Lecture 1 Electrostatics: Coulomb’s law and electric field (E-field)

1. Lecture 1 Electrostatics: Coulomb’s law and electric field (E-field)

2. Introduction • Electromagnetism  study of the properties of charge, one of the fundamental properties of nature. • Two types of charge; positive and negative. • Charge can neither be created or destroyed but positive and negative charges can act to cancel each other. • Unit of charge is the Coulomb (C) the usual mathematical symbol is Q • The movement of charge constitutes an electric current. If a charge dQ passes a given point in a time dt then a current is said to occur.

3. Definitions The point charge. Assumes all the charge is concentrated at a point having zero volume. If charge is spread over a finite volume then it approximates to a point charge if its physical extent is small compared to the distance(s) to other charges. Volume charge density. Charge spread over a finite volume. Density at a given point is (units Cm-3).  is not necessary constant. The total charge Q contained within a volume  is given by where the integral is performed over the volume .

4. Surface charge density.Charge is spread thinly over a surface or a sheet. The density at a given point is  (units Cm-2). The total chargeQ on a whole surface S is given by: where the integral is performed over the surface S. Line charge density. Charge is distributed along a line. The density at a given point is  (units Cm-1). The total charge Q in a total length L is given by where the integral is performed along the line L.

5. Electrostatics. Concerned with the properties of charges which are stationary. Although we will need to move charges when deriving equations for potential energy etc, the charges can always be taken to move infinitesimally slowly.

6. Forces between stationary charges (in vacuo) – Coulomb’s law – Electrostatic Force • Experiments show that an electric force exists between two charges. • The size of this force is proportional to the product of the magnitudes of the two charges and is inversely proportional to the square of their separation. • The force acts along the line joining the charges • It is repulsive for charges of identical (like) sign and attractive for opposite sign charges.

7. For two point charges Q1 and Q2 separated by a distance r in a vacuum, the electric force is described by Coulomb’s law (scalar form) in magnitude, along r in direction. (vector form) where is a unit vector along r direction

8. Alternative vector form 0 is a constant which gives the strength of the electric force. 0 is known as the permittivity of free space. 0=8.8542x10-12 C2m-2N-1 or Fm-1.

9. Principle of superposition For a system consisting of three or more charges, the electric force acting on any one charge is given by the vector summation of the individual forces due to all the other charges.

10. Worked Problem +3C +2C 5cm 10cm -5C Calculate the total force acting on the +2C charge

11. The electric field (E-field) • Frequently we wish to investigate the force (and subsequent motion) on an arbitrary charge due to a set of other known fixed charges. • Although Coulomb’s law can be used it is generally more convenient to think of the fixed charges as producing a field, the electric or E-field, which then exerts a force on any charge placed in the field. • Splits the physics into two parts – charges produce an electric field and then other charges interact with this field.

12. Definition of the E-field If a test charge Qt, placed at some position in space, experiences an electric force F then the E-field at that point is given by where the limit Qt0 is required so that Qt does not perturb the charges which produce F and E. The units of E are NC-1 or more usual Vm-1. Force, F, on charge, q, placed in field, E, is F=qE as long as q is sufficiently small.

13. E-field due to a point charge For a single point charge Q field points radially outwards from a positive charge and radially inwards towards a negative charge. +Q

14. For a collection of two or more point charges the principle of superposition can be applied to find the total E-field at a given point. • For continuous charge distributions the distribution is split up into an infinite number of infinitesimally small, equivalent point charges with the E-field then being given by a suitable integration (see 1st Year notes for a number of examples). • Other techniques for finding the E-field will be developed later in this course.

15. Worked Problem dE  dEZ P  r a dS dx x Find the E-field at a point P which is a distance ‘a’ from an infinite flat sheet of charge having a charge density ‘’.

16. Electric Field Lines • Allow the form of the E-field to be visualised in a limited sense. • The lines have the following properties • The tangent to the lines at any point gives the direction of the E-field at that point. • Lines start on positive charges and finish on negative ones. • The density of lines gives an indication of the field strength at a given point. Q+>Q- Point charge

17. Conclusions • Charge and relationship to current • Definitions of point, volume, surface and line charges • Coulomb’s law for two point charges • Superposition and forces between >2 point charges • Definition of E-field • E-fields resulting from one or more point charges • E-fields due to continuous charge distributions • E-field line diagrams.