Chapter 30 – Summary

 

For me, chapter 30 seems to collect some pretty disparate subjects.  The main features are the connection between the potential and the electric field, the component of E in a particular direction is - dV/dl, where l is the direction in question.  For example Ex = - dV/dx.  Next, a good deal of space is taken up with the way a battery establishes a potential difference in a circuit.  This leads to a definition of capacitance and resistance, C and R, which are measured in Farads and Ohms respectively.

 

1.         We already saw that the work done by the electric force on a charge q changed the potential energy of the charge according to the relationship ∫FAds = - (Uf - Ui).  Divide all the terms in the equation by q.  F/q is just the electric field produced at the location of the charge q as it moves from the initial to final locations.  Likewise, U/q is the potential at those locations.  In terms of E and V, we get

 

            ∫EAds = - (Vf - Vi).

 

2.         In the special case that we pick ds to be parallel to E and calculate the integral over a very small piece of the path so that E is constant over that piece, we can replace the integral by E Δs = - Δ V.  Now divide by Δs and as Δs gets smaller and smaller, the right side becomes - dV/ds which equals the component of E in the s direction.

 

3.         A positive charge placed at a location in space where the electric field is E and the potential is V moves in a very specific way in response to E and V.  First the initial motion is parallel to E and since E points in the direction of decreasing potential the charge moves from a region of higher potential to one of lower potential.  Moreover, if we move the charge sideways, that is perpendicular to E, the electric force does no work since F dot ds = 0.  Consequently, along a path that is everywhere perpendicular to the local electric field there is no change in the potential, that is, the sideways motion defines a path of constant potential.

 

4.         At every point in a region that has charges there exists an electric field and a potential.  The two quantities are related as shown above.  Moving from point A where the potential is VA to point B where the potential is VB entails moving through a potential difference of VB - VA.  Because the electric force is conservative, that difference in potential is independent of the path taken to connect point A to point B.  This leads immediately to Kirchoff’s Loop Law,

 

            Δ Vloop = Σ ΔV = 0.

This is true because going from A to B, regardless of the path, produces a potential change of VB - VA.  While returning on any path from B to A produces a potential difference of VA - VB. The net potential change is obviously zero demonstrating Kirchoff’s Loop Law.

 

5.         For a conductor in electrostatic equilibrium, that is no moving charges, the electric field inside the conductor has to be zero and the field at the surface must be perpendicular to the surface.  Consequently, there can be no potential difference between any two points on or inside a conductor!  If there were a potential difference, that would imply an electric field since ∫EAds = - (Vf - Vi).  No electric field implies no potential differences.  This is a very important property of conductors.

 

6.         Although the chemistry inside a battery is important, for our purpose a battery is a device that converts chemical energy into electrical energy.  The work done on a charge q by the battery as the battery moves the charge from the negative to the positive terminal increases the electric potential energy of the charge.  For an ideal battery all the chemical work shows up as electric potential energy.  In a real battery some of the chemical work gets lost during the process.

 

Wchemical = ΔUelectric = q ΔV battery.

 

ΔV battery is called the emf (electromotive force) of the battery and is denoted by E.  The source of energy that powers a circuit is the thing called the emf.  For this semester the primary sources will be batteries and later, alternating current from a wall outlet!

 

7.         Now if a wire is connected between the two terminals of a battery, from Kirchoff’s Loop Law the potential difference going from the negative end of the battery to the positive end is just ΔV battery and that potential difference is the same whether our path goes through the battery or through the wire.  Consequently,

 

ΔV battery = ΔV wire

 

8.         The potential difference in the wire establishes an electric field E in the wire.  The electric field produces a current,

 

I = JA = σ E A = σ (ΔV wire /L )A,

 

where σ is the conductivity and L is the length of the wire.  Remember that the inverse of the conductivity is the resistivity, ρ.  So in terms of the above quantities, the potential difference across the wire can be written as,

ΔV wire = I{ ρA/L} = I R (Ohm’s Law),

 

where the quantity R is called the resistance and depends on the material the wire is made of and the geometry of the wire.  The resistivity is a property of the material and does not depend on the geometry of the wire.  The units of R are Ohm’s = Volts/Amperes.

