Last week: Magnets & transformers A magnet is a material or object that produces a magnetic field. A method to detect a magnetic field is to scatter iron filings and observe their pattern. magnetic field An electromagnet is a wire coil in which the magnetic field is produced by the flow of an electric current. magnetic field current A transformer are two coils that transfer electrical energy from one circuit to another through magnetic coupling. A changing current in the first coil (the primary ) creates a changing magnetic field; in turn, this magnetic field induces a changing voltage in the second coil (the secondary).electrical energycircuitcurrentinduces Iron filings that have oriented in the magnetic field produced by a bar magnet & a coil

Course Summary Basic Concepts Matter: Atoms with a heavy nucleus and some electrons “circling” around. Electrons are negatively charged and very light. The nucleus is made of positive protons and neutral neutrons. The mass of the atom is proportional to the number of protons and neutrons in the nucleus. Phase changes of matter: When heat is added to - ice, the temperature of the ice increases till it reaches the melting temperature; - a mixture of ice and water, the temperature stays at the melting temperature, and more and more ice melts; -liquid water, the temperature of the water increases till it reaches the boiling Temperature (100 o C); -boiling water and water vapor, the temperature stays at the boiling temperature and more and more water evaporates -water vapor, the temperature of the vapor increases Mass: The mass of an object is the same, everywhere in the universe. It is measured in kg. Gravity: The gravitation force on a object is the same, everywhere on Earth, but less on the Moon, and greater on the Sun. It is measured in lbs or Newton.

Floating: The buoyancy force is opposite and equal to the weight force of the displaced liquid. Density of salt water > density of fresh water => object less submerged in salt water. Mechanical equilibrium: The net force is zero and the net torque is zero (balancing beam). Torque = (perpendicular force counter-clock-wise) * (displacement between axis and point of attack). Displacement =Change in position Velocity = Displacement / Time Motion with constant velocity: - Net force is zero (Newton’s first law). - Position versus time is linear (graph), - i.e. displacement = velocity * time - Example: Rock sinking in a viscous media, parachute: air resistance = - weightforce. Acceleration: Change in velocity / Time Motion with constant acceleration: -Net force is constant: Force = mass * acceleration (Newton’s second law). - Velocity versus time is linear, -i.e. velocity = acceleration * time -Example: Rock in free fall – y-axis points upwards – F=ma -> -m g = m a -> a = -g Energy: Kinetic energy = mass * velocity 2 /2 Potential energy = mass * gravity * height -example: ramp - kinetic energy= potential energy

Simple electronic circuits: a battery and a resistor in a closed loop (no short cut). Voltage drop = current * resistance Power consumption = resistance * current 2 Example: parallel light bulbs – serial light bulbs Magnets: detected with iron filing patterns - created by a loop current, - two poles (North and South), -equal poles repel, opposite poles attract (geographic North pole is a magnetic South pole) -there is a torque on a magnet in a magnetic field => electro motors - changing magnetic field induces a loop current

Complex systems: A system with a large throughput of - a fluid = turbulence, river networks - chemicals = flames & explosion - tension = fracture - electrical current = lightning, dielectric breakthrough - information = internet, social networks The throughput is large means “sudden appearance of a pattern or dynamics (self-organization)” This self-organization causes emergent properties. Today: Highlights from my research - Complex Systems

Complex Systems Research Project: Predicting the growth of ramified networks Alfred Hübler Center for Complex Systems Research University of Illinois at Urbana-Champaign Research supported in part by the National Science Foundation (DMS ITR)

We study: Growth of networks in a reproducible lab experiment, here: the structure of materials with high-voltage currents; which quantities a reproducible? We find: Materials produced in a high-voltage current develop open-loop, fractal structures which maximize the conductivity for the applied current. These fractal structures can be predicted with graph- theoretical models.