Chapter 21. = C and point charge


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1 Chapter What must the distance between point charge C and point charge C for the electrostatic force between them to be 5.70N? The magnitude of the force of attraction is given by Coulomb s law where we have taken the absolute value of the force and the charges. F 1 2 r 2 r 1 2 F C C 5.7N 1.39m 21.5 Of the charge, on a tiny sphere, a portion is to is to be transferred to a second nearby sphere. The spheres can be treated as particles. For what value of / maimizes the magnitude F of the electrostatic force between the two spheres. We begin with a drawing showing the charge on each sphere. () r To find the maimum, we take the derivative and set it eual to zero. F df d ( ) r 2 2 r 2 2 r In Fig 2123, four particles for a suare. The charges are 1 4 and 2 3 (a) What is / if the net electrostatic force on particles 1 and 3 is zero. (b) Is there any value of that makes the net electrostatic force on the each of the four particles zero? Eplain.
2 We begin by drawing the forces on 1 and 3. We know that the diagonal force will be repulsive and we know that the magnitude of the remaining forces are identical (since the charges are identical). To get them to all balance, the and must have opposite signs. 1 2 F F F F 41 1 F 21 2 F F 23 F 34 F 13 F 0 2a 2 cos45 ˆ i + i ˆ ˆ j F 23 + F 34 + F 13 2a sin45 ˆ j 2 0 2a cos45 + ˆ i + 2 2a sin45 + ˆ j 2 component 0 2a cos a 2 cos45 cos y component 2a sin45 2 sin We should now check the forces on charge 1.
3 F 41 F 21 F 31 F 0 2a 2 cos45 ˆ i + i ˆ ˆ j F 41 + F 21 + F 31 2a sin45 ˆ j 2 0 2a cos45 + ˆ i + 2 2a sin45 + ˆ j 2 component 0 2a cos a 2 cos45 2 cos y component 2a sin sin It does not appear possible to find a ratio that will allow the force to be zero on 1 and 3 at the same time. It is also not possible to make the net force on every particle zero at the same time In Fig , three charged particles lie on the ais. Particles 1 and 2 are fied in place. Particle 3 is free to move, but the net electrostatic force on it from particles 1 and 2 happens to be zero. If 12 23, what is the ratio 1 / 2? Again the forces must cancel. I will demonstrate another way to approach the problem Allow the s to be either positive or negative. et the charge at 3 be. 1 2 F2 F
4 F 1 F 2 1 ( ) 2 2 ( 23 ) 2 F 1 F 2 1 ( ) 2 2 ( 23 ) 2 1 ( ) 2 2 ( 23 ) ( )2 ( 23 ) 2 (2)2 () In Fig , particle 1 of charge +1nC and particle 2 of charge 3nC are held at separation 10.0cm on an ais. If particle 3 of unknown charge 3 is to located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the (a) and (b) y coordinates of particle 3. Particle 3 must be on the ais, to the left of 1 since it must be closer to 1 since particle 1 has a smaller charge. It must be in this region so that the net force can be zero. The sign of the charge of 3 determines which direction the forces due to 1 and 2 point, but their magnitudes must be eual and their directions are opposite. y We can write out the magnitudes and then set them eual
5 1 3 F 1 4rf F 2 4rf0( + ) 2 F 2 F rf 0( + ) 2 4rf ( + ) ( + ) ( ) 0 ( 12) ! ( 12) 1 2 2( 12)  1! ( 12) 1 2 2( 12) 13.6 cm Two tiny spherical water drops with identical charges of C, have a centertocenter separation of 1.00 cm. (a) What is the magnitude of the electrostatic force acting between them? (b) How many ecess electrons are on each drop, giving it its charge imbalance F 2 r 2 ( ) 2 ( m) N # electrons C C / e A current of A through your chest can send your heart in fibrillation, ruining through normal rhythm of heart beat and disrupting the flow of blood (and thus oygen) to your brain. If that current persists for 2.00 min., how many conduction electrons pass through your chest.
6 i Δ Δ t Δ iδ t 0.300A 120s 36C Δ #electrons C electron Δ # electrons C electron Earth s atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each suare meter of Earth s surface would intercept protons at the average rate of 1500 protons per second. What would be the corresponding electric current intercepted by the total surface area of the planet? We take the rate times the charge per proton time the surface area of the planet. i total 1500 p s m C 2 4π r 2 p e A Electrons and positrons are produced by the nuclear transformations of protons and neutrons known as beta decay. (a) If a proton transforms into a neutron, is an electron or a positron produced? (b) If a neutron transforms into a proton, is an electron or a positron produced. (a) If a proton transforms into a neutron, charge conservation reuires that a positron be produced. (b) If a neutron transforms into a proton, charge conservation reuires that an electron be produced Identify X in the following nuclear reactions (in the first, n represents neutron) We must conserve both the total charge and the total number of protons + neutrons.
7 1 H + 9 Be X + n 1p 4 p 5p 0p X 9 B 0n 5n 4n 1n 12 C+ 1 H X 6p 1p 7p X 13 N 6n 0n 6n 15 N + 1 H 4 He + X 7p 1p 2p 6p X 12 C 8n 0n 2n 6n Figure shows a long, nonconducting, massless rod of length, pivoted at its center and balanced with a block of weight W at a a distance from the left end. At the left and right ends of the rod are attached small conducting sphere with positive charges and 2, respectively. A distance h directly beneath each of these spheres is a fied sphere with a positive charge. (a) Find the distance when the rod is horizontal and balanced. (b) What value should h have so that the rod eerts no vertical force on the bearing when the rod is horizontal and balanced? + W h a) To proceed, we compute the torue about the central pivot. We will assume that torues that would cause clockwise acceleration are positive. Since the system is to remain at rest, the net torue must be zero. 0 ( 2 ) W 2 2 h h 2 0 W 2 W 2 h W h 2 b) We can compute the net force and set it eual to zero to find h
8 0 2 h h 2 W h 3 h 2 W 3 W h 2 W A Neutron consists of one up uark of charge +2e/3 and two down uarks each having charge e/3. If we assume that the down uarks are 2.6 # m apart inside the neutron, what is the magnitude of the electrostatic force between them d d F 3.78N 4rf0r In Fig , two tiny conducting balls of identical mass m and identical charge hang from nonconducting threads of length. Assume that the angle is so small that the tan can be replaced with the sin. (a) Show that the euation displayed is necessary for euilibrium. (b) If 120 cm, m10g and 5.0cm what is? We begin this problem by drawing a freebody diagram of the forces. Since we know that the balls are to each be in euilibrium, the forces must add to zero in each component direction. T F E m g
9 y direction 0 T cosθ mg T mg cosθ direction 0 F E T sinθ F E T sinθ...as epected. tanθ sinθ / 2 F E T sinθ E mg cosθ sinθ mgtanθ 2 2 mg / πε 0 mg 2 ( 2πε 0 mg )1/3 b) We invert this euation to find πε 0 mg 2 2πε 0 mg 3 2πε 0 mg C In the radioactive decay of E , a 238 U nucleus transforms to 234 Th and ejected 4 He. (these are nuclei, no atoms and thus electrons are not involved.). When the separation between 234 Th and 4 He is 9.0 # m, what are the magnitudes of (a) the electrostatic force between them and (b) the acceleration of the 4 He particle. He Th (2e)(90e) F 511.5N 4rf0r 2 4rf0(9.0 # m) 2 F 511.5N a He mhe 4 # (1.67 # kg) 7.66 # m/s 2
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