Aim of the lecture Coulombs Law: force between charges Gauss’ Law

1. Lecture 2 • Aim of the lecture • Coulombs Law: • force between charges • Gauss’ Law • Electric field and charge •   Main learning outcomes • familiarity with • forces between charges • Electric field relationship to charge • Calculation of fields and forces

2. BASICS • Coulomb’s Law • There is a force between two charges, Q1 and Q2 • charge magnitude q1 and q2 Coulombs • separated by a distance r • Force, F = keq1q2/r2 • ke is a constant which depends on the system of units • for the kinds of units we will use (SI), • F = (1/4pe0)q1q2/r2 • { for gravity almost the same formula: F=kgm1m2/r2 } • What does this mean? • Two charges will repel or attract each other • The force will be along the line joining the charges • The size of the force will drop rapidly with separation • Two positive charges repel each other • Two negative charges repel each other • A positive and a negative charge attract each other • In all cases the magnitude of the force on each charge is identical

3. + + + + - - - - F F

4. + - • The forces are equal • The acceleration need not be • Because the mass might be different • The charges need not be travelling directly towards each other • Because they might be affected by something else • Or they might ‘start off’ with some speed • When this happens the charges cannot collide • They will fly past each other • Or they might be in ‘orbit’

5. Which is what an atom is: • core of protons and neutrons • ‘shells’ of electrons outside • positive ‘nucleus’ • negative electron shells • held in orbit by the 1/r2 force • just like gravity holding planets • But be careful! • This is just an approximation • Accelerated charges radiate • Atom would decay! • A proper description requires • Quantum Mechanics • Not in this course • A lot of correct answers describing atomic behaviour can be derived using the 1/r2 force • Quantum Mechanics also uses the same 1/r2 form for the force

6. Some details: Force is a vector so the force between charges should be written: F = keq1q2/r2r where r is a unit vector along the line joining the charges the direction of F depends on the sign of q1 and q2

7. Some details: • F = keq1q2/r2 • e0 is called the permittivity of free space • m0 is called the permeability of free space • These are very basic properties of the vacuum • the speed of light c = 1/(e0m0)1/2

8. Some details: • The constant ke seems quite large, but the size of • charges on particles/objects is usually very small • ke ~ 9.0 x109 N/m2/C2 • But the charge on one electron = 1.6 10-19 C • So force between two electrons 1m apart is • F = 9 x109 x 1.6 x 10-19 x 1.6 x 10-19 = 2.3 x 10-28 N (but electron mass is also small!)

9. Some details: • A net Coulomb is a very large amount of charge • most objects easly have coulombs of charge in them, • but they have equal and opposite quantities of positive and negative • so the difference, or ‘net’ charge is always small • A Coulomb is a very large quantity of net charge. • But the forces holding atoms together are big because • the atoms is small, so the separation is tiny ~10-10m

10. Some details: • Charge is quantised. • it only ever comes in multiples of 1.6 x 10-19 Coulombs • It is impossible to have 2 x 10-19 C • An electron has one unit of charge • Scientists don’t know why it is 1.6 x 10-19 C • Another mystery! • So every object has a multiple of the basic charge • ….-3,-2,-1,0,1,2,3,4…. X (basic charge) • net charge cannot be changed (charge conservation) • Force is not quantised, because the distance between charges is not

11. The electric field, E, which is associated with a charge q, • is also a quantity with units and a direction: • E = F/q • where F would be the force on a test charge of one Coulomb • So • E = ke q/r2r • Where r is a vector pointing • away from q if q is positive • towards q if q is negative Note that Force = qE the force on a charge is equal to its charge times the electric field it is in

12. The forces and the electric fields for multiple charges • Add linearly Ftotal = Fq1 + Fq2 +…. Etotal = Eq1 + Eq2 + … • As vectors • (therefore taking sign into account) • Field lines are a visualisation of the vector field • The density of lines represents the magnitude • The direction represents the field direction. • But be careful! • It is not always possible to represent a 3-D field in 2-D • eg a simple single charge. • the density of lines in 3-D drops like 1/r2 • but drawn simply in 2-D it will look like 1/r – wrong! • Drawings of field lines are only a guide, you must • use the mathematics to get it right

13. Looking in 2-D this is what it looks like density of lines is 1/r The real situation in 3-D is likethis, the density of lines dropslike 1/r2

