---
title: "Free Electrical Oscillations: RLC"
book: "High School Physics"
subject: physics
language: en
chapter: 31
exercises: 15
source: https://one-course.com/books/physics/2/en/chapter/31-free-electrical-oscillations-rlc
---

# Chapter 31 — Free Electrical Oscillations: RLC

Turn the heavy dial of an attic radio and the stations file past the needle: a voice, violins, static, a voice again. Behind the panel there is no [motor](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-sign) and no computer — a [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid), a [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) whose interleaved plates the dial rotates, and a length of aerial wire. This chapter connects last chapter’s two [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) tanks ([Chapter 30](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#ch-g12-rc-rl-circuits)) head to head and finds the fastest pendulum ever built: charge swinging back and forth, up to millions of times per second, at a tempo the dial chooses.

## 31.1 An electrical pendulum: the LC circuit

Charge a [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) $C$ to a [voltage](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-voltage) $U_0$ — its plates then hold $\pm Q_0$ with $Q_0 = C U_0$ — and at $t = 0$ switch it onto an *ideal* [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) $L$, of zero [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance). No source anywhere in the loop: whatever happens next, the circuit does on its own.

![The LC circuit: a capacitor charged to Q_0, a switch, an ideal coil. Closing K lets the charge spill through the coil — and the coil, hating change, will not let it stop.](https://one-course.com/images/onecourse/chapters/physics-2/g12-rlc-oscillations/fig-ef767bffd34d.svg)

*The [LC circuit](#prop-g12-rlc-oscillations-equation): a [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) charged to $Q_0$, a switch, an ideal [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid). Closing $K$ lets the charge spill through the [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) — and the [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid), hating change, will not let it stop.*

**Proposition 31.1 (The LC equation).**

In the ideal LC loop the charge $q(t)$ of the upper plate obeys

$$
L\,\frac{d^2q}{dt^2} + \frac{q}{C} = 0,
\qquad\text{i.e.}\qquad
\frac{d^2q}{dt^2} = -\frac{1}{LC}\,q .
$$

**Proof.** Around the loop the [voltages](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-voltage) sum to zero ([Proposition 12.4](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#prop-g11-circuits-and-power-laws)): $u_L + u_C = 0$. The [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) gives $u_C = q/C$ and the [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) $u_L = L\,di/dt$ ([Definition 30.10](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-inductor)), with $i = dq/dt$ counting the charge arriving on the upper plate ([Proposition 30.3](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#prop-g12-rc-rl-circuits-cdudt)); substitute. ∎

**Theorem 31.2 (Free oscillations of the LC circuit).**

Released at $t = 0$ with $q(0) = Q_0$ and $i(0) = 0$, the circuit oscillates:

$$
q(t) = Q_0 \cos\!\left(\frac{2\pi t}{T_0}\right),
\qquad
i(t) = -I_0 \sin\!\left(\frac{2\pi t}{T_0}\right),
\qquad
T_0 = 2\pi\sqrt{LC},
$$

with peak current $I_0 = 2\pi Q_0/T_0 = Q_0/\sqrt{LC}$.

**Proof.** Verify by substitution. Differentiating, $i = dq/dt = -\frac{2\pi}{T_0} Q_0 \sin(2\pi t/T_0)$ — the stated $i(t)$ — and again $\frac{d^2q}{dt^2} = -(\frac{2\pi}{T_0})^2 q$, which equals $-q/(LC)$ exactly when $(2\pi/T_0)^2 = 1/(LC)$, i.e. $T_0 = 2\pi\sqrt{LC}$. And $q(0) = Q_0$, $i(0) = 0$. That no *other* function fits the equation and the start is admitted, exactly as for the spring ([Remark 28.9](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#rem-g12-oscillators-and-time-uniqueness)). ∎

**Definition 31.3 (Natural period and frequency).**

The *natural period* of an [LC circuit](#prop-g12-rlc-oscillations-equation) is $T_0 = 2\pi\sqrt{LC}$; its *natural frequency* is

$$
f_0 = \frac{1}{T_0} = \frac{1}{2\pi\sqrt{LC}} .
$$

Dimensional check: $1\,\mathrm{H} = 1\,\mathrm{V}\,\mathrm{s}/\mathrm{A}$ and $1\,\mathrm{F} = 1\,\mathrm{C}/\mathrm{V}$, so $LC$ carries $\mathrm{s} \times \mathrm{C}/\mathrm{A} = \mathrm{s}^{2}$ — $\sqrt{LC}$ is a time, as it must be.

