## Oscillation of a Mass-Spring System Compared with Electrical Oscillation in an L-C Circuit

### Mass-Spring System

• Kinetic energy = $\frac12mv_x^2$
• Potential energy = $\frac12kx^2$
• $\frac12mv_x^2+\frac12kx^2=\frac12kA^2$
• $v_x = \pm\sqrt{\frac km}\sqrt{A^2-x^2}$
• $v_x=\frac{dx}{dt}$
• $\omega=\sqrt{\frac km}$
• $x=A\cos(\omega t+\phi)$

### Inductor-Capacitor Circuit

• Magnetic energy = $\frac12Li^2$
• Electrical energy = $\frac{q^2}{2C}$
• $\frac12Li^2+\frac{q^2}{2C}=\frac{Q}{2C}$
• $i = \pm\sqrt{\frac1{LC}}\sqrt{Q^2-q^2}$
• $i=\frac{dq}{dt}$
• $\omega=\sqrt{\frac1{LC}}$
• $q=Q\cos(\omega t+\phi)$

## Uncharted 4: A Thief’s End

Surprisingly awesome!

No proverbial zombies or substances this time.  Deeper and more likeable characters, who you actually care about – nothing like “The Last of Us” with its intrusive “emotional bond development” crap.  Way better from moral standpoint.  A lot (a lot!) of details.  Very impressive facial expressions.  A real treat for vehicle driving fans.  Incredibly small number of bugs.  I didn’t feel irritated with the gamepad while shooting (like I did playing Uncharted 1-3).

The biggest con I can think of is the traditional (for the Uncharted series) total lack of (perceived) cooperation among opposing NPCs.  Of course, if one of the NPCs sees you, a bunch of others will attack you as well, but they won’t say anything – as if they just communicate telepathically.  Probably a design choice, because there’s already a lot of comments and remarks going on between the player and his allies.

Also, I didn’t particularly like the two ending scenes.  The thing is, all the three Drakes – who we meet in the game (Elena is somehow sill Fischer) – lie to each other, to Elena, and to others, at one point of the game or another.  And only Nathan has to pay for it, and only a little.  And the only guy in the whole game who is actively, seriously not happy about lying is… the villain!  Who dies in a deus-ex-machina way – as usual in the Uncharted series.

P.S. How come Rafe didn’t see Sam at Rossi???

P.P.S. Actually, that’s not the only curious thing about the game.

## Electromagnetic Induction

• Faraday’s law.  Induced emf in a closed loop $\mathcal{E}=-\frac{d\Phi_B}{dt}$, $\Phi_B$ – magnetic flux through the loop.
• Lenz’s law.  An induced current or emf always tends to oppose or cancel out the change that caused it.
• Motional emf.  $\mathcal{E}=\oint(\vec{v}\times\vec{B})\cdot d\vec{l}$.
• Induced electric fields.  When an emf is induced by a changing magnetic flux through a stationary conductor, there is an induced nonconservative electric field $\vec{E}$: $\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}$.
• Displacement current.  A time-varying electric electric field generates displacement current $i_D$, which acts as a source of magnetic field in exactly the same way as conduction current: $i_D=\epsilon\frac{d\Phi_E}{dt}$.
• Maxwell’s equations.  The relationships between electric and magnetic fields and their sources:

$\oint\vec{E}\cdot d\vec{A}=\frac{Q_{encl}}{\epsilon_0}$ (Gauss’s law for $\vec{E}$ fields)
$\oint\vec{B}\cdot d\vec{A}=0$ (Gauss’s law for $\vec{B}$ fields)
$\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}$ (Faraday’s law)
$\oint\vec{B}\cdot d\vec{l}=\mu_0(i_C+\epsilon_0\frac{d\Phi_E}{dt})_{encl}$ (Ampere’s law including displacement current).

## Magnetic Field and Magnetic Forces

• Magnetic forces: $\vec{F}=q\vec{v}\times\vec{B}$.
• Gauss’s law for magnetism: $\oint\vec{B}\cdot d\vec{A}=0$.
• Magnetic force on a conductor: $d\vec{F}=Id\vec{l}\times\vec{B}$.
• Magnetic torque: $\vec{\tau}=\vec{\mu}\times\vec{B}$, $\vec{\mu}=I\vec{A}$.  $U=-\vec{\mu}\cdot\vec{B}$.
• Magnetic field of a moving charge: $\vec{B}=\frac{\mu_0}{4\pi}\frac{q\vec{v}\times\hat{\vec{r}}}{r^2}$.
• The law of Biot and Savart: $d\vec{B}=\frac{\mu_0}{4\pi}\frac{Id\vec{l}\times\hat{\vec{r}}}{r^2}$.
• Magnetic field of a long, straight, current-carrying conductors: $\vec{B}=\frac{\mu_0 I}{2\pi r}$.
• Magnetic force between two long, parallel, current-carrying conductors: $\frac{F}{L}=\frac{\mu_0 I I\prime}{2\pi r}$.
• Magnetic field at the center of a current loop: $B=\frac{\mu_0 I}{2r}$.
• Long solenoid that has $n$ turns per unit length: $B=\mu_0 n I$.
• Ampere’s law: $\oint\vec{B}\cdot d\vec{l}=\mu_0 I_{encl}$.

## Neglecting Development of Your Pieces

Always start by playing out a center pawn, as this creates a line for developing a bishop.  Bring out the king knight very early – preferably to f3(f6).  By playing out the king knight and king bishop quickly, you make early castling possible and thus get your king out of any immediate danger.

Try to avoid placing your bishops on diagonals where they are blocked by your own pawns. Avoid, too, an excessive number of pawn moves – they contribute little or nothing to development.

Play over your games to see whether you are achieving the following minimum in the first ten moves: both center pawns advanced; both knights developed; both bishops developed; castling completed. This is an ideal goal which you may not always achieve, but it will help you to guard against moving the same piece repeatedly.

Managing the queen is a different matter. If you develop her too soon you will only expose her to harrying by enemy pieces of lesser value.

Reinfeld, Fred. “The Complete Chess Course: From Beginning to Winning Chess!”