### Learning Objectives

By the end of this section, you will be able to:

- Determine the angular frequency of oscillation for a resistor, inductor, capacitor $\left(RLC\right)$ series circuit
- Relate the $RLC$ circuit to a damped spring oscillation

When the switch is closed in the *RLC* circuit of Figure 14.17(a), the capacitor begins to discharge and electromagnetic energy is dissipated by the resistor at a rate ${i}^{2}R$. With *U* given by Equation 14.19, we have

where *i* and *q* are time-dependent functions. This reduces to

This equation is analogous to

which is the equation of motion for a *damped mass-spring system* (you first encountered this equation in Oscillations). As we saw in that chapter, it can be shown that the solution to this differential equation takes three forms, depending on whether the angular frequency of the undamped spring is greater than, equal to, or less than *b*/2*m*. Therefore, the result can be underdamped $(\sqrt{k\text{/}m}>b\text{/}2m)$, critically damped $(\sqrt{k\text{/}m}=b\text{/}2m)$, or overdamped $(\sqrt{k\text{/}m}<b\text{/}2m)$. By analogy, the solution *q*(*t*) to the *RLC* differential equation has the same feature. Here we look only at the case of under-damping. By replacing *m* by *L*, *b* by *R*, *k* by 1/*C*, and *x* by *q* in Equation 14.44, and assuming $\sqrt{1\text{/}LC}>R\text{/}2L$, we obtain

where the angular frequency of the oscillations is given by

This underdamped solution is shown in Figure 14.17(b). Notice that the amplitude of the oscillations decreases as energy is dissipated in the resistor. Equation 14.45 can be confirmed experimentally by measuring the voltage across the capacitor as a function of time. This voltage, multiplied by the capacitance of the capacitor, then gives *q*(*t*).

### Interactive

In the interactive circuit construction kit simulation below, you can add inductors and capacitors to work with any combination of R, L, and C circuits with both dc and ac sources. Then you can measure and evaluate current and voltage, and graph them as a function of time.

### Interactive

With the interactive DC circuit construction kit simulation below, you can add inductors and capacitors to work with any combination of R, L, and C circuits with both dc and ac sources. Then you can measure and evaluate current and voltage, and graph them as a function of time.

### Check Your Understanding 14.11

In an *RLC* circuit, $L=5.0\phantom{\rule{0.2em}{0ex}}\text{mH},C=6.0\mu \text{F},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R=200\phantom{\rule{0.2em}{0ex}}\text{\Omega}.$ (a) Is the circuit underdamped, critically damped, or overdamped? (b) If the circuit starts oscillating with a charge of $3.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{C}$ on the capacitor, how much energy has been dissipated in the resistor by the time the oscillations cease?