In Figure, the magnetic flux through the loop increases according to the relation ${\mathit{\phi }}_{\mathbf{B}}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathbf{7}\mathbf{.}\mathbf{0}\mathit{t}$, where${\mathit{\phi }}_{\mathbf{B}}$is in milli-Weber and t is in seconds. (a) What is the magnitude of the emf induced in the loop when t = 2.0 s? (b) Is the direction of the current through R to the right or left?

Magnetic flux through the loop is $\phi =6.0t^2+7.0t$

By using Faraday’s law and Lenz’s law, find the emf induced and the direction of the current.

Faraday'slaw of electromagnetic induction states, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.

Lenz's law states that the current induced in a circuit due to a change in a magnetic field is directed to oppose the change in flux and to exert a mechanical force that opposes the motion.

Formulae are as follows:

$\epsilon =-\frac{d\phi }{dt}$

Where,$d\phi$is magnetic flux, 𝜀 is emf, dt is time.

It is given that the changing flux through the loop is$\phi =6.0t^2+7.0t$. Hence, the emf induced in the loop can be calculated as,

Faraday’s law is,

$$\begin{gathered} \epsilon =-\frac{d\phi }{dt} \\ \epsilon =-\frac{d}{dt}\left[6.0t^2+7.0t\right] \\ \epsilon =-\left[2\times 6.0t+7.0\right] \end{gathered}$$

At t = 2 s

$$\begin{gathered} \epsilon =-\left[2\times 6.0\times 2+7.0\right] \\ \epsilon =-31\mathrm{mV} \end{gathered}$$

Hence, the magnitude of the emf induced in the loop is 31 mV.

According to Lenz’s law, the current i induced in a loop has the direction such that the current’s magnetic field B opposes the change in the magnetic field inducing the current i.

Here, the magnetic field increases in the direction out of the page. Thus, the current through the resistance in the left direction will produce the magnetic field on the page.

Hence, the direction of the current must be left.

Therefore, by using the concept of Len’s law and Faraday’s law, the emf induced and direction of current can be determined.