Charging a Capacitor

(instantaneous application of Kirchhoff’s rules to non-steady-state situation)

Use lower case v, i, q to denote time-varying voltage, current and charge

0

0 : 0

qiR

Ct q

dq qi

dt R RC

Initial current 0IR

fQ CFinal conditions, i=0

0

( ) (1 exp( ))

exp( ) exp( )

dq qi

dt R RCdq dt

q C RC

tq t C

RCdq t t

i Idt R RC RC

Time-constant

RC When time is small, capacitor charges quickly. For that either resistance or capacitance must be small (in either case current flows “easier”)

Discharging a capacitor

)exp()(

)exp()(

:0

0

tQ

tI

RC

tQtq

RC

q

dt

dqI

Qqt

IRC

q

Power distribution systems

Everything is connected in parallel

V=120 V (US and Canada)V=220-240 V (Europe, Asia)

Circuit Overloads and Short Circuits

Circuit breaker

Fuse

Utility power (kW*h) 3 61 (10 )(3600 ) 3.6 10kW h W s J

Magnetism

First observation ~2500 years agoin fragments of magnetized iron ore

Previously, interaction was thought in terms of magnetic polesThe pole that points North on the magneticfield of the Earth is called north poleWhen points South – south pole

By analogy with electric field bar magnetsets up a magnetic field in a space around it

Earth itself is a magnet. Compass needlealigns itself along the earth’s magnetic field

Earth as a magnet

Magnetic Poles vs Electric Charge

The interaction between magnetic poles is similar to the Coulomb interaction of electric charges BUT magnetic poles always come in pairs (N and S), nobody has observed yet a single pole (monopole).

Despite numerous searches, no evidence of magnetic charges exist. In other words, there are no particles which create a radial magnetic field in the way an electric charge creates a radial field.

Magnetic Field

)( BvEF q

Lorentz force acting on charge q moving with velocity v in electric field E and magnetic field B

Electric charges produce electric fields E and, when move, magnetic fields B

In turn, charged particles experience forces in those fields:

For now we will concentrate on how magnetic force affects moving charged particles and current-carrying conductors…

Like electric field, magnetic field is a vector field, B

Magnetic Forces on Moving Charges

sinF q v B q vB

Force F is perpendicular to the plane of v and B and numerically equal to

Direction of F is specified as follows

F q v B

(G) Gauss 10T 1

mA

N

m/sC

N(T) Tesla 1

:field magneticfor Unit

4

The right hand rule is a useful mnemonic for visualizing the direction of a magnetic force as given by the Lorentz force law. The diagrams above are two of the forms used to visualize the force on a moving positive charge. The force is in the opposite direction for a negative charge moving in the direction shown. One fact to keep in mind is that the magnetic force is perpendicular to both the magnetic field and the charge velocity, but that leaves two possibilities. The right hand rule just helps you pin down which of the two directions applies.

Measuring Magnetic Fields with Test Charges

( )F q E v B

Total force with both electric andmagnetic fields acting on the charge q

Example: Magnetic force on a protonBeam of protons moves at v=300000 m/sthrough a uniform field B=2.0 T at an angle30 degrees relative to the field direction

Alternative rule – direction of right-hand-threadscrew would advance when turned in the same direction as rotation of vector v toward B for a positive charge

Magnetic field does NO work; only the direction of the velocity changes, not its

magnitude!

B

v-

Which direction does the charge deflect?a)Upb)Downc)It keeps going straight

Application: The Mass Spectrometer

An atom or molecule is ionized by knocking one or more electrons off to give a positive ion. This is true even for things which you would normally expect to form negative ions (chlorine, for example) or never form ions at all (argon, for example). Mass spectrometers always work with positive ions.

The ions are accelerated so that they all have the same kinetic energy.

The ions are then deflected by a magnetic field according to their masses. The lighter they are, the more they are deflected.

The amount of deflection also depends on the number of positive charges on the ion - in other words, on how many electrons were knocked off in the first stage. The more the ion is charged, the more it gets deflected.

The beam of ions passing through the machine is detected electrically.

Magnetic Field Lines

IMPORTANT – Magnetic field lines are NOT lines of force !!!

The force is always perpendicularto magnetic field lines.

If magnetic fields exert forces on moving electric charges, then moving electric charges create magnetic fields, i.e. currents produce fields.

Follows right-hand rule: point thumb of right hand in direction of current - magnetic field curls around wire in direction of curled fingers

If the current flows in a loop, the magnetic field produced is like a bar magnetic - curl fingers of right hand in direction of current flow - north pole is in direction of thumb.

Sources of Magnetic Fields and Field Lines

Sources of Magnetic Fields and Field Lines

The electron spins on its axis, giving rise to a electron current in the direction of rotation. Think of the electron as a ball with charge distributed over its surface. When the ball spins, that charge is set in motion around the electron's spin axis, resulting in a magnetic field specific to the electron.

The electron is like a magnetic dipole, a miniature magnet, with a north end and a south end.

In most substances, electrons spin in random directions - magnetic fields cancel. For iron and other magnetic substances, the spin magnetism is not canceled. Can be permanently magnetized by placing in strong magnetic field and permanently aligning atoms - can be demagnetized by dropping magnet and jostling atoms out of alignment.

Electromagnetic produced by wrapping coil around iron bar - magnetic field produced that aligns atoms in bar - more coils or more current - larger magnetic field and greater atomic alignment