PHYSICS - CLUTCH CH 22: ELECTRIC FORCE & FIELD; GAUSS' LAW

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2 CONCEPT: ELECTRIC CHARGE e Atoms are built up of protons, neutrons and electrons p, n e ELECTRIC CHARGE is a property of matter, similar to MASS: MASS (m) ELECTRIC CHARGE (Q) - Mass Gravitational Force - More Mass More Gravity - Mass ONLY - Electric Charge Electric Force - More charge More Electric Force - Charge and ELEMENTARY charge e = C ( ) - Charge of protons = - Charge of electrons = The CHARGE of an object is the quantity of of protons and electrons in it: e p p e p p p p e e p p e e e e Notice these charges are in WHOLE MULTIPLES of e We call this charge. Q = (N p N e ) e - MOST materials are NEUTRAL Np Ne Page 2

3 EXAMPLE: CHARGE OF ATOM What is the charge of an atom with 16 protons and 7 electrons? PRACTICE: NUMBER OF ELECTRONS How many electrons make up C? EXAMPLE: ELECTRONS IN WATER Water weighs 1 kg/l, has a molecular weight of 18 g/mol, and has 10 electrons per molecule. a) How many electrons does 2L of water have? b) What charge do these electrons represent? PRACTICE: ADDING ELECTRONS How many electrons do you have to add to decrease the charge of an object by 16 C? Page 3

4 CONCEPT: CHARGING OBJECTS Materials come in two types: and - Conductors [ ALLOW / DON T ALLOW ] charges to move - Insulators [ ALLOW / DON T ALLOW ] charges to move - METALS are good conductors, PLASTICS are good insulators Rubbing objects together strips electrons from one and gives to the other - Fur + plastic rod = negative rod - Fur + glass rod = positive rod Like charges [ REPEL / ATTRACT ] and unlike charges [ REPEL / ATTRACT ] POLARIZATION = separation of charges NO NET CHARGE - This is how polarization in CONDUCTORS works: - This is how polarization in INSULATORS works: CONDUCTION = transfer of charges through physical contact NET CHARGE Page 4

5 CONCEPT: CHARGING BY INDUCTION INDUCTION charges object WITHOUT touching - Different than conduction which NEEDS touching Process of CHARGING BY INDUCTION: Steps for CHARGING BY INDUCTION 1. Connect neutral conductor to ground - Ground = and of charges 2. Bring charged rod near conductor pulls charges: [ INTO / FROM ] ground [ INTO / FROM ] conductor 3. Cut connection between ground and conductor Prevents escape of charges 4. Remove charged rod Now conductor is Page 5

6 CONCEPT: CONSERVATION OF CHARGE Charge [ CAN / CANNOT ] be created or destroyed Known as charge - Charge can only be MOVED from one object to another - This means if one object gains 1 C, the other object loses 1 C EXAMPLE 1: In the following scenario, each pair of conducting spheres is brought into contact and allowed to reach equilibrium. What is the amount of charge transferred, and the direction of transfer, in each of the cases? -1 C 3 C -3 C -5 C 3 C -2 C A B C EXAMPLE 2: Two charged, metal balls move around an insulated box, colliding and randomly exchanging charge. Initially, one ball has a charge of 1C while the other has a charge of 3C. After some time, you find that one ball has a charge of 2C. What is the charge of the other ball at this time? 1 C 3 C? -2 C Page 6

7 CONCEPT: COULOMB S LAW Electric forces can be or - Consequence of + and - charges COULOMB S LAW gives the force between charges: q 1 q 2 r - F = - k = (Coulomb s constant) - Force always points along - Like charges [ ATTRACT / REPEL ], unlike charges [ ATTRACT / REPEL ] EXAMPLE 1: What is the ratio of the electric to the gravitational forces in a hydrogen atom? - + EXAMPLE 2: If two identical charges are connected by a 5 cm wire with a 10 N tension, what is magnitude of the charges? q q Page 7

8 PRACTICE: CHANGING DISTANCE If the force between two charges is F when the distance is d, what will the force between the two charges be if they were moved to a distance of 2d? EXAMPLE: CHARGES IN A LINE Where should we put a 1C charge so that the force on it is zero? 2 C 3 C x 10 cm Page 8

9 PRACTICE: 3 CHARGES IN A LINE In which direction will the 1 C charge move? If it has a mass of 10 g, what will its initial acceleration be? 4 cm 1 C - 1C 2C x 10 cm EXAMPLE: CHARGES IN A TRIANGLE Rank all of the possible pairs of charges in the following figure by which pair has the greatest electric force. e d d d 2e 3e Page 9

