A rectangular loop of resistance $R=5 \Omega$ is placed in a uniform magnetic field such that the magnetic flux through the loop increases according to the relation $\phi=4 t^2-2 t$ where $\phi$ is in weber and $t$ is in second. The current induced at $t=3$ second is:

The magnetic flux through the loop is given by $\phi = 4t^2 - 2t$.

According to Faraday's law of electromagnetic induction, the induced electromotive force (emf) $\varepsilon$ is given by the negative rate of change of magnetic flux:

$\varepsilon = -\dfrac{d\phi}{dt}$

Differentiating $\phi$ with respect to $t$:

$\dfrac{d\phi}{dt} = \dfrac{d}{dt}(4t^2 - 2t) = 8t - 2$

Thus, the magnitude of the induced emf is $|\varepsilon| = |8t - 2|$.

The induced current $I$ is given by Ohm's law, $I = \dfrac{|\varepsilon|}{R}$, where $R = 5 \Omega$.

At $t = 3$ seconds:

$|\varepsilon| = 8(3) - 2 = 24 - 2 = 22 \text{ V}$

$I = \dfrac{22}{5} = 4.4 \text{ A}$

Answer: 4.4 A