The RC time constant τ for an RC circuit is given by the product of the resistance R and the capacitance C of the circuit: τ = RC.
Let's start by finding R, the resistance of the copper wire. The resistance of a cylindrical conductor is given by R = ρl/A, where ρ is the resistivity, l is the length, and A is the cross-sectional area. The cross-sectional area of a wire with diameter d is given by A = π(d/2)^2.
R = ρl/π(d/2)^2 = 1.72×10^−8 Ωm * 25m / π(0.255mm/2)^2 = 0.878 Ω.
Now let's find the capacitance C of the capacitor. The capacitance of a parallel plate capacitor is given by C = εA/d, where ε is the permittivity of the dielectric, A is the area of one plate, and d is the separation between the plates. In this case, the permittivity of the dielectric is ε = Kε0, where K is the dielectric constant and ε0 is the vacuum permittivity. The area A of a square plate with side length s is given by A = s^2.
C = Kε0 * A/d = 3.5 * 8.854×10^−12 C^2/Nm^2 * (10cm)^2 / 0.0125mm = 2.51 × 10^-9 F.
Finally, we can find the RC time constant τ:
τ = RC = 0.878 Ω * 2.51 × 10^-9 F = 2.20 × 10^-9 s = 2.20 ns.
So the RC time constant for this circuit is 2.20 nanoseconds.