Lecture6 : El. Potential (cont’d),...

Lecture6 : El. Potential (cont’d), CapacitorsToday:

• how to use equipotential surfaces to visualize how the electric potential varies in space.

• how to use electric potential to calculate the electric field.

• how to calculate electric potential in general

Ch. 24:

• The nature of capacitors, and how to calculate a quantity that measures their ability to store charge.

• How to analyze capacitors connected in a network.

• How to calculate the amount of energy stored in a capacitor.

© 2016 Pearson Education Inc.

Equipotential surfaces and field lines• An equipotential surface is a surface on which the electric

potential is the same at every point.

• Field lines and equipotential surfaces are always mutually perpendicular.

• Shown are cross sections of equipotential surfaces (blue lines)and electric field lines (red lines) for a single positive charge.


Equipotential surfaces and field lines for a dipole


Field and potential of two equal positive charges


Equipotentials and conductors• If the electric field had a tangential component at the surface

of a conductor, a net amount of work would be done on a test charge by moving it around a loop as shown here—which is impossible because the electric force is conservative.


Equipotentials and conductors• When all charges are at rest:

- the surface of a conductor is always an equipotential surface.

- the electric field just outside a conductor is always perpendicular to the surface.

- The electric field lines can’t go in loops, because there will be positive work done on a test charge going along such loop will. The charge will gain energy from nowhere. Violates energy conservation.


Qualitative problem about potential• The figure shows equipotential

contours drawn at 1kV intervals in the plane of three point charges, Q1 , Q2 , and Q3 . The values of the potentials are in kV as indicated for the +5, 0, and -5 kV contours. The positions of the charges are indicated by the dots. The letters are on the equipotential contours.

• For clarity, the equipotential lines are not drawn in the yellow regions.

a.) Charge Q2 is the largest negative charge.

b.) The magnitude of the electric field at i is stronger than at j.

c.) Q1 is a negative charge.

d.) Charge Q3 has the largest magnitude of all.

e.) The force on a proton at g points to the bottom of the page. f.) The electric field at k is zero.

Potential gradient• !" = −% & !(⃗ = −%)dx − %,dy −%.dz

• The components of the electric field can be found by taking partial derivatives of the electric potential:

• The electric field is the negative gradient of the potential:

The direction of the gradient is perpendicular to the equipotential surfaces. In this direction V changes the fastest! (the “steepest” direction). Let’s choose the coordinate system so that at that point (x,y,z) the steepest direction is x. Then % = −01

0) ̂3.© 2016 Pearson Education Inc.

% = −∇V = −010) ̂3 − 01

0, ̂6 − 010.78

How to find the potential?• Gauss Law ∮" # ⋅ %& =

()*+,-.

in a differential form (at a single point 0⃗ ):

• 12#2 + 14#4+ 15#5 ≡ ∇ ⋅ #(0⃗) = :(0⃗)/<= (requires the Gauss theorem)

• : is the volume charge density: > = ∫@ :(0⃗)%A0. & = 1B (boundary of V)

• For simplicity: Let’s have a region of space that has no charge (q=0). Then the potential V satisfies the Laplace equation:

∇CB = 0, F01CB

1GC+1CB

1HC+1CB

1IC= 0

We need to know its value on the boundary to solve this equation. Examples:

1. Cavity inside a conductor: V=V0 on the boundary, no charges in the cavity.

Solution: V=V0 inside the cavity.

Then E=0 inside as we know.


V0

How to find the potential? - 22. Two parallel conducting planes: V=V0 on 1, V=0 on 2

!

" = "$(1 − ()), Check that V satisfies +

,-+., +

+,-+(, +

+,-+0, = 0

and V(0)=V0, V(d)=0.

We can also find E and 2: ! = -3) , 2 = 5$! = 5$ -3

) .

v0

0

d12"$34"$

14"$

y

x

−2

2

How to find the potential? - 32. Spike (lightning rod): V=V0 , V=0 on ”clouds”


Ionization and corona discharge• At an electric-field magnitude of about 3 × 106 V/m or greater, air

molecules become ionized, and air becomes a conductor.

• For a charged conducting sphere, Esurface = Vsurface /R.

• For a sharp tip R is the radius of its curvature.

• Thus, if Em is the electric-field magnitude at which air becomes conductive (known as the dielectric strength of air), then the maximum potential Vm to which a spherical conductor can be raised is Vm = REm. You see why the Van der Graaf generator is a large sphere and not a cube or a sphere with spikes!


For ch. 23 problem solving videos• See prof. Tom Hemmick YouTube channel:https://www.youtube.com/watch?v=_XF78_y-EZY&list=PLFCadHQMw31ivfb9N9zpI3xd8jlnRBM8


https://www.youtube.com/watch?v=_XF78_y-EZY&list=PLFC-adHQMw31ivfb9N9zpI3xd8jlnRBM8

Capacitors• Any two conductors

separated by an insulator (or a vacuum) form a capacitor.

• When the capacitor is charged, it means the two conductors have charges with equal magnitude and opposite sign, and the net charge on the capacitor as a whole is zero.


Capacitors and capacitance• One common way to charge a capacitor is to connect the two

conductors to opposite terminals of a battery.

• This gives a potential difference Vab between the conductors that is equal to the voltage of the battery.

• If we change the magnitude of charge on each conductor, the potential difference between conductors changes; however, the ratio of charge to potential difference does not change.

• This ratio is called the capacitance C of the capacitor:


Parallel-plate capacitor• A parallel-plate capacitor consists of two parallel conducting

plates separated by a distance that is small compared to their dimensions.


Parallel-plate capacitor• The field between the plates of a parallel-plate capacitor is

essentially uniform, and the charges on the plates are uniformly distributed over their opposing surfaces.

• When the region between the plates is vacuum, the capacitance is: (remember !"# = %& = ' (

)*= +(

,)*

• The capacitance depends on only the geometry of the capacitor.

• The quantities A and d are constants for a given capacitor, and is a universal constant.


Units of capacitance• The SI unit of capacitance is the farad, F.

1 F = 1 C/V = 1 C2/N · m = 1 C2/J

• One farad is a very large capacitance.

• For the commercial capacitors shown in the photograph, C is measured in microfarads


Capacitors in series: Slide 1 of 3• Capacitors are in series if they are connected one after the

other, as illustrated.

• The equivalent single capacitor is shown on the next slide.


Capacitors in series: Slide 2 of 3


Capacitors in series: Slide 3 of 3• When several capacitors are connected in series, the

magnitude of charge is the same on all plates of all the capacitors.

• The potential differences of the individual capacitors add to give the total potential difference across the series combination: Vtotal = V1 + V2 + V3 + · · ·

• The equivalent capacitance of the series combination is given by:


Capacitors in parallel: Slide 1 of 3• Capacitors are connected in parallel between a and b if the

potential difference Vab is the same for all the capacitors.


Capacitors in parallel: Slide 2 of 3• This is the equivalent capacitor of two capacitors connected

in parallel.


Capacitors in parallel: Slide 3 of 3• When several capacitors are connected in parallel, the

potential differences are the same for all the capacitors.

• The charges on the individual capacitors add to give the total charge on the parallel combination:

Qtotal = Q1 + Q2 + Q3 + · · ·

• The equivalent capacitance of the parallel combination is given by:


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