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Lecture 4 Capacitance and Capacitors Chapter 16.6 16.10 Outline Definition of Capacitance Simple Capacitors Combinations of Capacitors Capacitors with Dielectrics
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Capacitance Introduction The capacitance C of a capacitor is the ratio of the charge (Q) on either conductor plate to the potential difference ( V) between the plates. Q C V General definition Units of capacitance are farads (F) 1F 1C/1V C(Earth) ~ 1 F C(adult) ~ 150 pF = 150 10 12 F History
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The Parallel-Plate Capacitor The capacitance of a parallel-plate capacitor whose plates are separated by air is: A C = є 0 d A is the area of one of the plates d is the distance between the plates є 0 is permittivity of free space More about capacitors
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Capacitors Problem: A parallel-plane capacitor has an area of A=5cm 2 and a plate separation of d=5mm. Find its capacitance. Unit conversion: A = 5 cm 2 = 5 10 4 m 2 d = 5 mm = 5 10 3 m C = є 0 A/d = 8.85 10 12 C 2 / (N m 2 ) 5 10 4 m 2 / 5 10 3 m = 8.85 10 13 C 2 /(N m)= 8.85 10 13 F = 0.885 pF N/C = V/m C/N = m/V, F=C/V C 2 /(N m)=C (C/N)/m = C (m/V)/m = C/V = F
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Combinations of Capacitors In real electric circuits capacitors can be connected in various ways. In order to design a circuit with desired capacitance, equivalent capacitance of certain combinations of capacitors can be calculated. There are 2 typical combinations of capacitors: Parallel combination Series combination
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Parallel Combination
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The left plate of each capacitor is connected to the positive terminal of a battery by a wire the left plates are at the same potential the potential differences across the capacitors are the same, equal to the voltage of the battery ( V). The charge flow ceases when the voltage across the capacitors equals to that of the battery and the capacitors reach their maximum charge. Q = Q 1 + Q 2 Q 1 = C 1 V Q 2 = C 2 V Q = C eq V C eq V = C 1 V + C 2 V C eq = C 1 + C 2 Examples
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Series Combination
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The magnitude of the charge is the same on all the plates. The equivalent capacitor must have a charge –Q on the right plate and +Q on the left plate. Q V = C eq V = V 1 + V 2 V 1 = Q/C 1 V 2 = Q/C 2 Q Q Q = + C eq C 1 C 2 1 1 1 = + C eq C 1 C 2 Examples
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Energy Stored in a Capacitor The work required to move a charge Q through a potential difference V is W = V Q. V = Q/C, Q is the total charge on the capacitor. The voltage on the capacitor linearly increases with the magnitude of the charge. Additional work increases the energy stored. W = ½ Q V = ½ (C V) V = ½C ( V) 2 = Q 2 /2C
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Capacitors with Dielectrics A dielectric is an insulating material. The dielectric filling the space between the plates completely increases the capacitance by the factor > 1, called the dielectric constant. If V 0 is the potential difference (voltage) across a capacitor of a capacitance C 0 and a charge Q 0 in the absence of a dielectric. Filling the capacitor with a dielectric reduces the voltage by the factor to V, so that V = V 0 / . C = Q 0 / V = Q 0 / V 0 / = Q 0 / V 0 = C 0
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Dielectric Strength For a parallel-plate capacitor: C = є 0 A/d The formula shows that the capacitance can be made very large by decreasing the plate separation. In practice, the lowest value of d is limited by the electric discharge through the dielectric. The discharge occurs when the electric field in the dielectric material reaches its maximum, called dielectric strength. Dielectric strength of air is 3 10 6 V/m.
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Summary Capacitance is defined as the charge over the potential difference Capacitance of parallel-plate capacitor is directly proportional to the plate area and inversely proportional to the plate separation The equivalent capacitance of a parallel combination of capacitors equals to the sum of individual capacitances The inverse equivalent capacitance of a series combination of capacitors equals to the sum of the inverse individual capacitances Placing a dielectric between the plates of a capacitor increases the capacitance by a factor , called the dielectric constant
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