Coulomb’s Law Chapter 21

Coulomb’s Law Magic? The two glass rods were each rubbed with a silk cloth and one was suspended by thread. When they are close to each other, they repel each other. The plastic rod was rubbed with fur. When brought close to the glass rod, the rods attract each other.

Electric Charge Coulomb’s Law Two charged rods of the same sign repel each other. Two charged rods of opposite signs attract each other. Plus signs indicate a positive net charge, and minus signs indicate a negative net charge.

Materials classified based on their ability to move charge Coulomb’s Law Materials classified based on their ability to move charge Conductors are materials in which a significant number of electrons are free to move. Examples include metals. The charged particles in nonconductors (insulators) are not free to move. Examples include rubber, plastic, glass. Semiconductors are materials that are intermediate between conductors and insulators; examples include silicon and germanium in computer chips. Superconductors are materials that are perfect conductors, allowing charge to move without any hindrance.

Coulomb’s Law Charged Particles. The properties of conductors and insulators are due to the structure and electrical nature of atoms. Atoms consist of positively charged protons, negatively charged electrons, and electrically neutral neutrons. The protons and neutrons are packed tightly together in a central nucleus and do not move. When atoms of a conductor like copper come together to form the solid, some of their outermost—and so most loosely held—electrons become free to wander about within the solid, leaving behind positively charged atoms (positive ions). We call the mobile electrons conduction electrons. There are few (if any) free electrons in a nonconductor.

Concept Check - Electrostatics Two charged balls are repelling each other as they hang from the ceiling. What can you say about their charges? 1. one is positive, the other is negative 2. both are positive 3. both are negative 4. both have the same charge

Concept Check - Electrostatics Two charged balls are repelling each other as they hang from the ceiling. What can you say about their charges? 1. one is positive, the other is negative 2. both are positive 3. both are negative 4. both have the same charge The fact that the balls repel each other only can tell you that they have the same charge, but you do not know the sign. So they can be either both positive or both negative.

Concept Check - Electrostatics From the picture, what can you conclude about the charges? 1. have opposite charges 2. have the same charge 3. all have the same charge 4. one ball must be neutral (no charge)

Concept Check - Electrostatics From the picture, what can you conclude about the charges? 1. have opposite charges 2. have the same charge 3. all have the same charge 4. one ball must be neutral (no charge) The PERIWINKLE and BLACK balls must have the same charge, since they repel each other. The RED ball also repels the PERIWINKLE , so it must also have the same charge as the PERIWINKLE (and the BLACK).

Coulomb’s Law Induced Charge. A neutral copper rod is electrically isolated from its surroundings by being suspended on a non-conducting thread. Either end of the copper rod will be attracted by a charged rod. Here, conduction electrons in the copper rod are repelled to the far end of that rod by the negative charge on the plastic rod. Then that negative charge attracts the remaining positive charge on the near end of the copper rod, rotating the copper rod to bring that near end closer to the plastic rod.

Concept Checks – Conductors A metal ball hangs from the ceiling by an insulating thread. The ball is attracted to a positive-charged rod held near the ball. The charge of the ball must be: 1. positive 2. negative 3. neutral 4. positive or neutral 5. negative or neutral

Concept Checks – Conductors A metal ball hangs from the ceiling by an insulating thread. The ball is attracted to a positive-charged rod held near the ball. The charge of the ball must be: 1. positive 2. negative 3. neutral 4. positive or neutral 5. negative or neutral Clearly, the ball will be attracted if its charge is negative. However, even if the ball is neutral, the charges in the ball can be separated by induction (polarization), leading to a net attraction. remember the ball is a conductor!

