5) Coulomb’s Law a)form

b) Units Two possibilities: - define k and derive q (esu) - define q and derive k (SI) √ “Define” coulomb (C) as the quantity of charge that produces a force of 9 x 10 9 N on objects 1 m apart.

For practical reasons, the coulomb is defined using current and magnetism giving k = x 10 9 Nm 2 /C 2 Permittivity of free space Then

c) Fundamental unit of charge e = x C

Example: Force between two 1 µC charges 1 mm apart F = kq 1 q 2 /r 2 = (10 -6 ) 2 /(10 -3 ) 2 N = 9000 N ~weight of 1000-kg object (1 tonne)) same as force between two 1-C charges 1 km apart

Example: Coulomb force vs gravity for electrons m, e FCFC FgFg F C = ke 2 /r 2 F N = Gm 2 /r 2 Ratio:

Example: Velocity of an electron in the Bohr Atom Coulomb force: F = kq 1 q 2 /r 2 (attractive) Circular motion requires: F = mv 2 /r So, v 2 = kq 1 q 2 /mr For r = 5.29 x m, v = 2.18 x 10 6 m/s

d) Superposition of electric forces Net force is the vector sum of forces from each charge q1q1 q2q2 q3q3 q F3F3 F2F2 F1F1 Net force on q: F = F 1 + F 2 + F 3 F

6) Electric Field - abstraction - separates cause and effect in Coulomb’s law a) Definition Units: N/C

b) Field due to a point charge F Q q0q0 r Coulomb’s law: Electric Field: direction is radial

c) Superposition of electric fields Net field is the vector sum of fields from each charge P E3E3 E2E2 E1E1 Net field at P: E = E 1 + E 2 + E 3 E q1q1 q2q2 q3q3

Example 16 µC4 µC q1q1 q2q2 P d D=3m Find d to give E = 0 at P P E1E1 E2E2

7) Electric Field Lines (lines of force) a) Direction of force on positive charge radial for point charges out for positive (begin) in for negative (end)

b) Number of lines proportional to charge Q 2Q

c) Begin and end only on charges; never cross E?

d) Line density proportional to field strength Line density at radius r: Lines of force model inverse-square law

8) Applications of lines-of-force model a) dipole

b) two positive charges

c) Unequal charges

d) Infinite plane of charge Field is uniform and constant to ∞, in both directions Electric field is proportional to the line density, and therefore to the charge density,  =q/A By comparison with the field from a point charge, we find: E q, A

e) Parallel plate capacitor (assume separation small compared to the size) E+E+ E-E- E=2E + E+E+ E-E- E R =0 E+E+ E-E- E L =0 Strong uniform field between: Field zero outside

Fringing fields near the edges

f) Spherically symmetric charge distribution Symmetry ==> radial number of lines prop. to charge Outside the sphere: as though all charge concentrated at the centre (like gravity)

9) Electric Fields and Conductors Excess charge resides on surface at equilibrium E1E1 E1E1 E2E2 Field inside is zero at eq’m; charges move until |E 1 | = |E 2 |

Closed conductor shields external fields E E = 0

Field outside conducting shell not shielded Field lines perpendicular at surface

Field outside grounded shell is shielded Field larger for smaller radius E = kq/r 2 (concentrated at sharp tips)

Demonstration: Van de Graaf generator - purpose: to produce high field by concentrating charge -- used to accelerate particles for physics expts - principle: charge on conductors moves to the surface