**Example 31.4 (First numbers).**

$L = 10\,\mathrm{mH}$ with $C = 1.0\,\text{µ}\mathrm{F}$: $\sqrt{LC} = 1.0 \times 10^{-4}\,\mathrm{s}$, so $T_0 = 0.63\,\mathrm{ms}$ and $f_0 \approx 1.6\,\mathrm{kHz}$ — an audible tone. $L = 1.0\,\mathrm{mH}$ with $C = 100\,\mathrm{pF}$: $T_0 = 2.0\,\text{µ}\mathrm{s}$, $f_0 \approx 0.50\,\mathrm{MHz}$ — a radio [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator). Only the *product* $LC$ sets the tempo.

**Remark 31.5 (A quarter period out of step).**

The current is a quarter [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) out of [phase](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-phase) with the charge: $i = 0$ when $q = \pm Q_0$ (plates full, flow reversing) and $i = \pm I_0$ when $q = 0$ (plates empty, flow at full tilt) — exactly the pendulum, motionless at the extremes and fastest at the bottom.

![Charge and current of the free LC circuit: the current peaks each time the charge crosses zero, a quarter period out of step.](https://one-course.com/images/onecourse/chapters/physics-2/g12-rlc-oscillations/fig-6b6eee0e7326.svg)

*Charge and current of the free [LC circuit](#prop-g12-rlc-oscillations-equation): the current peaks each time the charge crosses zero, a quarter [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) out of step.*

## 31.2 The energy waltz

**Proposition 31.6 (Conservation of the stored energy).**

In the ideal [LC circuit](#prop-g12-rlc-oscillations-equation) the total stored [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) ([Proposition 30.15](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#prop-g12-rc-rl-circuits-energy))

$$
E \;=\; E_C + E_L \;=\; \frac{q^2}{2C} + \tfrac12\,L i^2
\;=\; \frac{Q_0^2}{2C} \;=\; \tfrac12\,L I_0^2
$$

is constant: entirely in the [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) when $q = \pm Q_0$, entirely in the [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) when $i = \pm I_0$.

**Proof.** Differentiate: $\frac{dE}{dt} = \frac{q}{C}\frac{dq}{dt} +
Li\frac{di}{dt} = i\left(\frac{q}{C} + L\frac{d^2q}{dt^2}\right) = 0$ by [Proposition 31.1](#prop-g12-rlc-oscillations-equation). Its value is read at $t = 0$ (all [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor)), and again at the all-current instant $q = 0$ — whence $\tfrac12 L I_0^2 = Q_0^2/2C$, recovering $I_0 = Q_0/\sqrt{LC}$ by [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) alone. ∎

![One period of the waltz: energy sloshes from plates to coil and back twice per period, the sum pinned at Q_02/2C — the very figure of the spring–mass oscillator, relabeled.](https://one-course.com/images/onecourse/chapters/physics-2/g12-rlc-oscillations/fig-7f05345a5d8c.svg)

*One [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) of the waltz: [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) sloshes from plates to [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) and back twice per [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator), the sum pinned at $Q_0^2/2C$ — the very figure of the [spring–mass oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#prop-g12-oscillators-and-time-spring-period), relabeled.*

**Example 31.7 (The waltz, measured).**

$C = 1.0\,\text{µ}\mathrm{F}$ charged to $U_0 = 10\,\mathrm{V}$, switched onto $L = 10\,\mathrm{mH}$: [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) $E = \tfrac12 C U_0^2 = 5.0 \times 10^{-5}\,\mathrm{J}$, [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) $T_0 = 0.63\,\mathrm{ms}$, and a peak current $I_0 = U_0\sqrt{C/L} = 0.10\,\mathrm{A}$ reached a quarter [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) ($0.16\,\mathrm{ms}$) after the switch closes.

## 31.3 Real circuits: damped oscillations

Every real [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) is wound from [resistive](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-sign) wire: the honest loop is a series *RLC* circuit, and the [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance) taxes the waltz.

