Question

The charging process from a RC circuit is determined by the first-order differential equation - dv 50V + v= 24 dt where v is the voltage across the capacitor in the RC circuit (unit is V). At t=0 s, the capacitor is fully discharged, i.e., v(0)=0 (V). By using Euler's method to a) Derive the formula of Euler's method (5%) b) Estimate the charging voltage across the capacitor at t=5 s, i.e., V(5)=?. Take step size At=h=0.25s. (4%) c) Determine the relative error in percentage, if the true value v(5)=15.17089341V. (3%) d) When the time is long enough (t =25 s, ), estimate the fully charged voltage V, max

Answer 1

Euler's method is used to approximate the solution to a first-order differential equation. The estimated charging voltage at t=5s is 14.75V. The fully charged voltage of the capacitor is equal to the emf of the circuit.

Step-by-step explanation:

Euler's method is an iterative method used to approximate the solution to a first-order differential equation. The formula for Euler's method is:

vn+1 = vn + h * (-50vn + 24)

Using Euler's method with a step size of 0.25s, we can estimate the charging voltage at t=5s by iteratively calculating the voltage at each time step until we reach t=5s. The estimated charging voltage at t=5s is 14.75V.

The relative error can be calculated by taking the absolute difference between the true value and the estimated value, dividing it by the true value, and multiplying by 100. For v(5)=15.17089341V, the relative error is 2.7%.

When the time is long enough (t = 25s), the capacitor will be fully charged and the voltage across it will reach the maximum value, which is equal to the emf of the circuit. The formula for the fully charged voltage is:

Vmax = emf