PHYS 208 Lecture 13

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Exam 2 next week: Ch. 23s3-5, 24-26

DC Circuits with Capacitors

A few final words to explain the stuff he flew through last time.

Capacitors added to a circuit: once a capacitor is charged, there is no current

Charging Capacitor

Kirchhoff's laws still apply: $V-IR-{\frac {Q}{C}}=0$, where $I={\frac {\mathrm {d} Q}{\mathrm {d} t}}$

Solving for $Q$ gives:

$Q(t)=V_{battery}C\left(1-\mathrm {e} ^{-{\frac {t}{RC}}}\right)$

Current (derivative of $Q$) has an inverted behavior:

$I={\frac {\mathrm {d} Q(t)}{\mathrm {d} t}}=V_{battery}C\mathrm {e} ^{-{\frac {t}{RC}}}$

Note: $RC$ is called the time constant

Magnetism and Magnetic Forces

All magnets that we know of have two poles: North (+) and South (−)

Like in electric charge, opposite poles attract, like poles repel.

The laws of physics do not prevent one-poled magnetic materials, but none have been found. Instead, we use long, thin magnetic rods so that the poles start looking like "point poles"

• There is a force field around magnetic objects. Magnetic field lines look and behave almost exactly like electric field lines
• All permanent magnets have an even number of poles. breaking a magnet will result in a pole at each end of the smaller magnet
• Magnetic field affects electric charges in a current, so there's a new twist.

Magnetic Force

Very similar to Coulomb's law:

$\left|{\vec {F}}\right|=k_{mag}{\frac {q_{1}q_{2}}{d^{2}}}$

However, we will be talking about the interactions between magnets and moving electric charges, which is a little more complicated:

${\vec {F}}_{mag}=q{\vec {v}}\times {\vec {B}}$

Where ${\vec {B}}$ is the magnetic field, and the strength is measured in Tesla [T = N sec / C m]. Like Coulombs, 1 Tesla is a huge amount.

Magnetic fields apply a torque around the path of the moving electron with the axis along the electric field lines. Force is perpendicular to both field and velocity

Recall Cross Products

Charges' velocity (speed) does not increase; only their direction changes, so the trajectory of a charge in a uniform magnetic field is a circle:

Recall PHYS 218 Chapter 3#Circular Motion:

${\begin{aligned}m{\frac {v^{2}}{r}}&=qvB\\mv&=qBr\end{aligned}}$

Magnetic Flux

Magnetic Flux through a surface is almost identical to electric flux:

$\Phi =\oint {\vec {B}}\cdot \mathrm {d} {\vec {A}}$

Measured in Tesla-meters [T m] (or Webers)

Because magnetic field lines loop through a magnetic pole (there are no single-pole particles), the flux of magnetic field over any closed surface is always 0.

Exploiting Electric and Magnetic Properties

Velocity Selector

A magnetic field and an electric field at right angles to each other so the magnetic and electric forces on particles of a certain velocity are equivalent:

${\begin{aligned}q{\vec {E}}&=q{\vec {v}}\times {\vec {B}}\\{\frac {E}{B}}&=v\end{aligned}}$