Now that we have a feel for the state of the circuit in its steady state, let us obtain the expression for the current in the circuit as a function of time. Note that we can use the loop rule (going around counterclockwise) : E-vR-vL=0. Note as well that vR=iR and vL=Ldidt. Using these equations, we can get, after some rearranging of the variables and making the subsitution x=ER-i, dxx=-RLdt. Integrating both sides of this equation yields x=x0e-Rt/L. Use this last expression to obtain an expression for i (t). Remember that x=ER-i and that i0=i (0) = 0. Express your answer in terms of E, R, and L. You may or may not need all these variables. Use the notation exp (x) for ex.

Question

Physics

Dominik Small

9 November, 20:57

Now that we have a feel for the state of the circuit in its steady state, let us obtain the expression for the current in the circuit as a function of time. Note that we can use the loop rule (going around counterclockwise) : E-vR-vL=0. Note as well that vR=iR and vL=Ldidt. Using these equations, we can get, after some rearranging of the variables and making the subsitution x=ER-i, dxx=-RLdt. Integrating both sides of this equation yields x=x0e-Rt/L. Use this last expression to obtain an expression for i (t). Remember that x=ER-i and that i0=i (0) = 0. Express your answer in terms of E, R, and L. You may or may not need all these variables. Use the notation exp (x) for ex.

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Answers (1)

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Jillian Thornton · Answer 1

i (t) = (E/R) [1 - exp (-Rt/L) ]

Explanation:

E-vR-vL=0

E - iR - Ldi/dt = 0

E - iR = Ldi/dt

Separating te variables,

dt/L = di / (E - iR)

Let x = E - iR, so dx = - Rdi and di = - dx/R substituting for x and di we have

dt/L = - dx/Rx

-Rdt/L = dx/x

interating both sides, we have

∫-Rdt/L = ∫dx/x

-Rt/L + C = ㏑x

x = exp (-Rt/L + C)

x = exp (-Rt/L) exp (C) A = exp (C) we have

x = Aexp (-Rt/L) Substituting x = E - iR we have

E - iR = Aexp (-Rt/L) when t = 0, i (0) = 0. So

E - i (0) R = Aexp (-R*0/L)

E - 0 = Aexp (0) = A * 1

E = A

So,

E - i (t) R = Eexp (-Rt/L)

i (t) R = E - Eexp (-Rt/L)

i (t) R = E (1 - exp (-Rt/L))

i (t) = (E/R) (1 - exp (-Rt/L))