The uploaded image contains a physics and electronics exercise written in French, focusing on the experimental verification of the capacitance $C$ of a capacitor in a photographic flash device. Below, I will transcribe and break down the main components of the problem:

Exercise III Summary (Translated to English):

The goal of the exercise is to experimentally verify the capacitance $C$ of a capacitor in a photographic flash device.

Given Information:

1. On the label of the flash capacitor:
   • $100V$, $150\mu F$, and $-10^\circ C$ to $+55^\circ C$ (capacitor specifications).
2. The capacitor is charged using a DC voltage source $U = 12V$, through a resistor $R$ of known value, and discharged through a circuit. A diagram (provided in the exercise) shows the circuit configuration with a switch $K$.
3. At $t = 0$, the switch $K$ is closed. The time evolution of the current intensity $i(t)$ in the circuit is measured, as represented in a graph of $i(t)$ vs. $t$.

Questions (Paraphrased):

1. Oscilloscope Connection (0.5 points):

   • Show on the circuit diagram how to connect an oscilloscope to measure the voltage $u_C(t)$ across the capacitor.

2. Differential Equation (0.75 points):

   • Derive the differential equation satisfied by $u_C(t)$, the voltage across the capacitor.

3. Solution Form (0.75 points):

   • The differential equation solution is given as: $u_C(t) = A(1 - e^{-\frac{t}{\tau}})$
     • Determine $A$, $\tau$, and the parameters of the circuit.

4. Expression for $i(t)$: (0.75 points):

   • Derive the expression for $i(t)$ and verify that it is of the form: $i(t) = I_0 e^{-\frac{t}{\tau}}$

5. Dimensional Analysis (0.5 points):

   • Perform a dimensional analysis of the equation to verify its homogeneity in time.

6. Capacitance Value (2.0 points):

   • Using $R = 2.0 \, k\Omega$, estimate $\tau$ and calculate $C$.
   • Verify the given capacitor value $C = 150 \, \mu F$.

7. Capacitors in Parallel (0.75 points):

   • If a second capacitor $C_2 = 2C$ is connected in parallel with the first:
     • Determine the equivalent capacitance $C_{eq}$.

8. Stored Energy (0.75 points):

   • Derive the expression for the stored energy $W$ in the capacitor: $W = \frac{1}{2} C u_C^2$
   • Verify this for $u_C = 100 \, V$.

Step-by-Step Approach:

Would you like me to work on specific parts of this problem (e.g., derive the differential equation, solve for $C$, or verify energy expressions)? Let me know!

Relative Questions to Expand Learning:

1. What is the role of the resistor $R$ in the capacitor charging/discharging process?
2. How does the time constant $\tau$ relate to the circuit components $R$ and $C$?
3. Why is energy stored in capacitors represented as $W = \frac{1}{2} C u_C^2$?
4. What happens to the equivalent capacitance if capacitors are connected in series instead of parallel?
5. How can an oscilloscope help in visualizing the capacitor's behavior in the circuit?

Tip:

Always confirm that your derived equations have consistent physical units. Dimensional analysis is a quick and reliable tool for checking equation validity.