In Fig. 21-33, particles 2 and 4, of charge , are fixed in place on a y axis, at and . Particles 1 and 3, of charge , can be moved along the x axis. Particle 5, of charge, is fixed at the origin. Initially particle 1 is at and particle 3 is at. (a) To what x value must particle 1 be moved to rotate the direction of the net electric force on particle 5 by counterclockwise ? (b) With particle 1 fixed at its new position, to what xvalue must you move particle 3 to rotate back to its original direction?
a) The value of xneeded to rotate the direction of the net electric force on particle 5 by 300 counter-clockwise is .
b) With particle 1 fixed at its position, the value of x for particle 3 to rotate back the net force to its original position is .
a. Position of particle 2 and 4 is and .
b. Initially particle 1 is at and particle 3 is at , particle 5 is fixed at the origin
Using the concept of the angle between two vectors, we can get the relation of the two forces. Now, using the concept of Coulomb's law, we can get the distance relation in both cases. Thus, we can get the required value.
The magnitude of the electrostatic force between any two particles is (i)
The angle between two vectors in form of tangent angle is (ii)
We note that
In the initial (highly symmetrical) configuration, the net force on the central bead is in the –y direction and has magnitude , where is the Coulomb’s law force of one bead on another at distance. This is due to the fact that the forces exerted on the central bead (in the initial situation) by the beads on the x axis cancel each other; also, the force exerted “downward” by bead 4 on the central bead is four times larger than the “upward” force exerted by bead 2. This net force along the y axis does not change as bead 1 is now moved, though there is now a non-zero x component F(x).
The components of force are now related using equation (ii) as follows:
Now, bead 3 exerts a “leftward” force of magnitude F on the central bead, while bead 1 exerts a “rightward” force of magnitude F. Therefore, the net force relation to the original force is as follows:
Now, using equation (i), we can get the above equation (a) in terms of distance as follows:
Buthere r corresponds to the distance between bead 1 and the central bead. Hence, the required value of x for the rotation is .
To regain the condition of high symmetry (in particular, the cancellation of
x-components), bead 3 must be moved closer to the central bead, so that it, too, is at the distance r (as calculated in part (a)) away from the central bead.
Hence, the required value of x is .
In Fig. 21-26, particle 1 of charge +q and particle 2 of charge +4.00q are held at separation L=9.00cm on an x-axis. If particle 3 of charge q3 is to be located such that the three particles remain in place when released, what must be the (a) x and (b) y coordinates of particle 3, and (c) the ratio q3/q?
In crystals of the salt cesium chloride, cesium ions form the eight corners of a cube and a chlorine ion is at the cube’s center (Fig. 21-36). The edge length of the cube is . The ions are each deficient by one electron (and thus each has a charge of ), and the ion has one excess electron (and thus has a charge of ). (a)What is the magnitude of the net electrostatic force exerted on the ion by the eight ions at the corners of the cube? (b) If one of the ions is missing, the crystal is said to have a defect; what is the magnitude of the net electrostatic force exerted on the ion by the seven remaining ions?
Question: In Fig. 21-32, particles 1 and2 of charge are on a y-axis at distance d = 17.0 from the origin. Particle 3 of charge is moved gradually along the x-axis from x=0 to x=+5.0 m . At what values of x will the magnitude of the electrostatic force on the third particle from the other two particles be (a) minimum and (b) maximum? What are the (c) minimum and (d) maximum magnitudes?
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