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Step-by-step Solution
Step 1: Identify the given information
• Resistance of the coil, $R = 400 \,\Omega$.
• Magnetic flux linked with the coil, $\phi(t) = 50t^2 + 4$.
• We need to find the current in the coil at $t = 2\,\text{seconds}$.
Step 2: Write down Faraday's law of electromagnetic induction
According to Faraday’s law, the induced electromotive force (emf) $\varepsilon$ in a coil is given by
$\varepsilon = - \frac{d\phi}{dt}$,
where $\phi(t)$ is the magnetic flux linked with the coil.
Step 3: Compute the induced emf
Differentiate $\phi(t) = 50t^2 + 4$ with respect to $t$:
$\frac{d\phi}{dt} = \frac{d}{dt} (50t^2 + 4) = 100t.$
Hence,
$\varepsilon = -100t.$
Step 4: Calculate the induced emf at $t = 2\,\text{seconds}$
Substitute $t=2$ into $\varepsilon = -100t$:
$\varepsilon_{t=2} = -100 \times 2 = -200 \,\text{V}.$
Step 5: Find the current in the coil
Ohm's law states that the current $i$ is given by the ratio of the emf to the resistance:
$i = \frac{\varepsilon}{R}.$
At $t = 2\,\text{seconds}$:
$i_{t=2} = \frac{-200}{400} = -0.5\,\text{A}.$
The negative sign indicates the direction of current is opposite to the chosen reference direction. The magnitude of the current is $0.5\,\text{A}$.
Final Answer
Therefore, the current in the coil at $t = 2\,\text{seconds}$ is $0.5\,\text{A}$ (in magnitude).
