To solve the problem step by step, we will follow the process of calculating the energy stored in the capacitor, the charge on the capacitor, and the heat produced during the charging process.
### Step 1: Calculate the initial energy stored in the capacitor
The formula for the energy stored in a capacitor is given by:
\[
U = \frac{1}{2} C V^2
\]
Where:
- \( U \) is the energy in joules,
- \( C \) is the capacitance in farads,
- \( V \) is the voltage in volts.
Given:
- \( C = 10 \, \text{mF} = 10 \times 10^{-3} \, \text{F} \)
- \( V = 2 \, \text{V} \)
Substituting the values:
\[
U = \frac{1}{2} \times (10 \times 10^{-3}) \times (2^2) = \frac{1}{2} \times 10 \times 10^{-3} \times 4 = 20 \times 10^{-3} = 20 \, \text{mJ}
\]
### Step 2: Calculate the initial charge on the capacitor
The charge \( Q \) on a capacitor is given by:
\[
Q = C \times V
\]
Substituting the values:
\[
Q_{\text{initial}} = 10 \times 10^{-3} \times 2 = 20 \, \text{mC}
\]
### Step 3: Calculate the final energy stored in the capacitor after connecting to the 4V battery
Using the same energy formula:
\[
U_{\text{final}} = \frac{1}{2} C V^2
\]
Where \( V = 4 \, \text{V} \):
\[
U_{\text{final}} = \frac{1}{2} \times (10 \times 10^{-3}) \times (4^2) = \frac{1}{2} \times 10 \times 10^{-3} \times 16 = 80 \times 10^{-3} = 80 \, \text{mJ}
\]
### Step 4: Calculate the final charge on the capacitor
Using the charge formula again:
\[
Q_{\text{final}} = C \times V
\]
Substituting the values:
\[
Q_{\text{final}} = 10 \times 10^{-3} \times 4 = 40 \, \text{mC}
\]
### Step 5: Calculate the charge flow through the battery
The charge flow \( \Delta Q \) is given by:
\[
\Delta Q = Q_{\text{final}} - Q_{\text{initial}} = 40 \, \text{mC} - 20 \, \text{mC} = 20 \, \text{mC}
\]
### Step 6: Calculate the work done by the battery
The work done \( W \) by the battery is given by:
\[
W = \Delta Q \times V
\]
Where \( V = 4 \, \text{V} \):
\[
W = 20 \times 10^{-3} \times 4 = 80 \, \text{mJ}
\]
### Step 7: Calculate the heat produced in the process
The heat \( Q_h \) produced can be calculated using:
\[
Q_h = W - (U_{\text{final}} - U_{\text{initial}})
\]
Substituting the values:
\[
Q_h = 80 \, \text{mJ} - (80 \, \text{mJ} - 20 \, \text{mJ}) = 80 \, \text{mJ} - 60 \, \text{mJ} = 20 \, \text{mJ}
\]
### Final Answer
The total heat produced in the second charging process is:
\[
\boxed{20 \, \text{mJ}}
\]
