physics

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The magnetic field in the region shown below is increasing by 5 T/s. The circuit shown (with R=25 , C=55 µF, dimensions 1 m by 2 m) is inserted into the region as shown. The capacitor is initially uncharged.

1. How long will it take the capacitor to charge to 25% of its final charge?

Please help I tried T=RC but not right formula!

  • no picture -

    I do not have a picture of your circuit so find this hard to follow. However for an RC circuit at initial voltage V when the switch is closed:
    V = i R + (1/C) integral i dt
    with i = 0 at t = 0
    0 = R di/dt + i/C
    0 = RC di/dt + i
    let i = Io (e^kt)
    di/dt = k Io e^kt
    0 = RC kIo e^kt + i
    so
    i = -RC k Io e^kt = Io e^kt
    so k = -1/RC
    so if in form i = Io e^-t/T
    T indeed = RC
    Now
    The voltage around this loop = rate of change of magnetic flux
    E = 5*2square meters = 10 volts
    so we put a ten volt force on our RC circuit
    i = Io e^-t/RC
    Initially all the voltage appears on the resistor because there is no charge yet on the capacitor
    Io = E/R = 10/R
    so
    i = (10/R)e^-t/RC
    charge on capacitor Q= integral i dt
    = (10 /R) (-RC e^-t/RC + k)
    when t = 0, Q = 0 so
    k = RC
    so
    Q = 10 C (1 - e^-t/RC)
    Q final = 10 C
    when does Q = (10/4)C ?
    .25 = 1 - e^-t/RC
    e^-t/RC = .75
    -t/RC = ln .75 = -.2877
    so
    t = .2877 R C

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