**Definition 31.8 (Pseudo-periodic and aperiodic regimes).**

In a series RLC circuit the resistor dissipates [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) by Joule heating ([Chapter 12](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#ch-g11-circuits-and-power)): the [free oscillations](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) are *[damped](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping)*. Two regimes, read off an [oscilloscope](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-oscilloscope):

- *[pseudo-periodic](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-period)* (small $R$ ): the circuit still oscillates inside a shrinking envelope, with a repeat time — the *pseudo-period* — close to $T_0 = 2\pi\sqrt{LC}$ when the [damping](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping) is light;
- *aperiodic* (large $R$ ): the charge creeps back to zero without ever overshooting — no oscillation at all.

The exact mirror of the [damped](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping) [mechanical oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) ([Definition 28.12](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping)).

![The two fates of a real RLC circuit: oscillations dying inside an exponential envelope (dashed), or a creep to zero with no oscillation. Lightly damped, the pseudo-period stays T_0.](https://one-course.com/images/onecourse/chapters/physics-2/g12-rlc-oscillations/fig-a4ca16d2b0ee.svg)

*The two fates of a real [RLC circuit](#def-g12-rlc-oscillations-regimes): oscillations dying inside an exponential envelope (dashed), or a creep to zero with no oscillation. Lightly [damped](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping), the [pseudo-period](#def-g12-rlc-oscillations-regimes) stays $\approx T_0$.*

**Remark 31.9 (Maintaining oscillations).**

To keep the [amplitude](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) alive, feed the circuit exactly the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) the [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance) burns, once per cycle and *in step* with the swing: an amplifier does for the RLC loop what the escapement’s push does for the pendulum ([Chapter 28](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#ch-g12-oscillators-and-time)). Every quartz watch and every radio transmitter is such a maintained [oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator), running at its [natural frequency](#def-g12-rlc-oscillations-period) $f_0$.

## 31.4 The mechanical twin

**Proposition 31.10 (The electrical–mechanical dictionary).**

The LC equation is the spring–mass equation ([Proposition 28.8](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#prop-g12-oscillators-and-time-spring-period)) word for word:

| [LC circuit](#prop-g12-rlc-oscillations-equation) |  | spring–mass glider |
| --- | --- | --- |
| [2pt] charge $q$ | $\longleftrightarrow$ | position $x$ |
| current $i = dq/dt$ | $\longleftrightarrow$ | velocity $v = dx/dt$ |
| [inductance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-inductor) $L$ | $\longleftrightarrow$ | mass $m$ |
| inverse [capacitance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) $1/C$ | $\longleftrightarrow$ | [stiffness](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-spring) $k$ |
| [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance) $R$ | $\longleftrightarrow$ | [friction](https://one-course.com/books/physics/2/en/chapter/6-forces-and-the-principle-of-inertia#def-g10-inertia-inventory) |
| $L\,\frac{d^2q}{dt^2} + \frac{q}{C} = 0$ | $\longleftrightarrow$ | $m\,\frac{d^2x}{dt^2} + kx = 0$ |
| [2pt] $T_0 = 2\pi\sqrt{LC}$ | $\longleftrightarrow$ | $T_0 = 2\pi\sqrt{m/k}$ |
| [2pt] $E_C = q^2/2C$ | $\longleftrightarrow$ | $E_p = \tfrac12 kx^2$ |
| $E_L = \tfrac12 Li^2$ | $\longleftrightarrow$ | $E_k = \tfrac12 mv^2$ |

Same equation, same solutions: every result about either [oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) translates instantly into the other language.

**Proof.** Compare [Proposition 31.1](#prop-g12-rlc-oscillations-equation) with $m\,d^2x/dt^2 = -kx$: substituting $q \to x$, $L \to m$, $1/C \to k$ turns one into the other, and $LC \to m/k$ turns $2\pi\sqrt{LC}$ into $2\pi\sqrt{m/k}$. ∎

**Example 31.11 (Tuning a radio).**

An aerial feeds a whisper of *every* station into an LC loop, but only the one broadcasting near $f_0$ pushes in step and builds up — [resonance](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-resonance) ([Definition 28.13](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-resonance)). The dial rotates the plates of a variable [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor): changing $C$ moves $f_0 = 1/(2\pi\sqrt{LC})$, and the needle hands the earphone one station at a time. On the transmitting side, a maintained [oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) ([Remark 31.9](#rem-g12-rlc-oscillations-maintain)) sets the broadcast [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator); in a watch, a quartz crystal — a mechanical twin of breathtaking [stiffness](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-spring) — plays the same role.

