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PHYS1008 Electric Potential

Electrostatic potential surface of chlorobenzene

Oliver Kamp, Carsten Biele, Gregor Fels

Objectives: By the end of this you should understand
  • the reasons for using potential
  • the difference between potential and field
  • the potential for a point charge, dipoles and flat plates
  • The Oscilloscope
  • how to use electric potential in simple cases
  • the analogy of electric charge as a fluid
  • Equipotentials
  • How Conductors and Insulators modify fields and potentials
  • Electrical Breakdown and the Van der Graaf generator
  • the basic idea of capacitance
  • how to calculate it in simple cases.
  • how capacitors store energy

We introduced Potential Energy as an alternative to Force, since it was a much easier way to handle problems: scalar quantities are easier than vectors.


e.g.
Simplest case is uniform electric field. Two oppositely charged flat plates: fields will cancel outside them, but add in between. We found the field is
\color{red}{ E = 4\pi k\sigma }
so the potential is
\color{red}{ V = - 4\pi k\sigma d = - \frac{{\sigma d}}{{\varepsilon _0 }}}

since the field is a constant in this case. (Exactly the same argument as gravitational PE near the earth's surface). Minus sign arises since we define

\color{red}{ U = - \int {\vec F.d\vec s} }

In this case the flow chart becomes much easier. (remember, E is a constant)

The potential for a single point charge comes from Coulomb's law:

\color{red}{ E = \frac{{kq_1 }}{{r_1 ^2 }} \Rightarrow V = \frac{{kq_1 }}{{r_1 }}}
(k = 9x109 N m² C-2as before.)
We can use potential as we used PE: e.g.

The Oscilloscope

the most useful scientific instrument in existence.
Allows any time dependent quantity which can be converted into an electric signal to be measured. e.g
  • Sound
  • Nerve Impulses
  • Signals in circuits
  • Radio signals from Space (SETI)
  • Cathode is usually at ~-10 kV, anode at 0 V.
  • Deflection at X- and Y- plates is controlled by voltage (i.e. charge) on them.
  • This is the basic technology of the (old-fashioned!) TV: however there the steering of the beam is done (partly) by mag. fields.


Some important special cases of potential:
Potential due to a dipole (two equal and opposite charges).

What do we get off axis (i.e. at an angle θ to the direction of the dipole)?
\color{red}{ V = \frac{{kqd\cos \left( \theta \right)}}{{r^2 }}}
What does this mean
e.g a hollow sphere:

field E = 0 inside, exactly like a point charge outside.

\color{red}{ E\left( {\vec r} \right) = \left\{ {\begin{array}{*{20}c} {0\left( {r < R} \right)} \\ {\frac{{kq_1 }}{{r^2 }}\hat r\left( {r > R} \right)} \\ \end{array}} \right.}
Note that we have usually said the the zero of potential is arbitrary: however here we have defined the cathode to be at -30 kV and the anode (and screen) at 0 V. It would work equally well with the cathode to at 0 kV and the anode (and screen) at +30 kV. Why don't we do this?
  1. It is just a convention, chosen by the manufacturers
  2. The potential is measured with respect to the outside world, so + 30 kV would tend to fry small kids who were at 0 V.
  3. It is much easier to produce large negative voltages, since electrons are easier to move that protons.

Potential ⇔ Field

As with P.E. and force, we can go from potential to field. Formally
\color{red}{ E = - \frac{{dV}}{{dx}}}

This means (for a linearly increasing potential),

|E| = V/d

We can have problems getting the sign of the PE correct: Sign of work done and potential.
e.g. an electron is moved from a place where the potential is 10 V to where it is -10 V. What is the work done by the external applied force?
Why didn't we talk about gravitational field and gravitational potential when we dealt with gravity?

Equipotentials

Surfaces of constant electric potential. What are the equipotential surfaces called in geography?


The equipot. surfaces get more complicated with more charges:
e.g. a dipole

The 3 charges that we consider earlier

even in this case the equipots are perp to the field lines

Obviously the parallel plate system has equipots (almost) || to the plates

Conductors and Insulators

The surface of a conductor is always an equipotential.
If it wasn't then there would be an electric field acting on any charges, given a force so the charges would move so as to make it an equipotential.

Also the inside of a conductor is an equipotential.

A conductor is put between two charged plates.
Which of the diagrams shows the charges and fields correctly?


Insulators can't do this:
However, insulators often consist of dipoles,
and these dipoles can be lined up by a field.

