ab

hdr

ba

h

r

xr

dxdRR

drba

hdxx

h

babxr

a

b

h

2

02 )(

;)(

Calculating resistance

A variable cross-section resistor treated as a serial combination of small straight-wire resistors:

Example: Equivalent resistances

Series versus parallel connection

What about power delivered to each bulb?

2

2 2ab bc

P I R or

V VP

R R

2

2de

P I R or

VP

R

What if one bulb burns out?

Symmetry considerations to calculate equivalent resistances

No current through the resistor

I1

I1

I1

I1

I1

I1

I2

I2

I2

I2

I2

I2

rR

rIrIV

IIII

r

6

56

5)

3

1

6

1

3

1(

:b and abetween drop voltageTotal

2/ ;3/ :Currents

resistors All

121

Kirchhoff’s rules

To analyze more complex (steady-state) circuits:

1. For any junction: Sum of incoming currents equals to sum of outgoing currents

(conservation of charge)

Valid for any junction

2. For any closed circuit loop: Sum of the voltages across all elements of the loop is zero

(conservation of energy)

Valid for any close loop

- The number of independent equations will be equal to the number of unknown currents

0I

0V

Loop rule – statement that the electrostatic force is conservative.

Sign conventions for the loop rule

A single-loop circuit

y)consistenconly is (Important

negative is

given, numbers With the

0

21

21

1221

I

RRI

IRIR

Charging of a car battery

Complex networks

1 1 3

2 2 3

1 3 2

(1 ) ( )(1 ) 0 (1)

(1 ) ( )(2 ) 0 (2)

(1 ) (1 ) (1 ) 0 (3)

I I I

I I I

I I I

Find currents, potential differences and equivalent resistance

3

1

2

1

6

5

I A

I A

I A

Electrical Measuring Instruments

Galvanometer

Can be calibrated to measurecurrent (or voltage)

Example: Full-scale deflectionIfs =1 mA, internal coil resistanceRc =20

0.020fs cV I R V

( )fs c a fs shI R I I R

For max current reading Ia of 50mA

0.408

0.4sh

eq

R

R

( )v fs c shV I R R

For max voltage reading Vv =10V

9980

10,000sh

eq

R

R

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