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? What if one bulb burns out?

Symmetry considerations to calculate equivalent resistances No current through the resistor I1 I2

To analyze more complex (steady-state) circuits: Kirchhoff’s rules To analyze more complex (steady-state) circuits: 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 Loop rule – statement that the electrostatic force is conservative.

Sign conventions for the loop rule

A single-loop circuit Charging of a car battery

Complex networks Find currents, potential differences and equivalent resistance

Electrical Measuring Instruments Galvanometer Can be calibrated to measure current (or voltage) Example: Full-scale deflection Ifs =1 mA, internal coil resistance Rc =20 W

For max current reading Ia of 50mA For max voltage reading Vv =10V

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 Initial current Final conditions, i=0

Time-constant 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

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 Fuse Circuit breaker

Utility power (kW*h)

Magnetism First observation ~2500 years ago in fragments of magnetized iron ore Previously, interaction was thought in terms of magnetic poles The pole that points North on the magnetic field of the Earth is called north pole When points South – south pole By analogy with electric field bar magnet sets up a magnetic field in a space around it Earth itself is a magnet. Compass needle aligns 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.

In turn, charged particles experience forces in those fields: Magnetic Field 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