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Q29P

Expert-verifiedFound in: Page 626

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**In Fig. 21-33, particles 2 and 4, of charge ****, are fixed in place on a y axis, at **

a) The value of xneeded to rotate the direction of the net electric force on particle 5 by 30^{0} 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. **

Formulae:

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 .

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