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Physics 272: Electricity and Magnetism Mark Palenik Wednesday, June 27th Midterm • Reminder: Midterm 1 week from Thursday Will cover up through tomorrow’s lecture • There will be no lab. • Midterm will start at 9:50 here (room 112) • 2 hours long • We will review solutions afterward Not contradictory • Two things may have sounded contradictory 1. Potential difference is path independent 2. You can’t know the potential from E at just two points x Need Need x Don’t Need • You need the electric field in enough points to do an indefinite integral (i.e. a small region around both end points) Energy stored in a field • Rather than potential energy, we can talk about the energy of the EM field • This is useful because radiation (e.g. light) carries energy, and we may want to know how much • Instead of a change in potential energy, we can say that rearranging charges changes the field energy • What is the energy density of a field (Energy/m3)? iClicker question • To find the field energy density, let’s start by thinking about work • Consider a capacitor with two plates. The charge on each plate is +Q and –Q • How much work does it take to move the second plate a distance Ds (assume infinite plates) a) b) c) d) 0 Q/Ae0 Ds Q2/Ae0 Ds Q2/2Ae0 Ds Field energy • We could write • DK + DU = W, since DK = 0, DU = W • Or, instead of potential energy, we can think of it as field energy • DK + DUfield = W, DK = 0, DUfield = W • This means DUfield = Q2/2Ae0 Ds = ½e0E2ADs = ½e0E2DV Energy density • So, ∆𝑈 ∆𝑉 1 2 = ∈0 𝐸 2 • Not a rigorous proof, but the result is correct! Volume, not potential Topics for today • Magnetic fields: – Why do we have magnetic fields – Electron current – Biot-Savart law – If time, how do magnetic fields arise from relativity So, why magnetism? • We all know a compass needle points North • Force called magnetism pulls on it so it always points that way • What is magnetism? Why do we have it? • Moving charges produce magnetism because of relativity! Electric fields and relativity • Coulomb’s law applies to charges at rest • Static charges produce fields that sit in space • But relativity says nothing can move faster than light, so moving a charge takes a finite time for its field to propagate • Also, relativity says things get squished in the direction of motion. Relativity gives rise to magnetism • Relativity says: Changes in electric field get delayed, electric fields and objects get squished when they move, and also that time runs differently for a moving observer. • Because of relativistic effects on charges and fields, we observe magnetic fields – Moving charges produce magnetic fields! Maxwell’s equations Maxwell’s equations are fully relativistic, even though they were written decades before Einstein came up with the theory of Relativity. r div( E ) = Ñ × E = e0 Current running Radiation out of the page div( B) = Ñ × B = 0 ¶B ¶B curl(E) = Ñ ´ E = curl(E) = Ñ ´ E = ¶t ¶t é ¶E ù curl( B ) = Ñ ´ B = m é J + e ¶E ù curl( B ) = Ñ ´ B = m0 ê J + e 0 0ê 0 ú ú ¶ t ¶t û ë û ë Current density Currents are source of B Electron current • The “J” in Maxwell’s equations is “current density”. We can start with the idea of electron current. • Electron current is number of electrons per second that pass through a section of wire. Labeled i (lower case). • Conventional current is the amount of charge passing through a section of wire per second. Labeled I (upper case). Conventional current • Originally, it was thought that positive charges that moved (Ben Franklin just guessed) • Conventional current points in the direction that positive charges flow • Opposite of direction that negative charges flow • Current is a vector, so you can also think of it as Charge*electron current Current is a vector • Electron current and conventional current are both vectors. • Because they have opposite sign, they point in opposite directions • 𝐼 = 𝑞𝑖 where q is charge of electron Electron current Conventional current iClicker • A wire has a circular cross section with an area of 10-6 m2. • The space between electrons in each direction is 10-10 m (So each electron occupies a “box” that is 10-30 m3) • How many electrons are in a section of wire 1 m long? a) b) c) d) e) 1030 1024 1016 1036 I have no idea Circular surface iClicker • A wire has a circular cross section with an area of 10-6 m2. • The space between electrons in each direction is 10-10 m (So each electron occupies a “box” that is 10-30 m3) • The electrons are moving at 2 m/s • How many electrons per second pass through a surface in the wire a) b) c) d) 1024 2x1024 2x1030 1048 Circular surface Electron current • Electron current, i = nAv = electron density*Cross sectional area of wire*velocity of electrons • Because of friction-like losses, an electric field is needed just to keep electrons moving at constant speed • v = mE where m is a constant of the metal (electron mobility) Biot-Savart Law • Similar to Coulomb’s law, gives the magnetic field