In the given figure the magnetic flux through the loop increases according to the relation $$\phi_B(t) = 10t^2 + 20t$$, where $$\phi_B$$ is in milliwebers and $$t$$ is in seconds. The magnitude of current through $$R = 2 \, \Omega$$ resistor at $$t = 5$$ s is _________ mA.

We need to find the magnitude of the current flowing through the resistor in a loop where the magnetic flux is changing over time.

1. Identify the Given Parameters

From the problem statement page, we have:

• Magnetic flux function: $$\phi_B(t) = 10t^2 + 20t$$ (where $$\phi_B$$ is in milliwebers, $$\text{mWb}$$)
• Resistance ($$R$$) = $$2\ \Omega$$
• Time ($$t$$) = $$5\text{ s}$$

2. Apply Faraday's Law of Induction

According to Faraday's Law, the magnitude of the induced electromotive force ($$\varepsilon$$) is equal to the time rate of change of the magnetic flux through the loop:

$$\varepsilon = \left| \frac{d\phi_B}{dt} \right|$$

Differentiating the flux equation with respect to time ($$t$$):

$$\frac{d\phi_B}{dt} = \frac{d}{dt}(10t^2 + 20t) = 20t + 20 \text{ mV}$$

Now, calculate the induced emf at the specific time $$t = 5\text{ s}$$:

$$\varepsilon = 20(5) + 20 = 100 + 20 = 120\text{ mV}$$

3. Calculate the Induced Current ($$I$$)

Using Ohm's Law, the magnitude of the induced current running through the circuit is:

$$I = \frac{\varepsilon}{R}$$

Substitute our calculated emf and the given resistance into the formula:

$$I = \frac{120\text{ mV}}{2\ \Omega} = 60\text{ mA}$$

Conclusion

The magnitude of the current through the resistor at $$t = 5\text{ s}$$ is 60 mA.