Magnetic Sources The Biot-Savart Law Magnetic fields go around the wire – they are perpendicular to the direction of current Magnetic fields are perpendicular to the separation between the wire and the point where you measure it Sounds like a cross product! r I ds Permeability of free space The Amp is defined to work out this way

Sample Problem A loop of wire consists of two quarter circles of radii R and 2R, both centered at a point P, and connected with wires going radially from one to the other. If a current I flows in the loop, what is the magnetic field at point P? ds ds I ds Do one side at a time First do one of the straight segments ds and r-hat are parallel No contribution to the integral Other straight segment is the same P ds R 2R Now do inner quarter loop Outer loop opposite direction, similar

Ampere’s Law (original recipe) Suppose we have a wire coming out of the plane Let’s integrate the magnetic field around a closed path There’s a funky new symbol for such an integral Circle means “over a closed loop” The magnetic field is parallel to direction of integration ds ds cos r d What if we pick a different path? I We have demonstrated this is true no matter what path you take Wire need not even be straight infinite wire All that matters is that current passes through the closed Ampere loop

We will later realize that this formula is imperfect Understanding Ampere’s Law If multiple currents flow through, add up all that are inside the loop Use right-hand rule to determine if they count as + or – Curl fingers in direction of Ampere loop If thumb points in direction of current, plus, otherwise minus The wire can be bent, the loop can be any shape, even non-planar 5 A 2 A 1 A 4 A 7 A We will later realize that this formula is imperfect

Using Ampere’s Law Ampere’s Law can be used – rarely – to calculate magnetic fields Need lots of symmetry – usually cylindrical A wire of radius a has total current I distribu-ted uniformly across its cross-sectional area. Find the magnetic field everywhere. I End-on view Draw an Ampere loop outside the wire – it contains all the current Magnetic field is parallel to the direction of this loop, and constant around it Use Ampere’s Law But we used a loop outside the wire, so we only have it for r > a

Using Ampere’s Law (2) Now do it inside the wire Ampere loop inside the wire does not contain all the current The fraction is proportional to the area a End-on view

Field Inside a Solenoid It remains only to calculate the magnetic field inside We use Ampere’s law Recall, no significant B-field outside Only the inside segment contributes There may be many (N) current loops within this Ampere loop Let n = N/L be loops per unit length L Works for any shape solenoid, not just cylindrical For finite length solenoids, there are “end effects” Real solenoids have each loop connected to the next, like a helix, so it’s just one long wire

A Tesla meter2 is also called a Weber (Wb) Magnetic Flux Magnetic flux is defined exactly the same way for magnetism as it was for electricity A cylindrical solenoid of radius 10 cm has length 50 cm and has 1000 turns of wire going around it. What is the magnetic field inside it, and the magnetic flux through it, when a current of 2.00 A is passing through the wire? A Tesla meter2 is also called a Weber (Wb)

Electromotive Force (EMF) Faraday’s Law Electromotive Force (EMF) Suppose we have some source of force on charges that transport them Suppose it is capable of doing work W on each charge It will keep transporting them until the work required is as big as the work it can do + – q The voltage difference at this point is the electromotive force (EMF) Denoted E

Right hand rule for Faraday’s Law: Motional EMF Suppose you have the following circuit in the presence of a B-field Charges inside the cylinder Now let cylinder move Moving charges inside conductor feel force Force transport charges – it is capable of doing work This force is like a battery - it produces EMF v  B W v L B v is the rate of change of the width W We can relate this to the change in magnetic flux Right hand rule for Faraday’s Law: EMF you get is right-handed compared to direction you calculated the flux

Power and Motional EMF Resistor feels a voltage – current flows v L R Where does the power come from? Current is in a magnetic field B To get it to move, you must oppose this force You are doing work The power dissipated in the resistor matches the mechanical power you must put in to move the rod

How to Make an AC Generator Have a background source of magnetic fields, like permanent magnets Add a loop of wire, attached to an axle that can be rotated Add “commutators” that connect the rotating loop to outside wires Rotate the loop at angular frequency  Magnetic flux changes with time This produces EMF To improve it, make the loop repeat many (N) times A

Sample Problem A rectangular loop of wire 20 cm by 20 cm with 50 turns is rotated rapidly in a magnetic field B, so that the loop makes 60 full rotations a second. At t = 0 the loop is perpendicular to B. (a) What is the EMF generated by the loop, in terms of B at time t? (b) What B-field do we need to get a maximum voltage of 170 V? The angle is changing constantly with time After 1/60 second, it must have gone in one full circle  loop of wire The flux is given by: The EMF is given by:

Comments on Generators The EMF generated is sinusoidal in nature (with simple designs) This is called alternating current - it is simple to produce This is actually how power is generated Generators extremely similar to motors – often you can use a single one for both Turn the axle – power is generated Feed power in – the axle turns Regenerative braking for electric or hybrid cars Generators: When current does not flow, there is little resistance to turning the axle When current does flow, magnetic fields produce forces that resist turning the axle Motors: When power is demanded, they require a lot of electric current When power is not needed, little power keeps them going

Ground Fault Circuit Interruptors Fuses/circuit breakers don’t keep you from getting electrocuted But GFI’s (or GFCI’s) do GFCI Under normal use, the current on the live wire matches the current on the neutral wire Ampere’s Law tells you there is no B-field around the orange donut shape Now, imagine you touch the live wire – current path changes (for the worse) There is magnetic field around the donut Changing magnetic field means EMF in coiled wire Current flows in coiled wire Magnetic field produced by solenoid Switch is magnetically turned off

Inductance Self-Inductance A Consider a solenoid L, connect it to a battery Area A, lengthl, N turns What happens as you close the switch? Lenz’s law – loop resists change in magnetic field Magnetic field is caused by the current “Inductor” resists change in current A l + – E

Inductors An inductor in a circuit is denoted by this symbol: L An inductor satisfies the formula: L is the inductance Measured in Henrys (H) L Kirchoff’s rules for Inductors: Assign currents to every path, as usual Kirchoff’s first law is unchanged The voltage change for an inductor is L (dI/dt) Negative if with the current Positive if against the current In steady state (dI/dt = 0) an inductor is a wire I + – L E

Energy in Inductors Is the battery doing work on the inductor? L + – L E Integral of power is work done on the inductor It makes sense to say there is no energy in inductor with no current Energy density inside a solenoid? Just like with electric fields, we can associate the energy with the magnetic fields, not the current carrying wires

RL Circuits An RL circuit has resistors and inductors Suppose initial current I0 before you open the switch What happens after you open the switch? Use Kirchoff’s Law on loop Integrate both sides of the equation R I L + –

RL Circuits (2) Where did the energy in the inductor go? How much power was fed to the resistor? Integrate to get total energy dissipated It went to the resistor Powering up an inductor: Similar calculation L E + – R

Sample Problem An inductor with inductance 4.0 mH is discharging through a resistor of resistance R. If, in 1.2 ms, it dissipates half its energy, what is R? I L = 4.0 mH R

Inductors in Series Inductors in Parallel L2 L1 For inductors in series, the inductors have the same current Their EMF’s add Inductors in Parallel For inductors in parallel, the inductors have the same EMF but different currents L1 L2

Parallel and Series - Formulas Capacitor Resistor Inductor Series Parallel Fundamental Formula