10 EXAMPLE: CHARGES IN A PLANE Find the net force on the 3 C charge in the following figure. y - 2C 8 cm 3C 10 cm 1C x EXAMPLE: EXPLOITING SYMMETRY IN ELECTRIC FORCES For each of the following, what is the direction of the net force on the 1 C charge: 1 C 1 C d d d d 2 C 2 C 2 C -2 C Page 10

11 PRACTICE: DIRECTION OF NET FORCE What is the direction of the net force on the charge at the center of the square in the following figure? -2C 2C 2C -2C 2C EXAMPLE: ELECTROSCOPE Two identical charges at the end of an electroscope s leaves each have a mass of 50 g. If the electroscope leaves are deflected by 30 o as shown in the figure, what is the charge at the end of each leaf? 0.5m 30 o 30 o 0.5m 50 g 50 g Page 11

12 CONCEPT: ELECTRIC FIELD How do electric charges act across a distance? ANSWER: through ELECTRIC FIELDS A single charge will produce an electric field that a second charge can feel q 1 q 2 Charge feels a force in an ELECTRIC FIELD, E - F = - Units = N/C EXAMPLE: A 2C and a 3C charge are separated by some distance d such that the electric field at the 2C charge is 10 N/C. What is the force on the 2C charge? PRACTICE: BALANCING GRAVITY A 1.5 C charge, with a mass of 50g, is in the presence of an electric field that perfectly balances its gravity. What magnitude does the electric field need to be, and in what direction does it need to point? Page 12

13 CONCEPT: ELECTRIC FIELD DUE TO POINT CHARGES Positive charges produce fields [ OUTWARD / INWARD ], negative charge produce fields [ OUTWARD / INWARD ] A single charge will produce an ELECTRIC FIELD of magnitude - E = EXAMPLE 1: At a distance d, the electric field is 20 N/C. At a distance w, the electric field is 10 N/C. What is the ratio, d/w? EXAMPLE 2: Two charges lie on the x-axis as shown below. At what point on the x-axis is the electric field zero? 2C -3C 7 cm x Page 13

14 PRACTICE: ELECTRIC DIPOLE If two equal charges are separated by some distance, they form an electric dipole. Find the electric field at the center of an electric dipole, given by the point P in the following figure, formed by a 1C and a 1C charge separated by 1 cm. 1C P 1 cm - 1C x EXAMPLE: ELECTRIC FIELD ABOVE 2 CHARGES What is the electric field at the point above the two charges, indicated as P in the following figure? P 8 cm 1C - 1C x 5 cm 5 cm Page 14

15 PRACTICE: FIELD AT THE CENTER OF 4 CHARGES 4 charges are arranged as shown in the following figure. Find the electric field at the center of the arrangement, indicated by the point P. 1C -3C P -1C 4C EXAMPLE: PENDULUM IN ELECTRIC FIELD A pendulum is at equilibrium in a uniform electric field as shown in the following figure. If the electric field magnitude is 100N/C, what is the charge on the end of the pendulum, q? 15 O E q 30g Page 15

16 PRACTICE: BALANCING MASS IN UNIFORM ELECTRIC FIELD In the following figure, a mass m is balanced such that its tether is perfectly horizontal. If the mass is m and the angle of the electric field is θ, what is the magnitude of the electric field, E? q, m E Page 16

17 CONCEPT: CAPACITORS Two parallel plates of equal and opposite charge produce a uniform electric field between them - UNIFORM = same magnitude everywhere; magnitude doesn t change with time These are known as CAPACITORS - Between plates, E = - Outside of plates, E = - ε 0 is the VACUUM PERMITTIVITY, and ε 0 = EXAMPLE 1: The electric field between two parallel plates is 1000 N/C. If the plates have an area of 5 cm 2, what is the charge on each plate? EXAMPLE 2: A capacitor produces an electric field E when it is formed by two plates that have charges Q and Q. What happens to the electric field is the charge doubles, but the area of the plates half? Page 17

18 PRACTICE: KINEMATICS IN A CAPACITOR An electron moves into a capacitor at an initial speed of 150 m/s. If the electron enters exactly halfway between the plates, how far will the electron move horizontally before it strikes one of the plates? Which plate will it strike? 150 m/s E h = 5 cm Page 18

19 CONCEPT: ELECTRIC FIELD LINES Recall that field lines point [ FROM / TO ] positive charges and [ FROM / TO ] negative charges. EXAMPLE: Draw the field lines for an electric dipole. PRACTICE: FIELD LINES OF TWO IDENTICAL, NEGATIVE CHARGES Draw the field lines for a pair of identical, negative charges. Page 19

20 EXAMPLE: FIELD LINES OF ELECTRIC QUADRUPOLE Draw the electric field lines for the four charges shown below. This arrangement is known as an electric quadrupole. +q -q -q +q Page 20