## 31.5 Exercises

**Exercise 31.1 ★.**

Compute $T_0$ and $f_0$ for (a) $L = 10\,\mathrm{mH}$, $C = 1.0\,\text{µ}\mathrm{F}$; (b) $L = 1.0\,\mathrm{mH}$, $C = 100\,\mathrm{pF}$. By what factor do the frequencies differ?

**Solution of Exercise 31.1.**

(a) $\sqrt{LC} = 1.0 \times 10^{-4}\,\mathrm{s}$: $T_0 = 0.63\,\mathrm{ms}$, $f_0 \approx 1.6\,\mathrm{kHz}$. (b) $\sqrt{LC} = 3.16 \times 10^{-7}\,\mathrm{s}$: $T_0 = 2.0\,\text{µ}\mathrm{s}$, $f_0 \approx 0.50\,\mathrm{MHz}$ — a factor $\sqrt{10^5} \approx 316$.

**Exercise 31.2 ★.**

Using $1\,\mathrm{H} = 1\,\mathrm{V}\,\mathrm{s}/\mathrm{A}$ and $1\,\mathrm{F} = 1\,\mathrm{C}/\mathrm{V}$, show [unit](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) by [unit](https://one-course.com/books/physics/2/en/chapter/1-orders-of-magnitude-measuring-the-universe#def-g10-orders-of-magnitude-unit) that $\sqrt{LC}$ is a time. Why does the $2\pi$ change nothing?

**Solution of Exercise 31.2.**

$LC$: $(\mathrm{V}\,\mathrm{s}/\mathrm{A})(\mathrm{C}/\mathrm{V}) = \mathrm{s} \times \mathrm{C}/\mathrm{A} =
\mathrm{s} \times \mathrm{s} = \mathrm{s}^{2}$, so $\sqrt{LC}$ is in seconds. $2\pi$ is a pure number — no dimension to change.

**Exercise 31.3 ★.**

A $470\,\mathrm{nF}$ [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) is charged to $6.0\,\mathrm{V}$. Compute $Q_0$ and the stored [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy); describe, without computation, what follows when it is switched onto an ideal [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid).

**Solution of Exercise 31.3.**

$Q_0 = CU_0 = 2.8\,\text{µ}\mathrm{C}$; $E = \tfrac12 CU_0^2 =
\tfrac12 \times 4.7 \times 10^{-7} \times 36 \approx 8.5\,\text{µ}\mathrm{J}$. The charge oscillates at $f_0 = 1/(2\pi\sqrt{LC})$, current a quarter [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) out of [phase](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-phase), [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) sloshing between plates and [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid), total constant.

**Exercise 31.4 ★.**

An [LC circuit](#prop-g12-rlc-oscillations-equation) follows $q(t) = 2.0\cos(2\pi t/T_0)$ in microcoulombs, with $T_0 = 4.0\,\mathrm{ms}$. Read off $Q_0$; compute $f_0$, $q(T_0/2)$, $i(0)$, and the peak current $I_0 = 2\pi Q_0/T_0$.

**Solution of Exercise 31.4.**

$Q_0 = 2.0\,\text{µ}\mathrm{C}$; $f_0 = 1/T_0 = 250\,\mathrm{Hz}$; $q(T_0/2) = -2.0\,\text{µ}\mathrm{C}$; $i(0) = 0$ (cosine flat at $t=0$); $I_0 = 2\pi \times 2.0 \times 10^{-6}/4.0 \times 10^{-3} \approx 3.1\,\mathrm{mA}$.

**Exercise 31.5 ★.**

At which instants of the cycle is the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) entirely in the [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor)? Entirely in the [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid)? What is the mechanical counterpart of each moment for a pendulum?

**Solution of Exercise 31.5.**

All in the [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) when $q = \pm Q_0$, $i = 0$ ($t = 0$, $T_0/2$, $T_0$, …): the pendulum’s extremes. All in the [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) when $q = 0$, $i = \pm I_0$ ($t = T_0/4$, $3T_0/4$, …): the pendulum’s lowest, fastest point.

**Exercise 31.6 ★★.**

Verify by substitution that $q(t) = Q_0\cos(2\pi t/T_0)$ satisfies $L\,d^2q/dt^2 + q/C = 0$ exactly when $T_0 = 2\pi\sqrt{LC}$.

**Solution of Exercise 31.6.**

$d^2q/dt^2 = -(2\pi/T_0)^2\,Q_0\cos(2\pi t/T_0) = -(2\pi/T_0)^2 q$, so $L\,d^2q/dt^2 + q/C = q\,[\,1/C - L(2\pi/T_0)^2\,] = 0$ for all $t$ exactly when $(2\pi/T_0)^2 = 1/(LC)$, i.e. $T_0 = 2\pi\sqrt{LC}$.

