Circuits, cont. Calculating the current in the circuit is called circuit analysis Two types of circuits: DC stands for direct current The current is of constant magnitude and direction at the source AC stands for alternating current (Ch. 22) Source current changes magnitude and/or direction The current can be viewed as the motion of the positive charges traveling through the circuit Section 19.4

Circuits, cont. There must be a complete circuit for charge flow There must be a return path from the resistor for the current to return to the voltage source If the circuit is open, there is no current flow anywhere in the circuit Section 19.4

Circuit Analysis The main tools of circuit analysis are equivalence, Ohm’s Law, and Kirchhoff’s Rules Loop Rule: the change in potential energy of a charge as it travels around a complete circuit loop must be zero Junction Rule: the amount of current entering a junction much be equal to the current leaving it

Kirchhoff’s Loop Rule Conservation of energy is the heart of the Loop Rule Consider the electric potential energy of a test charge moving through the circuit The test charge gained energy when it passed through the battery It lost energy as it passed through the resistor The energy is converted into heat energy inside the resistor Section 19.4

Kirchhoff’s Loop Rule, cont. Since PEelec = q V, the loop rule also means the change in the electric potential around a closed circuit path is zero ΔV = ε – I R = 0 For the entire loop Assumes wires have no resistance Section 19.4

Power In the resistor, the energy decreases by Power is energy per unit time Applying Ohm’s Law, P = I² R = V² / R The circuit converts chemical energy from the battery to heat energy in the resistors Power is only dissipated through resistance Section 19.4

Kirchhoff’s Junction Rule Conservation of charge is the heart of the Junction Rule The points where the currents branch are called junctions or nodes Since charge cannot be created or destroyed, all charge entering a junction must leave the junction Section 19.4

Kirchhoff’s Junction Rule, cont. Since all charge entering a junction must leave the junction, the current entering must equal the current exiting the junction IT = I1 + I2 Both of Kirchhoff’s Rules may be needed to solve a circuit Section 19.4

Direction of Current Current directions are chosen arbitrarily The signs of the potentials in Kirchhoff’s loop rule will depend on assumed current direction Moving parallel to the current produces a potential drop across a resistor The sign of the currents in Kirchhoff’s junction rule will depend on assumed current direction Currents entering a junction are positive, currents leaving are negative Section 19.4

Direction of Current, cont. After solving the equations, the sign of the current indicates the direction if the current is positive the assumed direction of the current is correct If the current is negative, the direction of the current Is opposite the assumed direction Section 19.4

Using Kirchhoff’s Rules Reduce all equivalences first Apply the junction rule as many times as gives independent equations Use the loop rule to get as many equations as unknowns in the system Section 19.4

Example:

Capacitors in Circuits Charge takes time to move through wire V is felt at the speed of light, however Change in potential across capacitor is V = q / C Charge on capacitor is time dependent Section 19.5

Current and Capacitors No charge flows from one plate to the other in a capacitor. There is a net current through a capacitor, though. Negative charge flows off of one plate Net accumulation of positive charge on top plate Negative charge flows onto the other plate from the opposite direction. Positive flow in one direction is the same as negative flow in the other direction

RC Circuits When the switch is open, No current flows (open circuit), I = 0 Voltage across resistor, VR = I R = 0 Voltage across capacitor, VC = 0 Charge on capacitor, Q = C VC = 0 Section 19.5

RC Circuits, cont. Just after the switch is closed, (t = 0) Charge takes time to move so Q = 0 VC = Q / C = 0 VR = ε Current in circuit, I = V / R = ε / R Section 19.5

RC Circuits, cont. After the switch is closed for a long time, (t = ∞) Charge builds up until potential across capacitor equals emf of battery, VC = ε Q = C VC = C ε VR = 0 Current in circuit, I = 0 “A long time” is determined by the time constant, τ Section 19.5

Time Constant, τ = RC Time constant is set by components of the circuit τ = RC R and C are the total resistance and capacitance, respectively, of the circuit Time constant is independent of voltages or current through the circuit Section 19.5

Circuit Analysis Main tools are equivalence, Ohm’s Law, and Kirchhoff’s Rules Loop Rule: the change in potential around a complete circuit loop must be zero Junction Rule: the amount of current entering a junction much be equal to the current leaving it Current direction is set arbitrarily at the beginning and checked for correctness at the end. Circuit loops (for loop rule) have direction Looping across a battery from + to – gives “-ε” Looping across a resistor in direction opposite the current gives “+IR”

Example:

Electrostatics Summary Fields and Forces Coulomb’s Law Electric Flux and Gauss’s Law Conductors and Insulators Energy and Potential Conservation of energy Current and Circuits Drift velocity Capacitors, Resistors, and Batteries Kirchhoff’s Rules Series and Parallel Section 19.5

Magnetic Fields and Forces Chapter 20 Magnetic Fields and Forces Magnetic Levitation: http://www.youtube.com/watch?v=m-Al7GAnH8Q

Magnet Field The magnetic field is analogous to the electric field The symbol for the magnetic field is The magnetic field lines can be deduced from the pattern of the iron filings The filings are small, needle-shaped, permanent magnets Section 20.1

