Chapter 15: The Electric Field Section 15.1: A Model of the Mechanisms for Electrostatic Interactions Action-At-A-Distance How can Object A affect Object B if they are not literally touching? Well, it's clearly possible: Both gravitational and and electrostatic forces are demonstrable Still leaves you with a giant shoulder-shrug over how it actually works What Is A Field? Start with the concrete and work up to the abstraction Question: What happens to Object A because of Object B? (Could be A and B have mass, or maybe they have charge either way) Answer: Object A gets pushed/pulled by the force exerted on it by Object B (duh, right?) You can describe the outcome qualitatively or quantitatively Qualitatively, Then: What Is a Field? A description: It answers the question "What happens..." You do not need to test gravity with every tennis ball you see; you know that every tennis ball will behave the same way It doesn't even need to be a tennis ball, does it? It does not have to be a real thing at all, it can be the idea of a thing Gravity pulls your real tennis ball and your imaginary tennis ball exactly the same way: straight down to the ground Quantitatively: What Is a Field? Draw the answer to the question "What happens...", and for gravity you've got a straight line pointing at the center of the Earth If you were standing on Mars, your tennis ball would behave precisely the same way: It would fall toward the center of Mars How do you distinguish between them? Mars' gravity is 3.7 m/s 2, and Earth's is 9.8 m/s 2 So, you can say the shape of each field is the same, but the field strengths are different Gravitational Field Due to a Single Object With Mass Let's get beyond the g=9.8m/s 2 at or near the surface of the Earth Drop that tennis ball from higher and higher above the surface, and it still falls straight "down" But when you get far enough away, you can't use F = mg anymore, you have to use F = GmM/r 2, and the acceleration is no longer 9.8m/s 2 F = m[gm/r 2 ], and now the acceleration is [GM/r 2 ], which represents the effect on m due to M (call it the field strength!) Electric Field Due to a Single Point Charge You are "dropping" charges instead of masses, but they are behaving exactly the same way Qualitatively: Electric field lines are radial, exactly like the gravitational field lines Fixed +Q: Field lines are radially directed outward (+q pushed away from +Q) Fixed Q: Field lines are radially directed inward (+q pulled towards Q) page 01/05
Electric Field Due to a Point Charge: Quantitatively F = q[kq/r 2 ] = qe (This is a vector equation!) Q: Fixed charge that creates the field (may be + or ) q: Test charge used to "see" the field (default is always positive!) Electric Field Lines If E = F/q, then the vector E has exactly the same direction as the force vector! Again: Superposition when there is more than a single charge Graphically indicate field strength with line density: More lines means a stronger field Superposition of Charges What if you have more than one fixed charge (a distribution of charges)? Add them up: E = E1 + E2 +... + En This is vector addition! Infinite Sheet of Charge = Uniform E Field Distribute total charge +Q over some area A (not literally infinite) E is constant, perpendicular away form plane If you used charge Q, direction would be toward plane A uniform field which is not distance-dependent is a useful thing to have in your back pocket Section 15.3: The V Field Potential vs Potential Energy Electric potential electric potential energy Potential = (electric PE)/charge, or work/charge Why? Well, why not? Turns out to be a genius idea Removes the test charge from consideration, leaves only the fixed distribution Voltage Is a Scalar Energy (in general) is a scalar property Electrical potential energy is not an exception Potential = energy/charge = Joules/Coulomb = Volts (just call it voltage!) Superposition Is Still The Rule For anything other than a point charge, add up the discrete elements Potential Difference Most of the time, you want the relative (as opposed to the absolute) Recall that for gravity, an arbitrary reference level could be chosen for zero PE Gravity: Mass falls "down," from higher to lower PE ( PE > 0) Positive charge falls "down," from higher to lower voltage ( V > 0) Negative charge falls "up," from lower to higher voltage ( V < 0) Equipotential Surfaces Literally, equal potential (so, same voltage) Lines of equipotential are normal to the electric field lines Lines get more widely spaced with increasing distance from source charge page 02/05