**Exercise 31.7 ★★.**

A [long-wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) station broadcasts at $198\,\mathrm{kHz}$. With $C = 220\,\mathrm{pF}$, what [inductance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-inductor) tunes to it?

**Solution of Exercise 31.7.**

$L = \dfrac{1}{4\pi^2 f_0^2 C} = \dfrac{1}{39.5 \times
(1.98 \times 10^{5})^2 \times 2.2 \times 10^{-10}} \approx 2.9\,\mathrm{mH}$.

**Exercise 31.8 ★★.**

$C = 1.0\,\text{µ}\mathrm{F}$ charged to $10\,\mathrm{V}$ discharges into $L = 10\,\mathrm{mH}$. Compute the stored [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy), the peak current (by [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) conservation), and the instant it is first reached.

**Solution of Exercise 31.8.**

$E = \tfrac12 CU_0^2 = 5.0 \times 10^{-5}\,\mathrm{J}$; $\tfrac12 LI_0^2 = E$ gives $I_0 = \sqrt{2E/L} = \sqrt{1.0 \times 10^{-2}} = 0.10\,\mathrm{A}$, first reached at $T_0/4 = \tfrac14 \times 0.63\,\mathrm{ms} \approx
0.16\,\mathrm{ms}$.

**Exercise 31.9 ★★.**

Explain why $E_C$ and $E_L$ oscillate at [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) $2f_0$, not $f_0$. Show that at $t = T_0/8$ (started from $q = Q_0$) the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) is split exactly in half.

**Solution of Exercise 31.9.**

$E_C \propto \cos^2(2\pi t/T_0)$, and $\cos^2$ repeats every *half* turn: two fillings of the [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) per [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator), hence $2f_0$. At $t = T_0/8$ the [phase](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-phase) is $\pi/4$: $\cos^2(\pi/4) = \tfrac12$, so $E_C = E_L = E/2$.

**Exercise 31.10 ★★.**

On an [oscillogram](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-oscilloscope) of a lightly [damped](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping) [RLC circuit](#def-g12-rlc-oscillations-regimes), successive maxima read $4.0\,\mathrm{V}$ then $3.0\,\mathrm{V}$. Assuming the ratio stays constant, predict the sixth maximum after the $4.0\,\mathrm{V}$ one. What fraction of the *[energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy)* survives each [pseudo-period](#def-g12-rlc-oscillations-regimes)? What can you say of the [pseudo-period](#def-g12-rlc-oscillations-regimes) itself?

**Solution of Exercise 31.10.**

Ratio $3.0/4.0 = 0.75$ per [pseudo-period](#def-g12-rlc-oscillations-regimes): sixth maximum $4.0 \times 0.75^6 \approx 0.71\,\mathrm{V}$. [Energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) $\propto$ [amplitude](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) squared: $0.75^2 \approx 56\%$ survives each [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator). The [pseudo-period](#def-g12-rlc-oscillations-regimes) stays $\approx T_0 = 2\pi\sqrt{LC}$: light [damping](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping) shrinks the [amplitude](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator), hardly the tempo.

**Exercise 31.11 ★★.**

An [LC circuit](#prop-g12-rlc-oscillations-equation) has $L = 0.50\,\mathrm{H}$ and $C = 20\,\text{µ}\mathrm{F}$. Compute $T_0$; then, using the dictionary ([Proposition 31.10](#prop-g12-rlc-oscillations-analogy)), find the [stiffness](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-spring) $k$ giving a $0.50\,\mathrm{kg}$ glider the same [period](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator), and check it.

**Solution of Exercise 31.11.**

$T_0 = 2\pi\sqrt{0.50 \times 2.0 \times 10^{-5}} = 2\pi \times
3.16 \times 10^{-3} \approx 20\,\mathrm{ms}$. Dictionary: $k = m/(LC) =
0.50/1.0 \times 10^{-5} = 5.0 \times 10^{4}\,\mathrm{N}/\mathrm{m}$; check $2\pi\sqrt{m/k} = 2\pi\sqrt{1.0 \times 10^{-5}}$ — the same $20\,\mathrm{ms}$.