Magnetic Field Lines The magnetic poles are indicated at the ends of the bar magnet Called north and south The magnetic poles are analogous to positive and negative charges The north poles of the filings are attracted to the south pole of the bar magnet Section 20.1

Magnetic Field from Current Moving charges produce magnetic fields An electric current consists of moving charges, so it will produce a magnetic field The iron filings show the magnetic field pattern due to the current Section 20.1

Right-Hand Rule 1 Used to find direction of magnetic field due to current in a wire Point the thumb of your right hand in the direction of the current You thumb will be parallel to the wire Curling the fingers of your right hand around the wire gives the direction of the magnetic field Section 20.1

Plotting Field Lines Field lines are three-dimensional A large dot (•) indicates the tip of the vector when it points out of the plane A cross (×) denotes the tails of the vector when it points into the plane Section 20.1

Charges and Magnetic Fields The electric current can be modeled as a collection of positive electric charges The charges would be moving with a velocity parallel to the current direction The direction of the magnetic field is given by the right-hand rule A positive charge moving to the left produces the same magnetic field as a negative charge moving to the right Section 20.1

Magnetic Field and Current Loop Treat the loop as many small pieces of wire Apply the right-hand rule to find the field from each piece of wire Applying superposition gives the overall pattern shown in fig. 20.9B Section 20.1

Two Parallel Wires What is the direction of the magnetic field in regions 1, 2 and 3? 1 I 2 I 3

Two Parallel Wires

Magnetic Forces & Bar Magnets To determine the total force on the bar magnet, you must look at the forces on each pole The total force is zero The total torque is non-zero The torque acts to align the bar magnet along the magnetic field Section 20.2

Magnetic Moment The bar magnet possesses a magnetic moment The bar magnet acts similar to an electric dipole The poles of the magnet act as a “magnetic charge” The north pole of one magnet will attract the south pole of another magnet Unlike poles attract Like poles will repel Similar to electric charges Section 20.2

Comparing Electric and Magnetic Fields and Forces Similarities Behavior between poles/charges Behavior in field lines Relation to distance Differences North and south magnetic poles always occur in pairs An isolated magnetic pole has never been found Magnetic fields do not terminate Magnetic force is perpendicular to motion of charge Section 20.2

Force on Moving Charge Magnetic force acts on individual moving charges The force depends on the velocity of the charge If the charge is not moving, there is no magnetic force If a positive charge, q, is moving with a given velocity, v, in an external magnetic field, B, then the magnitude of the force on the charge is The angle θ is the angle between the velocity and the field Section 20.3

Right Hand Rule 2 Used to determine the direction of the force on moving charges Point the figures of your right hand in the direction of the velocity and curl them in the direction of the field Your thumb points in the direction of the force Section 20.3

Motion of a Charged Particle A charged particle moves parallel to the magnetic field The angle between the velocity and the field is zero Therefore, the force is also zero Since sin θ = 0 Section 20.3

Motion of a Charged Particle, 2 A charged particle moves perpendicular to the magnetic field The angle between the velocity and the field is 90o Therefore, the force is The particle will move in a circle Section 20.3

Motion of a Charged Particle, 2 The circle lies in the plane perpendicular to the magnetic field lines The radius of the circle can be calculated from noting there must be a centripetal force acting on the particle Section 20.3

Motion of a Charged Particle, 3 A charged particle moves neither parallel nor perpendicular to the magnetic field The angle between the velocity and the field varies The path of the particle is helical The charged particle will spiral around the magnetic field lines Section 20.3

Example: Bubble Chamber Used in particle physics to investigate decay products What is the sign of the charge? What is the relative mass?

Right Hand Rules, Summary Right-hand rule number 1: Finding the direction of the magnetic field from an electric current Place the thumb of your right hand along the direction of the current Curl your fingers; they will then give the direction of the magnetic field as the field lines encircle the current Right-hand rule number 2: Finding the direction of the magnetic force on a moving charge, q Point the fingers of your right hand along the direction of the velocity Curl your fingers in the direction of the field Curl your fingers through the smallest angle that connects the velocity and the field If q is positive, the magnetic force is parallel to your thumb. If q is negative, the magnetic force is in the opposite direction Section 20.3

Earth’s Magnetic Field The Earth acts like a very large magnet A compass needle aligns with its north magnetic pole pointing approximately toward the Earth’s geographic north pole So the Earth’s geographic north pole is actually a south magnetic pole Section 20.9

Earth’s Magnetic Field, cont. The location of the Earth’s south magnetic pole does not correspond exactly with the geographic north pole The Earth’s south magnetic pole moves slowly Currently at about 40 km/year The Earth’s magnetic field has completely reversed direction The field is probably produced by electric currents in the core Section 20.9

Cosmic Rays Charged particles from space are called cosmic rays Their motion is affected by the Earth’s magnetic field At the equator, the particles are deflected away from the Earth’s surface At the poles, the particles follow a helical path and spiral into the poles They interact with the Earth’s atmosphere and produce aurora Section 20.9