of a current carrying wire • 𝐵= 𝜇0 𝑞𝑣𝑥𝑟 4𝜋 𝑟 2 𝜇0 4𝜋 10−7 • = = 𝑞 𝜇0 𝑙 𝑣𝑥𝑟 4𝜋 𝑟 2 𝑡𝑒𝑠𝑙𝑎 𝑚2 𝑐𝑜𝑢𝑙𝑜𝑚𝑏 𝑚/𝑠 𝜇0 𝐼 𝑥𝑟 𝑑𝑙 4𝜋 𝑟 2 1 𝜇0 ∈ 0 = 2 𝑐 𝑑𝑙 = also, • Inside the integral charge*velocity bocomes charge/length * velocity, which is simply current! • dl means that the integration takes place over the length of the wire. • 𝐼 𝑥𝑟 is a “cross product” Cross Products • Two ways to think about cross products. First: purely algebraic Unit vectors in x,y,z directions 𝑖 𝑗 𝑘 • 𝐴𝑥𝐵 = 𝑑𝑒𝑡 𝐴𝑥 𝐴𝑦 𝐴𝑧 𝐵𝑥 𝐵𝑦 𝐵𝑧 = 𝐴𝑦 𝐵𝑧 − 𝐵𝑦 𝐴𝑧 , 𝐵𝑥 𝐴𝑧 − 𝐴𝑥 𝐵𝑧 , 𝐴𝑥 𝐵𝑦 − 𝐵𝑥 𝐴𝑦 • The magnitude of a cross product is|𝐴||𝐵|sin(𝜃) • The cross product is perpendicular to both 𝐴 and 𝐵 • Since 𝐼 𝑥𝑟 appears in the cross product, magnetic field is perpendicular to current and r Right hand rule • There are a few different ways to do the right hand rule • The one I like for 𝐴𝑥𝐵 – Using your right hand, point thumb in the direction of A – Point fingers in direction of B – Hand will push in the direction of 𝐴𝑥𝐵 – Note, this only gives you direction, not magnitude • Also, keep in mind, we use a “right handed” coordinate system, so 𝑋𝑥𝑌 = 𝑍 • Can use |𝐴||𝐵|sin(𝜃) for magnitude iClicker: cross products • If our X and Y axes are labeled as below, which way does the Z axis point? y x a) Out of the page b) Into the page iClicker B field • Remember, the Biot-Savart law is: 𝜇0 𝐼 𝑥𝑟 𝑑𝑙 4𝜋 𝑟 2 • Assuming the conventional current through the blue wire points to the right, which way does the magnetic field point at the x? x a) b) c) d) To the right Up Out of the page Into the page I (conventional current) A second right hand rule for wires • We don’t usually just want to know B at one point next to a wire. We want to know field direction everywhere around the wire. • Field lines twist around wire, so: • Thumb in the direction of current, fingers curl in direction of field The Magnetic Effects of Currents Conclusions: • The magnitude of B depends on the amount of current • A wire with no current produces no B • B is perpendicular to the direction of current • B under the wire is opposite to B over the wire Oersted effect: discovered in 1820 by H. Ch. Ørsted How does the field around a wire look? Hans Christian Ørsted (1777 - 1851) Simple Circuits Thinner filament wire Tungsten filament Inert gas Use socket The Magnetic Effects of Currents Make electric circuit: Compass needle is a magnetic dipole. Magnetic dipoles want to align themselves along field lines What is the effect on the compass needle? What if we switch polarity? What if we run wire under compass? What if there is no current in the wire? Use short bulb Net field In this picture, there are two components to the field. Field from wire is strongest, since the wire is very close. • 𝐵𝑛𝑒𝑡 = 𝐵𝑤𝑖𝑟𝑒 + 𝐵𝑒𝑎𝑟𝑡ℎ this is what the needle wants to point along – you will figure out the specifics in lab and recitation The Magnetic Effects of Currents Make electric circuit: Needle deflection at different currents: change light bulb (to long one) short-circuit: two batteries, no light bulb Relativity gives rise to magnetism • Relativity says: Changes in electric field get delayed, electric fields and objects get squished when they move, and also that time runs differently for a moving observer. • Let’s do a simple example to see how this can create magnetism. • In reality, electric and magnetic fields are two parts of a single relativistic object called the Faraday tensor (don’t worry, we won’t talk about it!) Electric field observed when moving • Take two metal wires side by side. Electrons flow in each of them (protons are fixed). Wires are neutral, so electron density = Proton density. Take a ride on an electron. The electron sees the other electrons as fixed and protons as moving What the electron sees Protons Electrons Remember, relativity also says that objects contract in the direction of motion. iClicker question • The moving electrons in a neutral wire see protons as moving. This means (because distances contract in relativity) that the density of protons they see is a) Greater than the electron density b) Less than the electron density c) The same as the electron density Protons Electrons Electrons see higher proton density • Because electrons are moving, they see the distance between protons as contracted (as well as the protons themselves, since protons have a finite, but small size). • The distance between protons is smaller, so the proton density is higher. Electrons see this iClicker question • The fact that electrons see a higher proton density means that the two wires will a) Attract b) Repel c) Nothing The protons see each wire as neutral. The electrons see each wire as positively charged. Therefore, the electrons will be attracted to the other wire. So, do we need a wire? • We could make similar arguments if we had just streams of flowing electrons (instead of electrons and protons) – In the moving frame, the electron density is still lower, so there is less repulsive force • Individual moving charges will also produce a magnetic field