Concept Checks – Conductors (2) Two neutral conductors are connected by a wire and a charged rod is brought near, but does not touch. The wire is taken away, and then the charged rod is removed. What are the charges on the conductors? ? 1. 0 0 2. + – 3. – + 4. + + 5. – –

Concept Checks – Conductors (2) Two neutral conductors are connected by a wire and a charged rod is brought near, but does not touch. The wire is taken away, and then the charged rod is removed. What are the charges on the conductors? While the conductors are connected, positive charge will flow from the blue to the green ball due to polarization. Once disconnected, the charges will remain on the separate conductors even when the rod is removed. ? 1. 0 0 2. + – 3. – + 4. + + 5. – –

Insulators and Conductors + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + A conductor allows charge introduced anywhere within it to flow freely and redistribute. Nonconductor Conductor

Insulators and Conductors + Q + + + + + + + + + + + + + + + + + + A conductor allows charge introduced anywhere within it to flow freely and redistribute. + + + + + + + + + + Q/2 Q/2 + + + + + + + +

Concept Check – Coulomb’s Law What is the magnitude of the force F2? 1. 1.0 N 2. 1.5 N 3. 2.0 N 4. 3.0 N 5. 6.0 N Q F1 = 3N F2 = ?

Concept Check – Coulomb’s Law What is the magnitude of the force F2? 1. 1.0 N 2. 1.5 N 3. 2.0 N 4. 3.0 N 5. 6.0 N The force F2 must have the same magnitude as F1. This is due to the fact that the form of Coulomb’s Law is totally symmetric with respect to the two charges involved. The force of one on the other of a pair is the same as the reverse. Note that this sounds suspiciously like Newton’s 3rd Law!! Q F1 = 3N F2 = ?

Concept Check – Electric Force Two uniformly charged spheres are firmly fastened to and electrically insulated from frictionless pucks on an air table. The charge on sphere 2 is three times the charge on sphere 1. Which force diagram correctly shows the magnitude and direction of the electrostatic forces: Answer: 5. The magnitude of the electrostatic force exerted by 2 on 1 is equal to the magnitude of the electrostatic force exerted by 1 on 2. If the charges are of the same sign, the forces are repulsive; if the charges are of opposite sign, the forces are attractive.

Concept Check – Electric Force Two uniformly charged spheres are firmly fastened to and electrically insulated from frictionless pucks on an air table. The charge on sphere 2 is three times the charge on sphere 1. Which force diagram correctly shows the magnitude and direction of the electrostatic forces: Answer: 5. The magnitude of the electrostatic force exerted by 2 on 1 is equal to the magnitude of the electrostatic force exerted by 1 on 2. If the charges are of the same sign, the forces are repulsive; if the charges are of opposite sign, the forces are attractive.

Concept Check – Coulomb’s Law (2) If we increase one charge to 4Q, what is the magnitude of F1? 1. 3/4 N 2. 3.0 N 3. 12 N 4. 16 N 5. 48 N Q F1 = 3N F2 = ? 4Q Q F1 = ? F2 = ?

Concept Check – Coulomb’s Law (2) If we increase one charge to 4Q, what is the magnitude of F1? 1. 3/4 N 2. 3.0 N 3. 12 N 4. 16 N 5. 48 N Originally we had: Now we have: which is 4 times bigger than before. Q F1 = 3N F2 = ? 4Q Q F1 = ? F2 = ?

Concept Check – Coulomb’s Law (3) The force between two charges separated by a distance r is F. If the charges are pulled apart to a distance 3r, what is the force on each charge? 1. 9 F 2. 3 F 3. F 4. 1/3 F 5. 1/9 F Q F r ? 3r

Concept Check – Coulomb’s Law (3) The force between two charges separated by a distance r is F. If the charges are pulled apart to a distance 3r, what is the force on each charge? 1. 9 F 2. 3 F 3. F 4. 1/3 F 5. 1/9 F Originally we had: Now we have: which is 1/9 as big as before. Q F r F/9 3r

Concept Check – Coulomb’s Law (4) A hydrogen atom is composed of a nucleus containing a single proton, about which a single electron orbits. The electric force between the two particles is 2.3 x 1039 greater than the gravitational force! If we can adjust the distance between the two particles, can we find a separation at which the electric and gravitational forces are equal? 1. Yes, we must move the particles farther apart. 2. Yes, we must move the particles closer together. 3. No, at any distance

Concept Check – Coulomb’s Law (4) A hydrogen atom is composed of a nucleus containing a single proton, about which a single electron orbits. The electric force between the two particles is 2.3 x 1039 greater than the gravitational force! If we can adjust the distance between the two particles, can we find a separation at which the electric and gravitational forces are equal? 1. Yes, we must move the particles farther apart. 2. Yes, we must move the particles closer together. 3. No, at any distance Both the electric and gravitational forces vary as the inverse square of the separation between two bodies. Thus, the forces cannot be equal at any distance. Answer: 3. Both the electric and gravitational forces vary as the inverse square of the separation between two bodies. Thus, the forces cannot be equal at any distance.

