Lecture notes, lecture 3 PDF

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PH504 – Part 3 Electric potential, potential energy. 1. Introduction: energy In the previous lecture we considered electrostatics in terms of the electric force. A different approach is in terms of energy. This is particularly useful for situations where conversion to different forms of energy (e.g. kinetic) occurs. In addition, for a number of situations, it is easier to find the electric potential (which is a scalar quantity) due to a charge distribution than the E-field which is a vector quantity. The E-field can subsequently be determined once the electric potential is known (see below) . Remember: curl E = 0 . The mutual potential energy of a charge system The potential energy of a system of charges depends upon its spatial configuration. The difference in potential energy DU between two configurations is given by the work done by external forces to change the system from one configuration to the other (this is done infinitesimally slowly so that there is no change in the kinetic energy). If DU is positive then the new configuration has a greater potential energy than the old configuration. Sometimes the (absolute) potential energy ( U) is given; this is relative to some standard configuration for which U=0 is assumed. U and DU for two point charges

Charges Q1 and Q2 are initially separated by a distance r1. An external force alters their separation to r2. What is DU? 1

The force needed to push the charges together is equal but opposite in direction to the electric force between them. Using work = force x distance

If Q1 and Q2 have the same sign and r22 point charges

Because of the superposition of forces the total potential energy is given by the summation of the individual potential energies. e.g. for three point charges:

For a collection of N point charges

where rij is the distance between charges i and j. The factor 1/2 compensates for each pair of charges being counted twice in the summation. An alternative way of writing the above result is in terms of the electric potential Vi produced at the site of charge i by the other (N-1) charges

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1 N U   qV 2 i1 i i where again the factor of ½ avoids counting the same interaction twice.

Moving a charge from A to B: DU U B  U A B    qo E ds A

DV 

DU q0

   Eds B

A

The equation can be modified to account for the case where there is a continuous charge distribution given by 

U

1 Vd  2

This form is particularly useful when calculating the mutual potential energy of a charged body. Substituting

div 0E =  and integrating by parts, the energy per unit volume is ue = 0E2 /2 ….integrated over all space gives U! No superposition principle.

Relationship between U and electric force 4

If the electric force is non-zero along one axis only (e.g. Fx) then

. More generally in three dimensions

In words 'the electric force is equal to the negative of the gradient of the potential energy'.

2. Electric potential (NOT potential energy) Since curl E = 0, field is irrotational. Hence potential, a scalar field, exists. E = – grad V

If the potential energy of a system varies by UAB as a test charge Qt is moved from point A to point B then the potential difference VAB between points A and B is defined by

VAB is related to UAB in a similar way to the relationship between Efield and electric force. The units of electric potential are J C-1º V (Volt) The previous equation gives the potential difference between the points B and A. The potential at a point can also be given assuming the zero point is known or specified. 5

If a charge Q is moved between points A and B then its potential energy will change by UAB=QVAB (Q should be sufficiently small so as not to perturb the charges which cause VAB).

Relationship between E-field and V We have

and

and also F = -ÑU . Hence E = -ÑV

… it is very easy to derive E from V !

Alternatively,

where the integral is a line along a path from point A to point B. For electrostatic fields VAB is independent of the path taken from A to B.

Potential due to a point charge Find the potential at a distance r1 from a point charge Q where the potential at infinity is taken as zero.

where the integral is performed in a radial direction so that E is parallel to r (cosq=1)

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The potential difference between two points at distances r1 and r2 from the point charge is

For a collection of point charges the potential at a given point is the algebraic sum of the individual potentials (electric potential obeys the superposition principle)

where V is the total potential a distance r1 from Q1, r2 from Q2 etc. If a charge system contains continuous distributions of charge then the potential may be found using a suitable integration. This is an alternative, and possibly simpler, method for finding the E-field as V is simply the algebraic sum of the individual potentials (not a vector sum as for E-field). E can be determined from the relationship E=-Ñ ÑV once V has been calculated.

3. Equipotential surfaces and E-Field lines Equipotential surfaces are those which connect points at the same potential. In practice we can only draw two-dimensional cross-sections of the equipotential surfaces. For a point charge the lines of force point radially outwards and the equipotential lines form a series of concentric circles. At all points the two types of lines are normal to each other.

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Proof that lines of E (or force) are always perpendicular to equipotentials In a direction tangential (along) an equipotential surface there can be no change in V. Hence there can be no component of E tangential to the surface (as E=-Ñ ÑV) and hence the only component of E must be normal to the surface (important).

Conclusions  Electric potential energy (U) and difference (DU)  U and DU for two or more point charges  Relationship between electric force and electric potential energy  Path independence of electric potential energy  Electric potential (definition and units)  Electric potential for a single point charge and multiple point charges  Electric potential due to continuous charge distributions  Relationship between electric potential and E-field E=-Ñ ÑV

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Equipotential surfaces (relationship to E-field) 8...