Question

The flux linked with a coil at any instant 't' is given by $\mathrm{\varphi }=10{\mathrm{t}}^2-50\mathrm{t}+250$. The induced emf at $\mathrm{t}=3\mathrm{s}$ is

Answer 1

Step 1: Given data:

According to the given data

$\mathrm{\varphi }=10{\mathrm{t}}^2-50\mathrm{t}+250$

$\mathrm{t}=3\mathrm{s}$

Step 2: Formula used

Faraday’s first law of electromagnetic induction states that the time($\mathrm{t}$) varies with magnetic flux and can induce an e.m.f ($\mathrm{E}$).

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

Where $\mathrm{\varphi }$ is the magnetic flux.

Step 3: Induced emf:

The time varies with magnetic flux can induce an e.m.f.

Therefore,

$$\begin{gathered} \mathrm{E}\left(\mathrm{t}\right)=-\frac{\mathrm{d\varphi }}{\mathrm{dt}} \\ \mathrm{E}\left(\mathrm{t}\right)=-\frac{\mathrm{d}(10{\mathrm{t}}^2-50\mathrm{t}+250)}{\mathrm{dt}} \\ \mathrm{E}\left(\mathrm{t}\right)=-(2\times 10\mathrm{t}-50+0) \\ \mathrm{E}\left(\mathrm{t}\right)=-(20\mathrm{t}-50) \end{gathered}$$

Here, substitute $\mathrm{t}=3\mathrm{s}$

$$\begin{gathered} \mathrm{E}\left(3\right)=-(20\times 3-50) \\ \mathrm{E}\left(3\right)=-(60-50) \\ \mathrm{E}\left(3\right)=-60+50 \\ \mathrm{E}\left(3\right)=-10\mathrm{V} \end{gathered}$$

Therefore, the induced emf at $\mathrm{t}=3\mathrm{s}$ is $-10\mathrm{V}$.