21 CONCEPT: DIPOLE MOMENT Recall that two equal charges with opposite signs form a dipole - Dipole moment p = q d - d points [ FROM / TO ] positive charge [ FROM / TO ] negative charge EXAMPLE 1: What is the (vector) dipole moment of the following dipole? y 0.5m 2C 1m -2C x A dipole in an electric field has a potential energy U = p E - This is because a dipole experiences a torque due to an electric field τ = p E EXAMPLE 2: The dipole depicted in the figure below is in a uniform electric field of 200 N/C. What is the potential energy of the dipole? What torque does the dipole experience? E 1 cm -1C 30 o 1C Page 21

22 CONCEPT: CONDUCTORS AND E-FIELDS Charges are [ ALLOWED / NOT ALLOWED ] to move within conductors - Electrons want to get as far apart as possible Electric field INSIDE conductor = CHARGE ARRANGEMENT in conductors: Charges ALWAYS move to on conductors EXAMPLE: A rectangular conductor with a positive charge Q is placed in a uniform electric field. Draw the electric field lines and charge distribution of the conductor. Page 22

23 CONCEPT: ELECTRIC FLUX Flux is a measure of HOW MUCH of a field passes through a surface - ELECTRIC FLUX is how much of the ELECTRIC FIELD passes through a surface ELECTRIC FLUX clearly depends upon the ANGLE of the surface to the electric field A θ Normal E Φ E = - Units are Nm 2 / C - where is between the ELECTRIC FIELD and the NORMAL of the area The TOTAL FLUX through a closed surface is the through the individual surfaces EXAMPLE: The electric flux through each surface of a cube is given below. What is the total flux through the cube? Φ 1 = 100 Nm 2 /C Φ 2 = 20 Nm 2 /C Φ 3 = 0 Nm 2 /C Φ 4 = 0 Nm 2 /C Φ 5 = 40 Nm 2 /C Φ 6 = 80 Nm 2 /C Page 23

24 EXAMPLE: FLUX THROUGH ANGLED SURFACE What is the electric flux through the surface depicted below? Note that the area of the surface is 1 m 2 and E = 100 N/C. Surface 30 EXAMPLE: FLUX THROUGH CUBE A cube of side length 2 cm is placed in an electric field of magnitude 100 N/C as shown below. What is the electric flux through each side of the cube? E EXAMPLE: FLUX THROUGH SPHERICAL SHELL BY POINT CHARGE What is the electric flux through a spherical shell of radius R due to a point charge, q, at the center? R q Page 24

25 PRACTICE: TOTAL ELECTRIC FLUX The electric flux through each surface of a cube is given below. Which surfaces of the cube does the electric field run parallel to? Φ 1 = 100 Nm 2 /C Φ 3 = 0 Nm 2 /C Φ 5 = 40 Nm 2 /C Φ 2 = 20 Nm 2 /C Φ 4 = 0 Nm 2 /C Φ 6 = 80 Nm 2 /C PRACTICE: FLUX THROUGH TWO SURFACES What is the total flux through the two surfaces depicted in the following figure? Note that surface 1 has an area of 50 cm 2 and surface 2 has an area of 100 cm 2, and E = 500 N/C. Surface 1 40 o E Surface 2 Page 25

26 CONCEPT: GAUSS LAW GAUSS LAW = the flux through a closed surface depends ONLY on the charge enclosed Φ E = EXAMPLE 1: What is the flux through the surface A? - 4C A 5C EXAMPLE 2: Using Gauss Law, find the electric flied due to a point charge q at some distance r. k = EXAMPLE 3: Using Gauss s law, find the electric field within a spherical conductor with some charge -Q. Page 26

27 PRACTICE: FLUX THROUGH MULTIPLE SURFACES Rank the flux through surfaces A, B and C in the figure below from greatest to smallest. B 3e -e C e A PRACTICE: FLUX THROUGH A PYRAMID The flux through the four surfaces of a pyramid are given below Φ 1 = 10 Nm 2 /C Φ 2 = 20 Nm 2 /C Φ 3 = 8 Nm 2 /C Φ 4 = 15 Nm 2 /C What is the charge enclosed by the pyramid? EXAMPLE: ELECTRIC FIELD DUE TO A HOLLOW SHELL What is the electric field at the follow three points due to the hollow shell shown below: a) r < a b) a < r < b c) r > b a q b Page 27

28 EXAMPLE: SURFACE CHARGE DENSITIES What is the surface charge density on the inner and outer surface of the hollow shell in the following figure? 3 cm 3 C 5 cm PRACTICE: ELECTRIC FIELD DUE TO A SHELL A spherical, conducting shell has a charge of 6C. If a 4C charge were placed at the center of the shell, what is the electric field at 4 cm? At 12 cm? - 6C 8 cm 4C Page 28