You Know, Like a Topographic Map Contour lines on a topo map show equal elevation You can tell from a glance whether a region has steep inclines or is fairly flat Just like a ball rolls downhill with greater acceleration when the hill is steeper, charges roll downhill (or uphill!) with greater acceleration when the potential gradient is steeper Section 15.4: Relating the E Field and the V This Is The Genius Part If F = qe and W = F x, then W = (qe) x If W = (qe) x = U and U = q V, then E = V/ x! Means what, exactly? Well, V = 0 means constant voltage, which in turn means zero E Means what, exactly? Ok, if V 0, then decreasing x increases the E field Section 15.5: Conductors in Electric Fields E Field of a Thin Spherical Conducting Shell Apply excess ( ) charge to a thin, hollow, metal sphere Like repels like: Charge will distribute evenly over the surface of the shell Outside the sphere: Effectively a point charge! Inside the sphere: E = 0 (E = F/q: add up all those vector forces!) means V = constant Grounding Why does this work? For the same reason that "water seeks its own level:" equal potential! Take that charged sphere, and wire it to a second (uncharged) sphere Charge will move to the uncharged sphere (like repels like), but only until the V equalizes: V1 = kq1/r1 = kq2/r2 = V2 Why won't all the excess charge move? Because like repels like If your second, uncharged sphere happens to be the Earth itself, the huge r means that effectively all (not literally all) excess charge is dissipated Faraday Cage Take an uncharged, hollow, conducting container (any shape is good, as long as it's hollow) Put it in an external E field (you choose how to come up with this) Why aren't you getting electrocuted?!?!?! Surprise! Net E field inside the uncharged container (Faraday cage) is exactly zero Section 15.6: Dielectric Materials in an Electric Field Polarizing Atoms (Or Molecules) Atoms are typically electrically neutral, (+) nucleus surrounded by ( ) electrons The e will typically distribute randomly/evenly surrounding the nucleus You can use an external Eo field to influence the location of the electrons You can keep the net charge = 0, but have (+) and ( ) ends of the atom page 03/05
Natural Dipoles Water (H 2 O) is a great example of a molecule that is polar even without application of an external E field The two hydrogens give up e to the oxygen, making the H end of the molecule (+2) and the O end of the molecule ( 2) Typically, the individual molecules are randomly oriented with respect to each other Dipole Moment Vector: Draw an arrow from the ( ) to the (+) center of a polar atom or molecule This vector represents the orientation of the polar molecule This is useful for visualizing what will happen to the molecule if an external E field is applied What Happens When You Line Them Up? Randomly oriented dipoles create no net electric field (everything cancels) Lined-up dipoles will create an electric field of their own If Eo is uniform, dipoles will create their own field E1 which points exactly opposite Within the dielectric medium, total E = Eo E1 (negative direction, not subtraction!) Not All Dielectric Materials Are the Same Dielectric constant: κ = Eo/E = Eo/(Eo E1) Vacuum: κ = 1 means that there is no E1! Vacuum does not affect Eo at all! The bigger κ, the greater E1 (as E1 approaches Eo, k approaches infinity!) Water: κ = 81 means that water is highly polarizable, and creates a substantial E1 when polarized Section 15.7: Capacitors Where Are We Headed With All This? Since we keep making connections with gravity and mechanical energy, keep going with this We use mechanical energy to perform mechanical work to do something useful and/or interesting We want to be able to use electrical energy to perform work to do other useful/interesting things We are trying to work out a way to store electrical PE to retrieve and use later Parallel Plate Capacitor Simplest configuration: parallel conducting plates Attach to a battery (which, unfortunately, we have no idea how it works yet) The battery's V will pull e from one plate and deposit on the other (magic!) The creates a uniform E field between plates (and zero E outside of plates!) Capacitance After you charge, you can disconnect the battery and plates remain charged; how much charge can you store? We know the E between the plates is constant, and depends on the total amount of charge: More q, more E You would have to increase the voltage to persuade more charges to move; more V creates more E page 04/05
So, more V puts more q on the plates: q = CV, where C is constant What Makes C a Constant? Several things affect the ability to store charge on a pair of plates Area: Larger plates can store more charge (sometimes the obvious actually is obvious) Plate separation: At a given V, increasing the separation decreases the E field (less E, less q) What's between the plates: Slip a dielectric in there, and you increase the ability to store charge! Higher κ, more q Energy of a Charged Capacitor You have to do electrical work to move charge from one plate to the other Every charge you move needs a little more work than the one before it: Slope is not constant! Exactly like stretching a spring: Every cm of stretch takes more force than the previous cm! Spring potential: U = ½kx 2 Energy to charge a capacitor: U = ½CV 2 (or U = q 2 /2C) Energy Density Just like density = mass/volume, energy density = energy/volume (or energy/mass) The idea is: How much energy do you get on a per-unit basis Ideally, you want a large return on your investment: Lots of energy for a small expenditure of mass In the context of capacitors: Mass not really relevant, size (volume) is page 05/05