**Exercise 31.12 ★★★.**

A tuner uses $L = 0.20\,\mathrm{mH}$ and a variable [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) sweeping $50\text{ to }500\,\mathrm{pF}$. Compute the two extreme frequencies. Show that $f_0 \propto 1/\sqrt{C}$, so tenfold in $C$ gives only $\sqrt{10} \approx 3.2$ in [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator).

**Solution of Exercise 31.12.**

$C = 500\,\mathrm{pF}$: $f_0 = 1/(2\pi\sqrt{1.0 \times 10^{-13}}) \approx
0.50\,\mathrm{MHz}$; $C = 50\,\mathrm{pF}$: $f_0 \approx 1.6\,\mathrm{MHz}$. At fixed $L$, $f_0 = (2\pi\sqrt{L})^{-1} C^{-1/2} \propto 1/\sqrt{C}$: tenfold $C$, only $\sqrt{10} \approx 3.2$ in $f_0$ — as found.

**Exercise 31.13 ★★★.**

A real [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) makes the circuit lose $5.0\%$ of its [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) each [pseudo-period](#def-g12-rlc-oscillations-regimes). After how many [periods](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) has the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) halved? What is the [amplitude](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) then, as a fraction of the start? At $f_0 = 1.0\,\mathrm{MHz}$, how long is that in microseconds?

**Solution of Exercise 31.13.**

$0.95^n = \tfrac12$: $n = \ln 2/\lvert\ln 0.95\rvert \approx 13.5$, so about $14$ [periods](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator). [Amplitude](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) $\propto \sqrt{E}$: down to $1/\sqrt{2} \approx 0.71$ of the start. At $1.0\,\mathrm{MHz}$, $T_0 = 1.0\,\text{µ}\mathrm{s}$: about $14\,\text{µ}\mathrm{s}$.

**Exercise 31.14 ★★★.**

A watch [oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) stores $E = 1.0 \times 10^{-9}\,\mathrm{J}$ and loses $2.0\%$ of it per cycle at $f_0 = 32\,768\,\mathrm{Hz}$. What average [power](https://one-course.com/books/physics/2/en/chapter/17-work-of-a-force#def-g11-work-of-force-power) must the maintaining circuit supply? Why does the push have to arrive in step with the oscillation?

**Solution of Exercise 31.14.**

Loss per cycle $0.020 \times 1.0 \times 10^{-9}\,\mathrm{J} = 2.0 \times 10^{-11}\,\mathrm{J}$, at $32\,768$ cycles per second: $P = 32\,768 \times 2.0 \times 10^{-11}
\approx 6.6 \times 10^{-7}\,\mathrm{W}$ — under a microwatt, years on a button cell. In step, because only a push at the right [phase](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-phase) *adds* [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) ([resonance](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-resonance)); out of step it would brake the swing.

**Exercise 31.15 ★★★.**

After each edge of a square [signal](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-signal), an experimenter’s wiring “rings” at $25\,\mathrm{MHz}$; the stray [inductance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-inductor) is about $1.0\,\text{µ}\mathrm{H}$. Estimate the stray [capacitance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) responsible. Which regime should a small added series resistor push the circuit into, and why does that cure the ringing?

**Solution of Exercise 31.15.**

$C = \dfrac{1}{4\pi^2 f_0^2 L} = \dfrac{1}{39.5 \times
(2.5 \times 10^{7})^2 \times 1.0 \times 10^{-6}} \approx 41\,\mathrm{pF}$. The resistor should push the loop into the [aperiodic regime](#def-g12-rlc-oscillations-regimes): the charge then settles without overshooting — no oscillation, no ringing.

## 31.6 Problem: Building a crystal radio

**Problem 31.1.**

Weekend problem — building a crystal radio: one coil, one variable capacitor, a diode and an earphone — no battery, no amplifier, and the evening news arrives on the energy of the wave itself

A kit from a grandparent’s attic: a [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) $L = 0.20\,\mathrm{mH}$ wound on a cardboard tube, a variable [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) whose plates rotate from $20\,\mathrm{pF}$ to $480\,\mathrm{pF}$, a crystal diode, an earphone, and $20\,\mathrm{m}$ of aerial wire. The stations to catch broadcast between $530\,\mathrm{kHz}$ and $1700\,\mathrm{kHz}$.

