Physics for Scientists and Engineers II, Summer Semester 2009 Physics 2220 Physics for Scientists and Engineers II

Physics for Scientists and Engineers II, Summer Semester 2009 Chapter 23: Electric Fields Materials can be electrically charged. Two types of charges exist: “Positive” and “Negative”. Objects that are “charged” either have a net “positive” or a net “negative” charge residing on them. Two objects with like charges (both positively or both negatively charged) repel each other. Two objects with unlike charges (one positively and the other negatively charged) attract each other. Electrical charge is quantized (occurs in integer multiples of a fundamental charge “e”). q =  N e (where N is an integer) electrons have a charge q = - e protons have a charge q = + e neutrons have no charge

Physics for Scientists and Engineers II, Summer Semester 2009 Material Classification According to Electrical Conductivity Electrical conductors: Some electrons (the “free” electrons) can move easily through the material. Electrical insulators: All electrons are bound to atoms and cannot move freely through the material. Semiconductors: Electrical conductivity can be changed over several orders of magnitude by “doping” the material with small quantities of certain atoms, making them more or less like conductors/insulators.

Physics for Scientists and Engineers II, Summer Semester 2009 Shifting Charges in a Conductor by “Induction” Negatively charged rod uncharged metal sphere Left side of metal sphere more positively charged Right side of metal sphere more negatively charged

Physics for Scientists and Engineers II, Summer Semester 2009 Coulomb’s Law (Charles Coulomb ) Magnitude of force between two “point charges” q 1 and q 2. r = distance between point charges Coulomb constant Permittivity of free space

Physics for Scientists and Engineers II, Summer Semester 2009 Charge Unit of charge = Coulomb Smallest unit of free charge: e = x C Charge of an electron: q electron = - e = x C

Physics for Scientists and Engineers II, Summer Semester 2009 Vector Form of Coulomb’s Law Force is a vector quantity (has magnitude and direction). unit vector pointing from charge q 1 to charge q 2 Force exerted by charge q 1 on charge q 2 (force experienced by charge q 2 ).

Physics for Scientists and Engineers II, Summer Semester 2009 Vector Form of Coulomb’s Law Force is a vector quantity (has magnitude and direction). unit vector pointing from charge q 2 to charge q 1 Force exerted by charge q 2 on charge q 1 (force experienced by charge q 1 ).

Physics for Scientists and Engineers II, Summer Semester 2009 Directions of forces and unit vectors + + q1q1 q2q2 + - q1q1 q2q2

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the Resultant Forces on Charge q 1 in a Configuration of 3 charges q3q q1q1 q2q2 a = 1cm q 1 = q 2 = +2.0  C q 3 =  C 0.5 cm

Physics for Scientists and Engineers II, Summer Semester 2009 Forces acting on q 1 q3q q1q1 q2q2 Total force on q 1 :

Physics for Scientists and Engineers II, Summer Semester 2009 Magnitude of the Various Forces on q 1 Note: I am temporarily carrying along extra significant digits in these intermediate results to avoid rounding errors in the final result.

Physics for Scientists and Engineers II, Summer Semester 2009 Adding the Vectors Using a Coordinate System q3q q1q1 q2q2 y x

Physics for Scientists and Engineers II, Summer Semester 2009 Adding the Vectors Using a Coordinate System y x

Physics for Scientists and Engineers II, Summer Semester 2009 …doing the algebra… F 1 has a magnitude of

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … another example using an even more mathematical approach q 1 = +3.0  C q 2 =  C ChargesLocation of charges x 1 =3.0cm ; y 1 =2.0cm ; z 1 =5.0cm x 2 =2.0cm ; y 2 =6.0cm ; z 2 =2.0cm In this example, the location of the charges and the distance between the charges are harder to visualize  Use a more mathematical approach!

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … another example using an even more mathematical approach d 12 =distance between q 1 and q 2.

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … mathematical approach We need the distance between the charges. d 12 is distance between q 1 and q 2. + x y z - q1q1 q2q2

Physics for Scientists and Engineers II, Summer Semester 2009 Distance between charges q 1 and q 2. Calculating the force on q 2 … mathematical approach

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … mathematical approach We need the unit vectors between charges. For example, the unit vector pointing from q 1 to q 2 is easily obtained by normalizing the vector pointing from from q 1 to q 2. + x y z - q1q1 q2q2

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … mathematical approach The needed unit vector:

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … mathematical approach You can easily verify that the length of the unit vector is “1”.

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … another example using an even more mathematical approach

Physics for Scientists and Engineers II, Summer Semester 2009 Calculating the force on q 2 … another example using an even more mathematical approach …and if you want to know just the magnitude of the force on q 2 :

Physics for Scientists and Engineers II, Summer Semester The Electric Field It is convenient to use positive test charges. Then, the direction of the electric force on the test charge is the same as that of the field vector. Confusion is avoided.

Physics for Scientists and Engineers II, Summer Semester The Electric Field + + Source charge test charge Q qoqo

Physics for Scientists and Engineers II, Summer Semester The Electric Field

Physics for Scientists and Engineers II, Summer Semester The Electric Field of a “Point Charge” q q q0q0 r

Physics for Scientists and Engineers II, Summer Semester The Electric Field of a Positive “Point Charge” q + q0q0 (Assuming positive test charge q 0 ) The electric field of a positive point charge points away from it. + P Force on test charge Electric field where test charge used to be (at point P).

Physics for Scientists and Engineers II, Summer Semester The Electric Field of a Negative “Point Charge” q - q0q0 (Assuming positive test charge q 0 ) The electric field of a negative point charge points towards it. - P Force on test charge Electric field where test charge used to be (at point P).

Physics for Scientists and Engineers II, Summer Semester The Electric Field of a Collection of Point Charges

Physics for Scientists and Engineers II, Summer Semester The Electric Field of Two Point Charges at Point P x y P q1q1 q2q2 ab y

Physics for Scientists and Engineers II, Summer Semester The Electric Field of Two Point Charges at Point P x y P q1q1 q2q2 r2r2 r1r1 ab y Pythagoras:

Physics for Scientists and Engineers II, Summer Semester The Electric Field of Two Point Charges at Point P x y P q1q1 q2q2

Physics for Scientists and Engineers II, Summer Semester The Electric Field of Two Point Charges at Point P

Physics for Scientists and Engineers II, Summer Semester The Electric Field of Two Point Charges at Point P Special case: q 1 = q and q 2 = -q AND b = a q -q + - E from + charge E from - charge

Physics for Scientists and Engineers II, Summer Semester The Electric Field of Two Point Charges at Point P Special case: q 1 = q and q 2 = q AND b = a q q + + E from + charge E from other + charge

Physics for Scientists and Engineers II, Summer Semester 2009 This is called an electric DIPOLE Special case: q 1 = q and q 2 = -q AND b = a q -q + - E from + charge E from - charge For large distances y (far away from the dipole), y >> a: E falls off proportional to 1/y 3 Fall of faster than field of single charge (only prop. to 1/r 2 ). From a distance the two opposite charges look like they are almost at the same place and neutralize each other.