Post on 25-Jul-2020
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.
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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.
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Equipotential surfaces and field lines for a dipole
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Field and potential of two equal positive charges
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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.
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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.
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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.
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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”
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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!
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For ch. 23 problem solving videos• See prof. Tom Hemmick YouTube channel:https://www.youtube.com/watch?v=_XF78_y-EZY&list=PLFCadHQMw31ivfb9N9zpI3xd8jlnRBM8
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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.
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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:
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Parallel-plate capacitor• A parallel-plate capacitor consists of two parallel conducting
plates separated by a distance that is small compared to their dimensions.
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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.
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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
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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.
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Capacitors in series: Slide 2 of 3
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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:
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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.
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Capacitors in parallel: Slide 2 of 3• This is the equivalent capacitor of two capacitors connected
in parallel.
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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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