Coulomb’s Law Coulomb’s law describes the electrostatic force (or electric force) between two charged particles. If the particles have charges q1 and q2, are separated by distance r, and are at rest (or moving only slowly) relative to each other, then the magnitude of the force acting on each due to the other is given by The electrostatic force on particle 1 can be described in terms of a unit vector r along an axis through the two particles, radially away from particle 2. where 𝜀0=8.85× 10 −12 C2/N∙m2 is the permittivity constant. The ratio 1/4𝜋𝜀0 is often replaced with the electrostatic constant (or Coulomb constant) 𝑘=8.99× 10 9 N∙m2/C2. Thus 𝑘=1/4𝜋𝜀0 .

Coulomb’s Law Force is a vector. The interaction between any two charges is independent of the presence of all other charges. Therefore, the net force on any one charge is the vector sum of all the forces exerted on it due to each of the other charges interacting with it independently.

Coulomb’s Law The electrostatic force vector acting on a charged particle due to a second charged particle is either directly toward the second particle (opposite signs of charge) or directly away from it (same sign of charge). If multiple electrostatic forces act on a particle, the net force is the vector sum (not scalar sum) of the individual forces. Two charged particles repel each other if they have the same sign of charge, either (a) both positive or (b) both negative. (c) They attract each other if they have opposite signs of charge.

Coulomb’s Law F21 F23 q1 (+) q2 (-) q3 (+) 𝐹 2 = 𝐹 21 + 𝐹 23 𝐹 2 = 𝐹 21 + 𝐹 23 𝐹 2 = 𝐹 23 −𝐹 21 𝐹 31 𝐹 31 q3 (+) Force is a vector. The interaction between any two charges is independent of the presence of all other charges. Therefore, the net force on any one charge is the vector sum of all the forces exerted on it due to each of the other charges interacting with it independently. 𝐹 32 𝐹 32 q1 (+) q2 (-)

Concept Check – Forces in 2D Which of the arrows best represents the direction of the net force on charge +Q due to the other two charges? +2Q +4Q +Q 1 2 3 4 5 d

Concept Check – Forces in 2D Which of the arrows best represents the direction of the net force on charge +Q due to the other two charges? +2Q +4Q +Q 1 2 3 4 5 d The charge +2Q repels +Q towards the right. The charge +4Q repels +Q upwards, but with a stronger force. Therefore, the net force is up and to the right, but mostly up. +2Q +4Q

Concept Check – Electric Force (3) Two balls with charges +Q and –4Q are fixed at a separation distance of 3R. Is it possible to place another charged ball Q0 anywhere on the line such that the net force on Q0 will be zero? 1. yes, but only if Q0 is positive 2. yes, but only if Q0 is negative 3. yes, independent of the sign (or value) of Q0 4. no, the net force can never be zero 3R +Q – 4Q

Concept Check – Electric Force (3) Two balls with charges +Q and –4Q are fixed at a separation distance of 3R. Is it possible to place another charged ball Q0 anywhere on the line such that the net force on Q0 will be zero? 1. yes, but only if Q0 is positive 2. yes, but only if Q0 is negative 3. yes, independent of the sign (or value) of Q0 4. no, the net force can never be zero A charge (positive or negative) can be placed to the left of the +Q charge, such that the repulsive force from the +Q charge cancels the attractive force from –4Q. 3R +Q – 4Q

Coulomb’s Law Multiple Forces: If multiple electrostatic forces act on a particle, the net force is the vector sum (not scalar sum) of the individual forces. Shell Theories: There are two shell theories for electrostatic force Answer: (a) left towards the electron (b) left away from the other proton (c) left

Sample Problem 21.01

Sample Problem 21.01

Sample Problem 21.01

Concept Check – Electric Force Two balls with charges +Q and +4Q are fixed at a separation distance of 3R. Is it possible to place another charged ball Q0 on the line between the two charges such that the net force on Q0 will be zero? 1. yes, but only if Q0 is positive 2. yes, but only if Q0 is negative 3. yes, independent of the sign (or value) of Q0 4. no, the net force can never be zero 3R +Q +4Q