So we can get small charges induced on the surface
If these are part of a solid, the charges on the dipoles will cancel out inside the solid, but add up on the surface

Breakdown voltage:

Reality raising its ugly head again!

We have treated field, charge and potential as if they can be adjusted arbitrarily. However, there is a maximum field in any material



The Van de Graaf generator.

The belt transfers charge at a very low voltage to the sphere: repulsion between the charges forces them to sit on the exterior of the sphere. Maximum voltage is given by
\color{red}{ \left. {\begin{array}{*{20}c} {E_{\max } = \frac{{kQ}}{{R^2 }}} \\ {V_{\max } = \frac{{kQ}}{R}} \\ \end{array}} \right\} \Rightarrow V_{\max } = RE_{\max } }
What would the maximum voltage be on a Van der Graaf generator if the sphere is 40 cm in radius, and the breakdown field in air is 3x106 Vm-1?

  1. 1200 V
  2. 1.2x104 V
  3. 1.2x106 V
  4. 1.2x108 V
What would the charge on a Van der Graaf generator if the sphere is 40 cm in radius, and the breakdown field in air is 3x106 Vm-1?
  1. 53 µC
  2. 533 µC
  3. 5330 µC
  4. 5.33 C

A thunderstorm:

works a bit like a Van der Graaf: positive charge is carried up by the up-drafts. Can get charges up to 100 C.

Lightning Photo by Gary Barnhart/Colorado

What would be the field at the ground in a thunderstorm,m if the base of the cloud, with the charge 0.3 C, has an area of 104
  1. 3.4 Vm-1
  2. 3400 Vm-1
  3. 3.4x104 Vm-1
  4. 3.4x106 Vm-1
What should you do if you are out on a lake in a canoe and a thunderstorm hits?
  1. Get into the water as fast as possible
  2. If you are in a aluminum canoe, get into the water as fast as possible
  3. If you are in a fibreglass canoe, get into the water as fast as possible
  4. Stay where you are and lie down

Electric Fish

An interesting example of applications of electric fields: if you are a fish and youi live in a muddy river, how do you see things?
You make a dipole field


this gives you a varying potential on your body and you can hunt with it!
Due to André Longtin , University of Ottawa and collaborators

Capacitance

FInal application of static electricity is Capacitors
Parallel Plate system: There is another way of writing the eqn. for the parallel plates:
  • V=  4πkσd
    
  • and
    σ = Charge  = Q 
         Area     A 
  • so
    Q =   A 
    V   4πkd
    


This tells you that if you put a certain amount of charge on parallel plates, there will be a certain fixed voltage. Capacity is measured in farads (which is an enormous unit)

What is the capacity of the earth? (R = 6500 km), assuming it is a conducting sphere.

  1. 720 f
  2. .72 f
  3. 7.2x10-4 f
  4. 7.2 μf

Charge as a fluid.

It's a bit hard to visualize capacitance: it helps to think of charge as a fluid.


Dielectrics

Usually a solid material is put between the plates of a capacitor: this has the effect of reducing the field for a given charge (the dipoles get rotated).

Dielectric constant

K = Eapplied
    Eeff
This
An application of this: the proximity detector. By measuring the capacitance, (via the voltage) can decide if a dielectric object has entered.
Can even do this by regarding object as one plate of capacitor

Another demonstration of the same effect (Thank you, Andrew Schagen!): the wahprobe
.

Energy in capacitors:

We can store energy in capacitors
Work done in adding charge dQ is V dQ, but V = Q/C, so
U = ½CV²

(Same calculation as we did for P.E. stored in stretched spring).


e.g an inventor proposes to build a car where the energy is stored in a parallel plate capacitor with a area of 100m², separation d = 10μ, dielectric constant K = 10.
  1. What is the capacitance?
  2. What is the maximum voltage? (take the breakdown field to be 3x107 V m-1
  3. How much energy can be stored?
  4. Would you invest in it?

Energy Density

e.g. Assume that you are trying to store energy at the same density as in gasoline: what field E would you need? (1 litre [~10-3 m³]~ 4x107 J), ε₀ = 8.85x10-12 C²N-1m-2 or k = 9x109 N m² C-2.
  1. 105 V/m
  2. 107 V/m
  3. 109 V/m
  4. 1011 V/m

So why do you use capacitors to store energy? Energy can be released very rapidly, but non-destructively. e.g. flash for camera has all of energy released in ~ 1 ms.

An example of energy storage in capacitors. This uses a 27 mF capacitor charged to 4 kV: how much energy does this correspond to?

This sets up electrostatics: we now want to go on and look at current electricity