**Part I — The tuning loop.**

1. The [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) , charged, is left to discharge into the [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) : write the loop law and turn it into the differential equation for $q(t)$ .
2. Verify by substitution that $q(t) = Q_0\cos(2\pi t/T_0)$ solves it, provided $T_0 = 2\pi\sqrt{LC}$ .
3. Check by dimensional analysis that $\sqrt{LC}$ is a time.
4. Dial set to $C = 330\,\mathrm{pF}$ : compute $T_0$ and $f_0$ . Is the loop tuned inside the band?
5. Find the [capacitance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) that tunes to exactly $1.00\,\mathrm{MHz}$ .

**Part II — The variable [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor).**

6. Compute the [capacitance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) needed for $530\,\mathrm{kHz}$ .
7. Compute the [capacitance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) needed for $1700\,\mathrm{kHz}$ .
8. Compute the two frequencies the kit’s $20\text{ to }480\,\mathrm{pF}$ [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) actually reaches. Does it cover the whole band?
9. Show that $f_0 \propto 1/\sqrt{C}$ at fixed $L$ ; by what factor must $C$ change to double $f_0$ ?
10. The plates open linearly, so $C$ falls steadily as the dial turns. Explain why the stations then crowd together at one end of the dial — and which end.

**Part III — [Energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) and [damping](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-damping).** Tuned to $1.00\,\mathrm{MHz}$ ($C = 127\,\mathrm{pF}$), the aerial momentarily charges the [capacitor](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) to $U_0 = 20\,\mathrm{mV}$.

11. Compute the charge $Q_0$ and the stored [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) $E$ .
12. Compute the peak current $I_0$ , using [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) conservation.
13. The [coil](https://one-course.com/books/physics/2/en/chapter/15-magnetism-and-magnetic-fields#def-g11-magnetic-fields-solenoid) ’s [resistance](https://one-course.com/books/physics/2/en/chapter/12-electric-circuits-and-power#def-g11-circuits-and-power-resistance) dissipates $4.0\%$ of the stored [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) per cycle: after how many cycles has the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) halved?
14. How long is that in microseconds? Why can this “ringing” alone not play the evening news?
15. Explain why the arriving [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) keeps the loop swinging only when tuned to that station — and what maintains the oscillation at the *transmitter* ( [Remark 31.9](#rem-g12-rlc-oscillations-maintain) ).

**Part IV — The mechanical twin.**

16. Using the dictionary ( [Proposition 31.10](#prop-g12-rlc-oscillations-analogy) ) with a glider of mass $m = 0.20\,\mathrm{kg}$ , compute the [stiffness](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-spring) $k$ of the spring matching the tuned circuit.
17. A stiff car spring has $k \approx 5 \times 10^{4}\,\mathrm{N}/\mathrm{m}$ . Comment: can a bench-top spring–mass system oscillate at megahertz?
18. Keep a reasonable $k = 100\,\mathrm{N}/\mathrm{m}$ instead: what mass oscillates at $1.00\,\mathrm{MHz}$ ?
19. What real object is such a featherweight, ultra-stiff [mechanical oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) , and where did you meet it earlier this year ( [Chapter 28](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#ch-g12-oscillators-and-time) )?
20. Spec sheet, one sentence: the band your radio sweeps, the [capacitance](https://one-course.com/books/physics/2/en/chapter/30-rc-and-rl-circuits#def-g12-rc-rl-circuits-capacitor) that picks $1.00\,\mathrm{MHz}$ , how long a lone ring lasts, and what plays the news anyway.

**Solution of Problem 31.1.**

**1.** Loop law: $u_L + u_C = 0$, with $u_C = q/C$, $u_L = L\,di/dt$, $i = dq/dt$: $L\,d^2q/dt^2 + q/C = 0$.

**2.** $d^2q/dt^2 = -(2\pi/T_0)^2 q$, so the equation reads $q\,[\,1/C - L(2\pi/T_0)^2\,] = 0$, true for all $t$ exactly when $T_0 = 2\pi\sqrt{LC}$.

**3.** $LC$: $(\mathrm{V}\,\mathrm{s}/\mathrm{A})(\mathrm{C}/\mathrm{V}) = \mathrm{s} \times
\mathrm{C}/\mathrm{A} = \mathrm{s}^{2}$ — $\sqrt{LC}$ is a time.

**4.** $LC = 6.6 \times 10^{-14}\,\mathrm{s}^{2}$: $T_0 = 2\pi \times 2.57 \times 10^{-7}
\approx 1.6\,\text{µ}\mathrm{s}$, $f_0 \approx 620\,\mathrm{kHz}$ — inside $530\text{ to }1700\,\mathrm{kHz}$.

