Electrical energy & CapacitancePHY232 – Spring 2008Jon Pumplinhttp://www.pa.msu.edu/~pumplin/phy232(original ppt courtesy of Remco Zegers)PHY232 - Electrical Energy & Capacitance2work…A force is conservative if the work done on an object when moving from A to B does not depend on the path followed. Consequently, work was defined as: W = PEi – PEf = -∆PEThis was derived in Phy231 for a gravitational force, but as we saw in the previous chapter, gravitational and Coulomb forces are very similar:Fg=Gm1m2/r122 with G=6.67x10-11 Nm2/kg2 Fe=keq1q2/r122 with ke=8.99x109 Nm2/C2Hence: The Coulomb force is a conservative forcePHY232 - Electrical Energy & Capacitance3work & potential energyconsider a charge +q moving in an E field from A to B over a distance D. We can ignore gravity (why?)What is the work done by the field?What is the change in PE?If initially at rest, what is its speed at B?WAB=Fdcosθ with θ the angle between F and direction of movement, so WAB=FdWAB=qEd (since F=qE)work done BY the field ON the charge (W is positive) ∆PE=-WAB=-qEd : negative, so the potential energy has decreasedConservation of energy:∆PE+∆KE=0 ∆KE=1/2m(vf2-vi2)1/2mvf2=qEdv=√(2qEd/m)PHY232 - Electrical Energy & Capacitance4work & potential energy IIConsider the same situation for a charge of –q. Can it move from A to B without an external force being applied, assuming the charge is initially (A) and finally (B) at rest?WAB=-qEd ; negative, so work must be done by the charge. This can only happen if an external force is appliedNote: if the charge had an initial velocity the energy could come from the kinetic energy (I.e. it would slow down)If the charge is at rest at A and B: external work done: -qEdIf the charge has final velocity v then external work done: W=1/2mvf2+|q|EdPHY232 - Electrical Energy & Capacitance5ConclusionIn the absence of external forces, a positive charge placed in an electric field will move along the field lines (from + to -) to reduce the potential energyIn the absence of external forces, a negative charge placed in an electric field will move along the field lines (from - to +) to reduce the potential energy+++++++++++++--------------------PHY232 - Electrical Energy & Capacitance6questiona negatively charged (-1 µC) mass of 1 g is shot diagonally in an electric field created by a negatively charge plate (E=100 N/C). It starts at 2 m distance from the plate and stops 1 m from the plate, before turning back. What was the initial velocity in the direction along the field lines? ----------------------2m1mXYPHY232 - Electrical Energy & Capacitance7answerNote: the direction along the surface of the plate does not play a role(there is no force in that direction!)Kinetic energy balanceInitial kinetic energy: 1/2mv2=0.5*0.001*(vx2+vy2)Final (at turning point) kinetic energy: 0.5*0.001*vx2Change in kinetic energy: ∆KE=-0.5*0.001*vy2 =-5x10-4vy2Potential energy balanceChange in Potential energy: ∆PE=-qEd=1x10-6*100*1=+10-4 JConservation of energy: ∆PE+∆KE=0 so -5x10-4vy2+10-4=0vy=0.44 m/sPHY232 - Electrical Energy & Capacitance8Electrical potentialThe change in electrical potential energy of a particle of charge Q in a field with strength E over a distance d depends on the charge of the particle: ∆PE=-QEdFor convenience, it is useful to define the difference in electrical potential between two points (∆V), that is independent of the charge that is moving: ∆V= ∆PE/Q=-|E|dThe electrical potential difference has units [J/C] which is usually referred to as Volt ([V]). It is a scalarSince ∆V= -Ed, so E= -∆V/d the units of E ([N/C] before) can also be given as [V/m]. They are equivalent, but [V/m] is more often used.PHY232 - Electrical Energy & Capacitance9Electric potential due to a single chargethe potential at a distance r away from a charge +q is the work done in bringing a charge of 1 C from infinity (V=0) to the point r: V=keq/rIf the charge that is creating the potential is negative (-q) then V=-keq/rIf the field is created by more than one charge, then the superposition principle can be used to calculate the potential at any point+rV1CV=keq/rPHY232 - Electrical Energy & Capacitance10example+1C-2 Cr1 ma) what is the electric field at a distance r?b) what is the electric potential at a distance r?a) E=E1-E2=ke(Q1/r2)-ke(Q2/[1-r]2)=ke(1/r2)-ke(-2/[1-r]2)= ke(1/r2+2/[1-r]2) Note the -: E is a vectorb) V=V1+V2=ke(Q1/r)+ke(Q2/[1-r])=ke(1/r-2/[1-r]) Note the +: V is a scalar12PHY232 - Electrical Energy & Capacitance11questiona proton is moving in the direction of the electric field. During this process, the potential energy …… and its electric potential ……b) increases, decreasesc) decreases, increasesd) increases, increasese) decreases, decreases ++-∆PE=-WAB=-qEd, so the potential energy decreases (proton is positive)∆V= ∆PE/q, so the electric potential that the proton feels decreasesNote: if the proton were exchanged for an electron moving in the same direction, the potential energy would increase (electron is negative), but the electric potential would still decrease since the latter is independent of the particle that is moving in the fieldPHY232 - Electrical Energy & Capacitance12equipotential surfacescompare with a mapPHY232 - Electrical Energy & Capacitance13A capacitoris a device to create a constant electric field. The potential difference V=Edis a device to store charge (+ and -) in electrical circuits.the charge stored Q is proportional to the potential difference V: Q=CVC is the capacitance, units C/V or Farad (F)very often C is given in terms of µF (10-6F), nF (10-9F), or pF (10-12F)Other shapes exist, but for a parallel plate capacitor: C=ε0A/d where ε0=1/(4 pi k) = 8.85x10-12 F/m and A the area of the plates +++++++++++++--------------------+Q-Qdsymbol for capacitorwhen used in electric circuitPHY232 - Electrical Energy & Capacitance14electric circuits: batteriesThe battery does work (e.g. using chemical energy) to move positive charge from the – terminal to the + terminal. Chemical energy is transformed into electrical potential energy.Once at the + terminal, the charge can move through an external circuit to do work transforming electrical potential energy into other formsSymbol used in electric circuits:+ -PHY232 -