(a) Explain the statement the capacitance of a capacitor is 5\(\mu\)F. (b)(i) State the factors upon which the capacitance of a parallel plate capacitor dep...
(a) Explain the statement the capacitance of a capacitor is 5\(\mu\)F.
(b)(i) State the factors upon which the capacitance of a parallel plate capacitor depend.
(ii) State how the capacitance depends on each of these factors stated in (b)(i).
(c) A series arrangement of three capacitors of values 8uF, 12\(\mu\)F, and 24\(\mu\)F is connected in series with 90-V battery.
(i) Draw an open-circuit diagram for this arrangement.
(ii) Calculate the effective capacitance in the circuit.
(iii) On closed circuit, calculate the charge on each capacitor when fully charged.
(iv) Determine the p.d across the 8\(\mu\)F capacitor.
(a) To say that the capacitance of a capacitor is \(5\,\mu\text{F}\) means that the capacitor stores a charge of \(5\,\mu\text{C}\) \((5\times10^{-6}\,\text{C})\) on each plate for every \(1\,\text{V}\) of potential difference applied across its plates. In other words, the ratio of the charge stored to the potential difference across the plates is \(5\,\mu\text{C V}^{-1}\), since \(C=\dfrac{Q}{V}\).
(b)(i) & (ii) The capacitance of a parallel-plate capacitor is given by \(C=\dfrac{\varepsilon A}{d}\). The factors on which it depends and the form of dependence are:
Factor
How the capacitance depends on it
Common (overlapping) area of the plates, \(A\)
Directly proportional: \(C\propto A\). Increasing the area increases the capacitance.
Distance between the plates, \(d\)
Inversely proportional: \(C\propto \dfrac{1}{d}\). Increasing the separation decreases the capacitance.
Permittivity (nature) of the dielectric, \(\varepsilon\)
Directly proportional: \(C\propto \varepsilon\). A dielectric of higher permittivity increases the capacitance.
(c)(i) Open-circuit diagram (switch open) showing the three capacitors in series with the \(90\,\text{V}\) battery:
Open-circuit diagram: the 8 uF, 12 uF and 24 uF capacitors connected in series with the 90 V battery, switch open.
(c)(ii) Effective capacitance. For capacitors in series the reciprocals add:
(c)(iii) Charge on each capacitor. In a series arrangement the same charge flows onto every capacitor, and it equals the charge on the equivalent capacitor:
The potential difference across the \(8\,\mu\text{F}\) capacitor is \(45\,\text{V}\). (As a check: \(V_{12}=360/12=30\,\text{V}\) and \(V_{24}=360/24=15\,\text{V}\); \(45+30+15=90\,\text{V}\), which equals the battery voltage.)
(a) To say that the capacitance of a capacitor is \(5\,\mu\text{F}\) means that the capacitor stores a charge of \(5\,\mu\text{C}\) \((5\times10^{-6}\,\text{C})\) on each plate for every \(1\,\text{V}\) of potential difference applied across its plates. In other words, the ratio of the charge stored to the potential difference across the plates is \(5\,\mu\text{C V}^{-1}\), since \(C=\dfrac{Q}{V}\).
(b)(i) & (ii) The capacitance of a parallel-plate capacitor is given by \(C=\dfrac{\varepsilon A}{d}\). The factors on which it depends and the form of dependence are:
Factor
How the capacitance depends on it
Common (overlapping) area of the plates, \(A\)
Directly proportional: \(C\propto A\). Increasing the area increases the capacitance.
Distance between the plates, \(d\)
Inversely proportional: \(C\propto \dfrac{1}{d}\). Increasing the separation decreases the capacitance.
Permittivity (nature) of the dielectric, \(\varepsilon\)
Directly proportional: \(C\propto \varepsilon\). A dielectric of higher permittivity increases the capacitance.
(c)(i) Open-circuit diagram (switch open) showing the three capacitors in series with the \(90\,\text{V}\) battery:
Open-circuit diagram: the 8 uF, 12 uF and 24 uF capacitors connected in series with the 90 V battery, switch open.
(c)(ii) Effective capacitance. For capacitors in series the reciprocals add:
(c)(iii) Charge on each capacitor. In a series arrangement the same charge flows onto every capacitor, and it equals the charge on the equivalent capacitor:
The potential difference across the \(8\,\mu\text{F}\) capacitor is \(45\,\text{V}\). (As a check: \(V_{12}=360/12=30\,\text{V}\) and \(V_{24}=360/24=15\,\text{V}\); \(45+30+15=90\,\text{V}\), which equals the battery voltage.)
