Chapter 27c-28a - RC Circuit Examples In an RC circuit what...

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RC Circuit Examples t q(t) C [1 e ] τ =ξ− For a charging capacitor, The full charge is, f qC = ξ We are asked for the time when, f q 0.99q 0.99C = In an RC circuit what multiple of the time constant gives the time taken for an initially uncharged capacitor to be charged to 99% of its full charge ? RC τ = t 0.99C C [1 e ] τ = ξ (t in q(t)) →∞ ξ So solve,
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t 0.99 1 e τ = t e 1 0.99 τ = t ln e ln(0.01) τ ⎛⎞ = ⎜⎟ ⎝⎠ t ln(0.01) 4.61 −= = τ t 4.61 = τ
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In the circuit shown R = 15 k Ω , and the ideal battery applies 12 V . The switch is closed, making the voltage across the capacitor rise to 5.00 V in 1.3 μ s . Example 2 For a charging capacitor, t qC [ 1 e ] τ =ξ− But, CC q V V C =→ = t C q V[ 1 e ] C τ == ξ So, What are a) the time constant of the series RC circuit and b) the capacitor’s capacitance? c) the current through the resistor at that time?
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t V [1 e ] τ =− ξ t V e1 τ ξ t V ln e ln 1 τ ⎛⎞ ⎜⎟ ξ ⎝⎠ tV ln 1 −= τξ t V ln 1 τ=− ξ With ξ =12 V and V = 5 V at t = 1.3 μ s, 6 1.3x10 s 5 ln 1 12 6 2.4x10 s τ = b) Since RC τ = 6 2.4x10 s C R 15000 τ == Ω 10 C 1.61x10 F 161 pF
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c) The current at that time is given by recalling that, tt dq d iC 1 e e dt dt R τ τ ⎛⎞ ⎡⎤ ξ == ξ = ⎜⎟ ⎢⎥ ⎣⎦ ⎝⎠ 1.3 2.4 12V i e 0.465 mA 15000 Ω
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Example 3 20 V 2.5 μ F The switch is closed for >>10 time constants for charging the capacitor. From when the switch is then opened how long does it take for the potential difference across the capacitor to become 2 V. A discharging capacitor obeys expression o t RC qq e = q qC V V C =→ = But, So, o t RC q q Ve CC == We solve this for the time when V = 2 V .
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With the switch initially closed for >> 10 time constants the capacitor becomes fully charged. Since for charging, o t RC q Ve C = o t RC VC e q = o t RC VC ln ln e q ⎛⎞ = ⎜⎟ ⎝⎠ o VC t ln qR C =− o VC tR C l n q We need q o the charge on the capacitor when the switch is opened and the resistance through which it discharges. 20 V 2.5 μ F t t qC [ 1 e ] C τ >>τ =ξ− ξ o = ξ For the resistance we have R( 6 6 ) | | 6 + Ω Ω
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20 V 2.5 μ F R1 2| | 6 =ΩΩ 12 6 R4 12 6 Ω⋅ Ω == Ω Ω+ Ω Then with o VC VC t RCln qC ⎛⎞ =− ⎜⎟ ξ ⎝⎠ o V2 V t (4 )(2.5 F)ln 20V =− Ω μ ξ t2 3 s = μ
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HITT–2/16/09 Closed Book Each network is connected across identical batteries at the same time. Rank order the time required to charge to 50% of the capacitor’s full charge greatest time first . 1) 1, 3, 2 Resistors & capacitors are identical 2) 3, 1, 2 3) 2, 3, 1 4) 1, 2, 3 5) 3, 2, 1
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Magnetic Fields You may have seen this method of visualizing the magnetic field lines surrounding a bar magnet .
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Chapter 27c-28a - RC Circuit Examples In an RC circuit what...

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