Concept Check – Electric Force Two balls with charges +Q and +4Q are fixed at a separation distance of 3R. Is it possible to place another charged ball Q0 on the line between the two charges such that the net force on Q0 will be zero? 1. yes, but only if Q0 is positive 2. yes, but only if Q0 is negative 3. yes, independent of the sign (or value) of Q0 4. no, the net force can never be zero A positive charge would be repelled by both charges, so a point where these two repulsive forces cancel can be found. A negative charge would be attracted by both, and the same argument holds. 3R +Q +4Q

Concept Check – Electric Force (2) Two balls with charges +Q and +4Q are separated by 3R. Where should you place another charged ball Q0 on the line between the two charges such that the net force on Q0 will be zero? 3R +Q +4Q R 2R 1 2 3 4 5

Concept Check – Electric Force (2) Two balls with charges +Q and +4Q are separated by 3R. Where should you place another charged ball Q0 on the line between the two charges such that the net force on Q0 will be zero? The force on Q0 due to +Q is: The force on Q0 due to +4Q is: Since +4Q is 4 times bigger than +Q, then Q0 needs to be farther from +4Q. In fact, Q0 must be twice as far from +4Q, since the distance is squared in Coulomb’s Law. 3R +Q +4Q R 2R 1 2 3 4 5

Sample Problem 21.02

Sample Problem 21.03

Charge is Quantized Electric charge is quantized (restricted to certain values). The charge of a particle can be written as ne, where n is a positive or negative integer and e is the elementary charge. Any positive or negative charge q that can be detected can be written as in which e, the elementary charge, has the approximate value

Charge is Quantized When a physical quantity such as charge can have only discrete values rather than any value, we say that the quantity is quantized. It is possible, for example, to find a particle that has no charge at all or a charge of +10e or -6e, but not a particle with a charge of, say, 3.57e. Answer: -15e

Sample Problem 21.04

Charge is Conserved The net electric charge of any isolated system is always conserved. If two charged particles undergo an annihilation process, they have equal and opposite signs of charge. If two charged particles appear as a result of a pair production process, they have equal and opposite signs of charge. A photograph of trails of bubbles left in a bubble chamber by an electron and a positron. The pair of particles was produced by a gamma ray that entered the chamber directly from the bottom. Being electrically neutral, the gamma ray did not generate a telltale trail of bubbles along its path, as the electron and positron did.

Charge is Conserved Nuclei contain positively charged protons and neutral neutrons. Nuclei are characterized by the number of protons and neutrons they contain.

Charge is Conserved The notation for a particular nucleus of element X is written: Examples: Masses and charges of atomic particles:

Alpha Decay

Alpha Decay When a nucleus decays by emitting an alpha particle, it loses two protons and two neutrons, Symbolically: Here, X is the parent nucleus and Y is the daughter.

- Decay – e- emission The basic process in beta decay converts a neutron into a proton and an electron: In fact, there is a third particle emitted, which has no electric charge and little, if any, mass, called the neutrino. Therefore, a nucleus that decays via beta decay loses a neutron and gains a proton.

Beta Decay – e+ emission If a nucleus emits a positron, a proton has become a neutron:

Gamma Decay

Gamma Decay A gamma ray is emitted when an excited nucleus returns to its ground state. Nuclei may become excited through alpha or beta decay, leading to a sequence such as this one: The asterisk indicates the excited nucleus.

Summary Electric Charge Coulomb’s Law Conductors and Insulators The strength of a particle’s electrical interaction with objects around it depends on its electric charge, which can be either positive or negative. Coulomb’s Law The magnitude of the electrical force between two charged particles is proportional to the product of their charges and inversely proportional to the square of their separation distance. . Conductors and Insulators Conductors are materials in which a significant number of electrons are free to move. The charged particles in nonconductors (insulators) are not free to move. Eq. 21-4 The Elementary Charge Electric charge is quantized (restricted to certain values). e is the elementary charge Conservation of Charge The net electric charge of any isolated system is always conserved. Eq. 21-12