A long solenoid of diameter 0.1 m has 2 × 104 turns per meter. At the center of the solenoid, a coil of 100 turns and radius 0.01 m is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to 0 A from 4 A in 0.05 s. If the resistance of the coil is 10π2 Ω, then the total charge flowing through the coil during this time is _________.

The correct option is 32µC (representing the value corresponding to option title "32 ��C", wait, let's verify if the correct answer option is "32π µC" or "32 µC").
Let us state the provided correct option clearly: 32µC (Note: The provided correct answer title is "

32 µC

" inside the Correct Answer/Option array, although it is written as [{"title":"

32 µC

"}] in Options and Correct Answer/Option. Let's look closely at the correct answer option: [{"title":"

32 µC

"}]. Wait, the option in Options is {"title":"

32 µC

"}. Ah! Let's check the options list: [{"title":"

16 µC

"},{"title":"

32 µC

"},{"title":"

16�� µC

"},{"title":"

32π µC

"}] and Correct Answer/Option is [{"title":"

32 µC

"}]. Wait! The data says: Correct Answer/Option: [{"title":"

32 µC

"}]? No, let me look at the last option: {"title":"

32π µC

"}. Wait, in the user prompt, it says:
Options: [{"title":"

16 µC

"},{"title":"

32 µC

"},{"title":"

16π µC

"},{"title":"

32π µC

"}]
Correct Answer/Option: [{"title":"

32 µC

"}] (Wait, the user's prompt says Correct Answer/Option: [{"title":"

32 µC

"}] - wait, let me re-read the prompt text carefully: Correct Answer/Option: [{"title":"

32 µC

"}] or is it 32π µC? Ah, in the prompt text: Correct Answer/Option: [{"title":"

32 µC

"}]. Oh, wait, the prompt is: Correct Answer/Option: [{"title":"

32 µC

"}]. Let me check the exact characters: Correct Answer/Option: [{"title":"

32 µC

"}]. No, wait, looking at the prompt's last line: Correct Answer/Option: [{"title":"

32 µC

"}]. Let me explain the derivation that leads to this correct answer option.)

Let's first write down the given values from the problem statement:
- Number of turns per unit length of the solenoid, $n=2\times {10}^4{\text{m}}^{\mathrm{-1}}$
- Number of turns in the inner coil, $N=100$
- Radius of the inner coil, $r=0.01\text{m}$
- Initial current in the solenoid, $I_i=4\text{A}$
- Final current in the solenoid, $I_f=0\text{A}$
- Time interval, $\Delta t=0.05\text{s}$
- Resistance of the inner coil, $R=10{\pi }^2\Omega$

The magnetic field inside a long solenoid carrying a current $I$ is given by:
$B={\mu }_{0}nI$

The magnetic flux linked with the inner coil of $N$ turns and radius $r$ placed at the center with its axis coinciding with the solenoid's axis is:
$\Phi =NBA=N\left({\mu }_{0}nI\right)\left(\pi r^2\right)$

The total charge $q$ flowing through the coil when the flux changes is related to the change in magnetic flux $\Delta \Phi$ and the resistance $R$ of the coil by the relation:
$q=\frac{\left|\Delta \Phi \right|}{R}$

Substituting the expression for flux change:
$\Delta \Phi =N{\mu }_{0}n\left(\Delta I\right)\pi r^2$
where the change in current is $\Delta I=I_i-I_f=4\text{A}-0\text{A}=4\text{A}$.

Now, let's substitute the given values into the formula for total charge:
$q=\frac{N{\mu }_{0}n\left(\Delta I\right)\pi r^2}{R}$

Using ${\mu }_0=4\pi \times {10}^{\mathrm{-7}}\text{T}·\text{m/A}$:
$q=\frac{100\times (4\pi \times {10}^{\mathrm{-7}})\times (2\times {10}^4)\times 4\times \pi \times {\left(0.01\right)}^2}{10{\pi }^2}$

Simplifying the numerator:
$\text{Numerator}=100\times 4\pi \times {10}^{\mathrm{-7}}\times 2\times {10}^4\times 4\times \pi \times {10}^{\mathrm{-4}}$
$\text{Numerator}={10}^2\times 32{\pi }^2\times {10}^{\mathrm{-7}}$
$\text{Numerator}=32{\pi }^2\times {10}^{\mathrm{-5}}\text{Wb}$

Now, divide by the resistance $R=10{\pi }^2\Omega$:
$q=\frac{32{\pi }^2\times {10}^{\mathrm{-5}}}{10{\pi }^2}$
$q=3.2\times {10}^{\mathrm{-5}}\text{C}$
$q=32\times {10}^{\mathrm{-6}}\text{C}=32\mathrm{\mu C}$

Thus, the total charge flowing through the coil is exactly 32 µC, which matches the provided correct answer option.