Question

The current i through a 4.6 Hinductor varies with time t as shown by the graph of Figure, where the vertical axis scale is set by ${\mathit{i}}_{\mathbf{s}}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathit{A}$ and the horizontal axis scale is set by ${\mathit{t}}_{\mathbf{s}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}\mathit{s}$ . The inductor has a resistance of$\mathbf{12}\mathit{\Omega }$.(a) Find the magnitude of the induced emf $\mathit{\epsilon }$ during time intervals 0 to 2 ms. (b) Find the magnitude of the induced emf $\mathit{\epsilon }$ during time intervals 2 ms to 5 ms. (c) Find the magnitude of the induced emf $\mathit{\epsilon }$during time intervals 5 ms to 6 ms. (Ignore the behavior at the ends of the intervals.)

Short answer

1. $\epsilon =1.6x{10}^{4}V$
2. $\epsilon =3.1x{10}^{3}V$
3. $\epsilon =2.3x{10}^{4}V$

Step-by-step solution

Given

$$\begin{gathered} i_s=8A \\ {t}_s=6ms \end{gathered}$$

Understanding the concept

According to Faraday’s law, the emf induced in a circuit is proportional to the time rate of change of the magnetic flux linking that circuit.

Formulae:

Induced emfis given by

$\epsilon =L\frac{di}{dt}$

(a) Calculate the induced emf for a time interval  0 to 2 ms

Induced emf $\epsilon$ is given by

$$\begin{gathered} \stackrel{,}{o}=L\frac{di}{dt} \\ i=0to7A \\ \epsilon =L\frac{di}{dt}=\frac{4.6\times \left(7-0\right)}{\left(2-0\right)\times {10}^{-3}} \\ \epsilon =1.6\times {10}^{4}V \end{gathered}$$

(b) Calculate the induced emf for time interval  2ms to5ms

For time interval 2ms to 5ms

$$\begin{gathered} 1=7t05A \\ \epsilon =L\frac{di}{dt}=\frac{4.6\times \left(5-7\right)}{\left(5-2\right)\times {10}^{-3}} \\ \epsilon =3.1\times {10}^3\hspace{0.17em}V \end{gathered}$$

(c) Calculate the induced emf for time interval  5ms to 6ms

For time interval 5ms to 6ms

$$\begin{gathered} 1=0to5A \\ \epsilon =L\frac{di}{dt}=\frac{4.6\times \left(5-0\right)}{\left(6-5\right)\times {10}^{-3}} \\ \epsilon =2.3\times {10}^{4}V \end{gathered}$$