CURRENT DECREASING TO ZERO: Previously we found the current immediately after a circuit is changed and the current a long time after. Now we want to know how the current I(t) depends on time, not just at t = 0 and at t=0, 1. Assume the switch is in position (b). Applying the loop rule around the right loop counterclockwise gives dl (t) EAVAV₁-1(t)R= -L- -1(t) R = 0 dt where I (t) is the current at time t. We can rewrite the differential equation as: dl R == I dt L 1 t R t (bet) = - = - 10 e ½ + = - = - 100 71 E d dt 00000 www The only function whose derivative is proportional to the function itself is the exponential. Substitute /(t) = A e-t/t into both the left and right hand side of the differential equation above and take the derivative (remember the chain rule). Solve for the time constant T₁. R The constant A is just the initial current at t = 0. Therefore the current as a function of time is I (t) = 1, e-t/¹¹ for the case of the current decaying from I, at t = 0 to zero at t = co.
CURRENT DECREASING TO ZERO: Previously we found the current immediately after a circuit is changed and the current a long time after. Now we want to know how the current I(t) depends on time, not just at t = 0 and at t=0, 1. Assume the switch is in position (b). Applying the loop rule around the right loop counterclockwise gives dl (t) EAVAV₁-1(t)R= -L- -1(t) R = 0 dt where I (t) is the current at time t. We can rewrite the differential equation as: dl R == I dt L 1 t R t (bet) = - = - 10 e ½ + = - = - 100 71 E d dt 00000 www The only function whose derivative is proportional to the function itself is the exponential. Substitute /(t) = A e-t/t into both the left and right hand side of the differential equation above and take the derivative (remember the chain rule). Solve for the time constant T₁. R The constant A is just the initial current at t = 0. Therefore the current as a function of time is I (t) = 1, e-t/¹¹ for the case of the current decaying from I, at t = 0 to zero at t = co.
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Transcribed Image Text:After this activity you should know: know the expressions for the current as a function of time for LR circuits for
currents increasing from zero and decaying to zero.. know the time constant for an LR circuit and how to obtain the
time constant from graphs of current versus time.
CURRENT DECREASING TO ZERO: Previously we found the current immediately
after a circuit is changed and the current a long time after. Now we want to know
how the current I(t) depends on time, not just at t= 0 and at t = 00,
1. Assume the switch is in position (b). Applying the loop rule around the right loop
counterclockwise gives
dl (t)
ZAVAV₁-1(t)R = -L- - I(t) R = 0
dt
where I (t) is the current at time t. We can rewrite the differential equation as:
dl
dt
R
-T
L
www
1
t
R
t
de (100 7 ) = - = - 10 0 ² 2 = -2 100 7
dt
TL
L
E
00000
b
ww
R
The only function whose derivative is proportional to the function itself is the exponential. Substitute 1(t) =
A e-t/TL into both the left and right hand side of the differential equation above and take the derivative (remember
the chain rule). Solve for the time constant TL.
The constant A is just the initial current at t = 0. Therefore the current as a function of time is / (t) = 1, e-t/t for
the case of the current decaying from I, at t = 0 to zero at t = 0.
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