Question

Show that if L, R, C, and E0 are constant, then the amplitude of the steady-state current in Example 10 is a maximum when γ =1/√(LC) What is the maximum amplitude?

Short answer

The amplitude of the steady-state current is maximum which is E₀/R.

Step-by-step solution

Definition Of LRC Series circuits:

LRC-SERIES CIRCUITS As mentioned in the introduction to this chapter, many different physical systems can be described by a linear second-order differential equation similar to the differential equation of forced motion with damping:

m (d²x/dt²) + ß (dx/dt) + kx= f(t)

Using LRC series to solve the problem:

According to Problem 54, the amplitude is E₀/Z which maximizes when Z minimizes. But Z= (X²+R²)^1/2 and so |x| = |L_γ-1/C_γ|; that is when X=0 and soL_{γ =}1/C_γ.

This implies,

γ =1/√(LC)

In this case the amplitude becomes,

E₀/Z= E₀/ (X²+R²)^1/2

=E₀/R