Question

The flux linked with a circuit is given by $\mathrm{\varphi }={\mathrm{t}}^3+3\mathrm{t}-7$. The graph between time (x-axis) and induced emf (y-axis) will be

• A. a straight line through the origin
• B. Straight-line through positive intercept
• C. Straight-line through negative intercept
• D. Parabola not through the origin

Answer 1

The correct option is D

Parabola not through the origin

Step 1: Given data

According to the given data, the flux is given as

$\varphi =t^3+3t-7$

Step 2: Formula used:

Faraday’s first law of electromagnetic induction states that the time varies with magnetic flux can induce an emf,

$E\left(t\right)=-\frac{\mathrm{d}\varphi }{\mathrm{dt}}$

Where $\varphi$ is the magnetic flux,

Step 3: The graph between time (x-axis) and induced emf (y-axis):

Therefore, the induced emf is,

$$\begin{gathered} E\left(t\right)=-\frac{d(t^3+3t-7)}{dt} \\ E\left(t\right)=-3t^2-3 \end{gathered}$$

As we can see, the degree of the function for induced emf is $2$. Thus it will trace out a parabola

At $t=0$

$$\begin{gathered} E\left(0\right)=-3(0^2+1) \\ E\left(0\right)=-3\times 1 \\ E\left(0\right)=-3 \end{gathered}$$

Hence, the $y$-intercept is $-3$ and not the origin.

Hence, the graph between time (x-axis) and induced emf (y-axis) will be the parabola not passing through the origin.

Hence, option D is correct.