An LC circuit is built with a 20 mH inductor and an 8.0 PF capacitor. The capacitor voltage has its maximum value of 25 V at t = 0 s.(a)How

Question

An LC circuit is built with a 20 mH inductor and an 8.0 PF capacitor. The capacitor voltage has its maximum value of 25 V at t = 0 s.(a)How much time does it take until the capacitor is fully discharged for the first time? (b)What is the inductor current at that time?

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Kim Chi 5 years 2021-07-24T22:51:44+00:00 1 Answers 149 views 0

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  1. vankhanh
    0
    2021-07-24T22:53:03+00:00
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    Answer:

    a) the required time is 0.6283 μs

    b) the inductor current is 0.5 mA

    Explanation:

    Given the data in the question;

    The capacitor voltage has its maximum value of 25 V at t = 0

    i.e V_m = V₀ = 25 V

    we determine the angular velocity;

    ω = 1 / √( LC )

    ω = 1 / √( ( 20 × 10⁻³ H ) × ( 8.0 × 10⁻¹² F) )

    ω = 1 / √( 1.6 × 10⁻¹³  )

    ω = 1 / 0.0000004

    ω = 2.5 × 10⁶ s⁻¹

    a) How much time does it take until the capacitor is fully discharged for the first time?

    V_m =  V₀sin( ωt )

    we substitute

    25V =  25V × sin( 2.5 × 10⁶ s⁻¹ × t )

    25V =  25V × sin( 2.5 × 10⁶ s⁻¹ × t )

    divide both sides by 25 V

    sin( 2.5 × 10⁶ × t ) = 1

    ( 2.5 × 10⁶ × t ) = π/2

    t = 1.570796 / (2.5 × 10⁶)

    t = 0.6283 × 10⁻⁶ s

    t = 0.6283 μs

    Therefore, the required time is 0.6283 μs

    b) What is the inductor current at that time?

    I(t) = V₀√(C/L) sin(ωt)

    { sin(ωt) = 1 )

    I(t) = V₀√(C/L)

    we substitute

    I(t) = 25V × √( ( 8.0 × 10⁻¹² F ) / ( 20 × 10⁻³ H ) )

    I(t) = 25 × 0.00002

    I(t) = 0.0005 A

    I(t) = 0.5 mA

    Therefore, the inductor current is 0.5 mA

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