Determine the energy stored in the inductor L as a function of time for the LR c

Question

Determine the energy stored in the inductor L as a function of time for the LR circuit of Fig. 30-6z. After how may time constants does the stored energy reach 99.9% of its maximum value? In the circuit of Fig. 30-27, determine the current in each resistor (I_1, I_2, I_3)at the moment the switch is closed, a long time after the switch is closed. After the switch has been closed for a long time, and then reopened, what is each current just after it is opened, after a long time? In Fig. 30-283, assume that the switch S has beer in position A for sufficient time so that a steady curt I_0 = V_0/R flows through the resistor R. At time t = 0, the switch is quickly switched to position B and the current through R decays according to I = I_0e^-t/tau. Show that the maximum emf E_max induced in the inductor during this time

Explanation / Answer

part a:

at the moment the switch is closed, inductor will behave as an open circuit

hence there will be no current across R3.

I3=0

then R2 and R1 are in series.

hence current in R1=current in R2=E/(R1+R2)

hence I1=I2=E/(R1+R2)

part b:

a long time after the switch is closed, inductor will behave as a closed circuit

hence R3 will be in parallel with R2.

equivalent resistance=R3*R2/(R3+R2)

this equivalent resistance is in series with R1.

then net resistance in the circuit=R=R1+(R2*R3/(R2+R3))

total current in the circuit=E/R

hence I1=E/R

by curent division , I2=I1*R3/(R2+R3)=E*R3/(R*(R2+R3))

where R=R2*R3/(R2+R3)

by current division , I3=I1*R2/(R2+R3)=E*R2/(R*(R2+R3))

part c:

after the switch is re-opened,

there is no path for current through I1 to flow

hence I1=0

as current in inductor can not change instantaneously,

at t=0+ after the switch is opened, I3 will remain same as it was before the switch as opened

hence I3=E*R2/(R*(R2+R3))

where R=R2*R3/(R2+R3)

using kirchoff's current law, I3+I2=0

==>I2=-I3

==>I2=-E*R2/(R*(R2+R3))

part d:

after a long time after the switch is re-opened, the current in inductor will have decayed through the resistors R2 and R3.

hence current in the circuit is zero

so I1=0

I2=0

I3=0

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