Electric Circuits. An elctric circuit contains a battery that produces a voltage of 60 volts, a resistor with a resistance of 13 ohms, and an inductor with an inductance of 5 henrys, as shown in the figure. Using calculus, it can be shown that the current I = I(t) (in amperes) t seconds after the switch is closed is I = 60/13(1 - e^-13t/5).
a.) Use this equation to express the time t as a function of the curren I.
b.) After how many seconds is the current 2A ?
a.) Use this equation to express the time t as a function of the curren I.
b.) After how many seconds is the current 2A ?
Answers & Comments
Asnwer:
a.) To express the time �t as a function of the current �I, you can rearrange the given equation:
�=6013(1−�−13�5)I=1360(1−e−513t)
Solve for �t in terms of �I.
1−�−13�5=1360�1−e−513t=6013I
�−13�5=1−1360�e−513t=1−6013I
−13�5=ln(1−1360�)−513t=ln(1−6013I)
�=−513ln(1−1360�)t=−135ln(1−6013I)
b.) Now, substitute �=2I=2 into the expression for �t to find the time when the current is 2A. Note that the logarithm argument must be positive for real solutions.
�=−513ln(1−1360(2))t=−135ln(1−6013(2))
Calculate �t using this expression.