel-plate capacitor is 8.854 mm, the surface charge density on the plates is 10 nC·m–2. There is air between the plates. What is the voltage between the plates?

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17.The capacitance of the Earth is 709.26·10-6F. The voltage between a nearby cloud and the Earth reached a value of 109V at the moment of the lightning flash. How much electrical energy was released during the lightning?

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18.A Leyden jar with radius 5 cm and height 20 cm has a glass wall thickness of 3 mm. The relative permittivity of the glass is 6. Calculate:

  • a.) the capacitance of the jar
  • b.) the work required to charge the jar to 30 000 V
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19. Determine the resulting capacitance of the capacitors (diagram), if the following holds:

kapacita-vodica-19z

kapacita-vodica-19z2.gif

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20.Calculate the resulting capacitance of the capacitors connected as in the diagram.


kapacita-vodica-20z1

kapacita-vodica-20z2 

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