 

9.         Imagine two conducting surfaces separated in space, that is not touching one another.  A charge Q is moved from one conductor to the other.  Now we have two conductors, one with charge Q and the other -Q.  Electric field lines run from the positive conductor to the negative conductor.  The electric field establishes a potential difference, -∫EAds = V+ - V- = ΔV.  The more charge transferred between the conductors, the stronger the electric field in the region around the conductors, the larger the potential difference going from one conductor to the other.  Typically Q is proportional to ΔV and the proportionality constant is called the capacitance, C.

 

            Q = C ΔV or C = Q/ΔV.

 

10.       For a parallel plate capacitor ΔV = Ed = (η/ε0)d = Q d/(Aε0).  Dividing this into Q gives the capacitance of a parallel plate capacitor,

 

            C = ε0 A/d.

 

The capacitance depends on the geometry of the two conductors used to establish the potential difference.  Not surprisingly, the geometry of the parallel plate capacitor leads to a particularly simple formula for the capacitance.

 

11.       Oftentimes more than one capacitor is in a circuit.  Therefore it is useful to know how the capacitance of a set of capacitors can be reduced to a single equivalent capacitor.  There are two cases of especial importance.  Capacitors connected in parallel to a source of emf and capacitors connected in series with a source of emf.

 

Imagine three capacitors connected in parallel with a source of emf.  This corresponds to each of the capacitors feeling the full potential difference established by the battery.  Now we want to replace those three capacitors with one capacitor that would have the same effect as the three.  Cequivalent = Q/ΔV.  The ΔV across the equivalent capacitor is just the potential difference of the battery, which is the same as being felt by each of the individual capacitors.  But the charge on the equivalent capacitor is the sum of the charges on the three capacitors,

 

            Qequivalent = C1 ΔV + C2 ΔV + C3 ΔV.

 

            Cequivalent = (C1 + C2 + C3 )ΔV/ΔV = C1 + C2 + C3 .

 

Obviously, this relationship can be generalized to any number of capacitors connected in parallel with a battery.

 

If three capacitors are connected in series, then by Kirchoff’s Loop Law the sum of the potential across the three capacitors has to equal the potential across the battery, ΔV = ΔV1 + ΔV2 + ΔV3 .  The key to thinking about capacitors in series is to see that the charge Q on each capacitor is the same!  The +Q on the first capacitor is balanced by the -Q on its other plate.  But that -Q comes from establishing a charge of +Q on the positive plate of the second capacitor which establishes a charge of + Q on the positive plate of the third capacitor in turn.  This gives,

 

            ΔV = Q/C1 + Q/C2 + Q/C3 = Q (1/C1 + 1/C2 + 1/C3).

 

But the charge on the equivalent capacitor is also Q so ΔV = Q/Cequivalent.  Setting the two expressions for ΔV equal to one another gives,

 

            1/Cequivalent = 1/C1 + 1/C2 + 1/C3.

 

12.       The last topic in the chapter is the energy stored in a capacitor.  Start with an uncharged capacitor and slowly drag charge from the negative plate to the positive plate until there is a charge Q on the positive plate and -Q on the negative plate.  The little bit of work I do to drag a charge Δq from the negative to the positive plate is just Δq V, where V is the potential difference across the capacitor when the charges on the plates is + q and - q respectively.  The potential difference between plates with positive and negative charge of q is just V = q/C.  Therefore the little bit of work done in moving Δq from the negative to positive plate is,

 

            ΔW = ΔUC = q Δq/C.

           

The total work I do in charging the capacitor is equal to the potential energy stored in the capacitor.  That is equivalent to integrating the above expression from 0 to Q giving,

 

            UC = ˝ Q2/C.

Using Q = VC, V=Ed, and C = ε0 A/d, the energy in the capacitor can be written in terms of the electric field E,

 

            UC = ˝ (ε0 E2) Ad,

 

but Ad is just the volume inside the capacitor, consequently the energy density, energy per volume, inside the capacitor, UC/Volume = uC = ε0 E2/2.

 

13.       The above equation for energy density stored in the electric field inside a capacitor is actually true anywhere that an electric field exists and not just inside a parallel plate capacitor! Later we will learn that electromagnetic waves consist of time varying electric and magnetic fields which transport energy according to the above equation.