14. The electric field from a dipole Dipoles have two equal but oppose charges

15. Define x = 0 halfway between two charges E x d On the x axis we can write: E = ke q2 { 1/(x+d/2) – 1/(x-d/2) } With some algebra, E = ke 2 qd/x3

16. Define x = 0 halfway between two charges E x d E = ke 2 qd/x3 The quantity qd is called the electric dipole moment Note that the electric field drops off much faster with distancefor a dipole. Like 1/x3 compared with 1/x2 for a single charge

17. Charged Plates • If two metal plates are given equal and opposite electric charge • The charges distribute evenly • The electric field simply points from one plate to the other • This is an important result for practical situations. - - - + - + + - + + + - + + + - + + + + - + + + + - + + + + + + + - + + + + + - + + +

18. At the edges it distorts Charged Plates • Why? • Think about the middle • The charge in the middle has the same other charges each side • So the charge in the middle cannot be different ‘left’ to ‘right’ • So the only configuration that works in the middle is an E line directly to the other plate. Strictly speaking the charges at the ends behave differently, but for plates which are large compared with their separation this is the correct field configuration

19. This is measurement of some real field lines which are quite straightin the middle even thoughthe spacing is largecompared with plate size

20. Potential • As a charge has a force on it when it is in an electric field • it will accelerate • it will gain energy • this is a conservative process, ie no energy is lost • if we want to push the charge back to where it started • we must use the same energy to slow it down and push it back • so the energy at each position in an electric field is defined • this is what we call a ‘potential’ • it is just like gravity and gravitational potential energy It takes a fixed and well defined amount of work to move a chargebetween different places in an electric field. This is not mysterious, it is just like gravity and height.

21. + • The potential energy in the case of an electric field is called • the electric potential • its units are volts • this is what we often just call VOLTAGE The difference in potential between two places in an electric field is related to the energy it takes to move a charge in it. W = qDV where W is the energy and DV is V2-V1 and q is the charge V1 V2 Work Done, W = (V2-V1)q

22. Equipotentials • In a diagram of an electric field • A set of points with the same voltage exists • these are equipotentials • Equipotentials • do not cross • cross the electric field lines at right angles • are surfaces in 3-D • can be correctly shown in 2-D as lines • Equipotential Lines • show the 2-D locus of points with the same voltage Equipotential lines (lines of constant voltage)

23. Dipole Field Lines and Equipotentials Equipotentials Field Lines In 2-D the field lines are a guide, The equipotentials are exact

24. Charged Plates • If two metal plates are given equal and opposite electric charge • The charges distribute evenly • The electric field simply points from one plate to the other • The equipotentials are planes between the plates - - - - + 3-D - + + - + + + + + - + + + + + - + + + + - 2-D + + + + + + + + - + + + + - + + + equipotentials

25. d E = DV/d V0/10 2V0/10 3V0/10 5V0/10 7V0/10 9V0/10 V0 0 Charged Plates • Note that the metal plates are equipotentials themselves • A voltage difference applied between two plates • Produces an electric field pointing from one plate to the other • A constant field gradient between the plates Voltage=V0 Voltage=0 This is a capacitor (see later) = V0/d

26. If we put a dipole in such a field • the forces on the charges are equal but opposite direction • so the dipole cannot translate • but it can rotate • with no means to loose energy it will oscillate + + - - Parallel plates can produce a uniform electric field The oscillation of dipoles in electric (or magnetic) fields is a common phenomena in physics. (because atoms often behave like dipoles)

27. Equipotential lines and surfaces can be complex or simple • and the methods exist today to calculate them • using numerical methods and computers. Complex equipotential patterns are common Equipotential surfaces round a charged cylinder are cylinders

28. For a single point charge, the number oflines passing through a sphere surrounding itcannot depend on the radius of the sphere. Or in fact on the shape of the surrounding surface Gauss’ Law • Idea behind Gauss’ Law already introduced • The field lines come from a charge, • so no extra lines appear away from it • and none disappear • the number of lines is a constant • total number of lines through any closed surface surrounding a charge must be constant We will use a more mathematical way to say this, but the above statement is what Gauss’ law is about. That’s Gauss’ Law

29. Flux The integral of E through a surface is called the Flux