In Fig. 21-26, particles 1 and 2 are fixed in place on an x axis, at a separation of .Their charges are and . Particle 3 with charge is to be placed on the line between particles 1 and 2, so that they produce a net electrostatic force on it. (a) At what coordinate should particle 3 be placed to minimize the magnitude of that force? (b) What is that minimum magnitude?
a)The coordinate value at which particle 3 needs to be placed to minimize the magnitude of that force is .
b) The magnitude of the minimum force is
Using the same concept of Coulomb's law, we can get the equation of the net force of the particles. Differentiating this equation will give the required value of x. Further, using this x-value, we can get the minimum force.
The magnitude of the electrostatic force between any two particles is .… (i)
Let be the distance between particle 1 and particle 3. Thus, the distance between particle 3 and particle 2 is . Both particles exert leftward forces on (so long as it is on the line between them), so the magnitude of the net force using equation (i) on is
Differentiating the above equation and equating it to zero, we can get the required valued of x as follows:
Hence, the required value of x is .
Substituting in the equation (a), we can get the net minimum force as
Hence, the value of the minimum force is
Two particles are fixed on an x-axis. Particle 1 of charge 40 is located at x=-2.0cm; particle 2 of charge Q is located at x=3.0cm. Particle 3 of charge magnitude is released from rest on the y axis at y=2.0cm. What is the value of Q if the initial acceleration of particle 3 is in the positive direction of (a) the x-axis and (b) the y-axis?
Figure 21-42 shows a long, non conducting, mass less rod of length L, pivoted at its center and balanced with a block of weight W at a distance x from the left end. At the left and right ends of the rod are attached small conducting spheres with positive charges q and 2q, respectively. A distance h directly beneath each of these spheres is a fixed sphere with positive charge Q.
(a)Findthe distance x when the rod is horizontal and balanced.
(b)What value should h have so that the rod exerts no vertical force onthe bearing when the rod is horizontal and balanced?
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?
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