5. (4 points) An electrical circuit consists of a resistor and a capacitor with capacitance C connected in series. A galvanic cell with electromotive force (emf) $\varepsilon$ and negligible internal resistance is connected to the ends of the circuit. Determine the amount of heat released in the resistor during the charging of the capacitor.

## Possible solution.

During the charging of the capacitor, the electric charge passing through the circuit is

$$
q=C \varepsilon
$$

In this process, the capacitor acquires potential energy

$$
W=\frac{C \varepsilon^{2}}{2}
$$

The work done by the galvanic cell in transferring the charge through the circuit is

$$
A=q \varepsilon=C \varepsilon^{2}
$$

The amount of heat released in the resistor is

$$
Q=A-W=\frac{C \varepsilon^{2}}{2}
$$
Answer: $Q=\frac{C \varepsilon^{2}}{2}$.

## Grading Criteria

| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for the charge that has passed through the circuit is written | $\mathbf{1}$ |
| The expression for the energy of the capacitor is written | $\mathbf{1}$ |
| The expression for the work of the source is written | $\mathbf{1}$ |
| Necessary transformations are performed, the final answer is obtained | $\mathbf{1}$ |
| Total points |  |