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Fundamentals Of Physics
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Illustration

Short Answer

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 .

See the step by step solution

Step by Step Solution

Step 1: Stating the given data

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

Step 2: Understanding the concept of Coulomb’s law

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)

Step 3: a) Calculation of the value of x to rotate net force by 300

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 .

Step 4: b) Calculation of the value of x to bring back the net force to the original position

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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