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To solve this problem, we need to calculate the induced EMF (electromotive force) in the loop at \( t = 2 \) seconds and determine the direction of the current in the resistor \( R \).

### Given Data
The magnetic flux \( \Phi \) through the loop is given by the equation:
\[
\Phi = 6t^2 + 7t
\]
where \( \Phi \) is in milli-Webers (mWb) and \( t \) is in seconds.

### Steps to Solve

1. **Find the Induced EMF**: According to Faraday's Law of Induction, the induced EMF \( \mathcal{E} \) in the loop is given by the rate of change of magnetic flux:
   \[
   \mathcal{E} = -\frac{d\Phi}{dt}
   \]
   
2. **Differentiate the Flux Function**:
   \[
   \Phi = 6t^2 + 7t
   \]
   Taking the derivative with respect to \( t \):
   \[
   \frac{d\Phi}{dt} = \frac{d}{dt}(6t^2 + 7t) = 12t + 7
   \]

3. **Calculate \( \mathcal{E} \) at \( t = 2 \) seconds**:
   Substitute \( t = 2 \) into the derivative to find the rate of change of magnetic flux:
   \[
   \frac{d\Phi}{dt} = 12(2) + 7 = 24 + 7 = 31 \, \text{mWb/s}
   \]
   Therefore, the induced EMF is:
   \[
   \mathcal{E} = -31 \, \text{mV}
   \]

   The negative sign indicates the direction of the induced EMF according to Lenz's Law.

4. **Determine the Direction of the Current in \( R \)**:
   - By Lenz's Law, the induced current will flow in such a direction as to oppose the change in magnetic flux.
   - Since the magnetic flux is increasing (positive derivative), the induced current will create a magnetic field that opposes this increase.
   - Based on the diagram, if we assume the magnetic field \( \vec{B} \) is pointing into the page (dot convention), the induced current will circulate in a direction that produces an opposing magnetic field.
   - Thus, the current will flow in a clockwise direction in the loop, causing it to go through \( R \) from left to right.

### Final Answer
1. **Induced EMF**: \( 31 \, \text{mV} \) (in magnitude)
2. **Direction of Current in \( R \)**: Left to right

Would you like further details on any part of the calculation, or do you have other questions?

### Further Related Questions
1. How would the answer change if the magnetic flux was given as a sinusoidal function of time?
2. What would happen to the induced EMF if the magnetic flux rate of change was zero?
3. How does the resistance \( R \) affect the current induced in the circuit?
4. What is the role of Lenz's Law in determining the direction of the induced current?
5. How does this problem relate to the concept of magnetic induction in electric generators?

### Tip
In problems involving induced EMF, always check if the flux is increasing or decreasing to apply Lenz's Law correctly.

El flujo magnético mostrado en la figura se incrementa de acuerdo a la relación Φ = 6t^2 + 7t, donde el flujo está en mili-Webbers y el tiempo en segundos. ¿Qué FEM se induce al cabo de 2 segundos, y cuál es la dirección de la corriente en R?

Explore how to calculate the induced EMF and determine the current direction in a resistor for a time-dependent magnetic flux Φ = 6t^2 + 7t. This problem uses Faraday's Law and Lenz's Law to solve for the EMF at t = 2 seconds and identify the direction of current flow in a circuit.

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