**5.** $C = \dfrac{1}{4\pi^2 f_0^2 L} = \dfrac{1}{39.5 \times
1.0 \times 10^{12} \times 2.0 \times 10^{-4}} \approx 127\,\mathrm{pF}$.

**6.** Same formula at $530\,\mathrm{kHz}$: $C \approx 451\,\mathrm{pF}$.

**7.** At $1700\,\mathrm{kHz}$: $C \approx 44\,\mathrm{pF}$.

**8.** $C = 480\,\mathrm{pF}$: $f_0 \approx 514\,\mathrm{kHz}$; $C = 20\,\mathrm{pF}$: $f_0 \approx 2.5\,\mathrm{MHz}$ — the whole band, with margin at both ends.

**9.** $f_0 = (2\pi\sqrt{L})^{-1}C^{-1/2} \propto 1/\sqrt{C}$; doubling $f_0$ requires dividing $C$ by $4$.

**10.** Since $f_0 \propto 1/\sqrt{C}$, equal steps of $C$ move $f_0$ slowly at large $C$ and fast at small $C$: a given spacing between stations then takes little dial travel — they crowd at the [high-frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) (low-$C$) end.

**11.** $Q_0 = CU_0 = 1.27 \times 10^{-10} \times 0.020 \approx
2.5 \times 10^{-12}\,\mathrm{C}$; $E = \tfrac12 CU_0^2 \approx 2.5 \times 10^{-14}\,\mathrm{J}$.

**12.** $\tfrac12 LI_0^2 = E$: $I_0 = U_0\sqrt{C/L} =
0.020\sqrt{6.35 \times 10^{-7}} \approx 16\,\text{µ}\mathrm{A}$.

**13.** $0.96^n = \tfrac12$: $n = \ln 2/\lvert\ln 0.96\rvert
\approx 17$ cycles.

**14.** At $1.00\,\mathrm{MHz}$, $T_0 = 1.0\,\text{µ}\mathrm{s}$: the ring halves in about $17\,\text{µ}\mathrm{s}$. A lone ring dies in microseconds; music needs the loop fed continuously.

**15.** The [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) pushes millions of times per second; only when its [frequency](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator) matches $f_0$ do the pushes arrive in step and build the swing — [resonance](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-resonance); off-tune stations push as often against as with it. At the transmitter, an amplifier returns the [dissipated](https://one-course.com/books/physics/2/en/chapter/18-mechanical-energy-and-its-conservation#def-g11-mechanical-energy-dissipation) [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) each cycle: a maintained [oscillator](https://one-course.com/books/physics/2/en/chapter/28-mechanical-oscillators-and-the-measurement-of-time#def-g12-oscillators-and-time-oscillator).

**16.** $LC = 2.54 \times 10^{-14}\,\mathrm{s}^{2}$: $k = m/(LC) =
0.20/2.54 \times 10^{-14} \approx 7.9 \times 10^{12}\,\mathrm{N}/\mathrm{m}$.

**17.** About $10^8$ times the car spring: no bench-top spring–mass system reaches megahertz — everyday stiffnesses and masses are hopelessly slow.

**18.** $m = \dfrac{k}{(2\pi f_0)^2} = \dfrac{100}
{(6.28 \times 10^{6})^2} \approx 2.5 \times 10^{-12}\,\mathrm{kg}$ — a few nanograms, a speck of dust.

**19.** A quartz crystal: a featherweight, ultra-stiff sliver singing at its [natural frequency](#def-g12-rlc-oscillations-period) — the timekeeper met earlier this year.

**20.** With $L = 0.20\,\mathrm{mH}$ and $20\text{ to }480\,\mathrm{pF}$ the loop sweeps about $0.51\text{ to }2.5\,\mathrm{MHz}$, covering the band; $C \approx 127\,\mathrm{pF}$ picks $1.00\,\mathrm{MHz}$; left alone the ring halves in $\approx 17\,\text{µ}\mathrm{s}$, but the station’s own [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave), resonating in step, re-feeds the loop — the news plays on the [energy](https://one-course.com/books/physics/2/en/chapter/9-energy-forms-and-conservation#def-g10-energy-conservation-energy) of the [wave](https://one-course.com/books/physics/2/en/chapter/8-signals-and-waves#def-g10-signals-and-waves-wave) itself.