To solve this problem, we will use the formula for the magnetic field inside a solenoid.$\\$The magnetic field $B$ inside a long solenoid is given by:$\\ B = \mu_0 \cdot n \cdot I \\$where:$\\$• $\mu_0 = 4\pi \times 10^{-7}$ T m A$^{-1}$ is the permeability of free space,$\\$• $n$ is the number of turns per unit length of the solenoid,$\\$• $I = 2.5$ A is the current flowing through the solenoid.$\\$First, calculate the number of turns per unit length $n: \\ n = \frac{N}{L} \\$where:$\\$• $N = 100$ is the total number of turns,$\\$• $L = 50$ cm $= 0.5$ m is the length of the solenoid.$\\$Substitute the values:$\\ n = \frac{100}{0.5} = 200$ turns/m$\\$Now, substitute $n$ and $I$ into the formula for $B: \\ B = \mu_0 \cdot n \cdot I \\ B = (4\pi \times 10^{-7}) \cdot 200 \cdot 2.5 \\$Calculate the expression:$\\ B = 4\pi \times 10^{-7} \cdot 500 \\$Simplify further:$\\ B = 2000\pi \times 10^{-7} \\$Convert $\pi$ to its approximate value (3.14):$\\ B \approx 2000 \times 3.14 \times 10^{-7} \\ B \approx 6280 \times 10^{-7} \\ B \approx 6.28 \times 10^{-4}$ T$\\$Therefore, the magnetic field at the center of the solenoid is $6.28 \times 10^{-4}$ T.$\\$This corresponds to Option 4.