Coulomb's Law

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Purpose The purpose of this experiment is to test Coulomb's Law which states that the force between two spherically symmetric charged objects is directly proportional to the product of the charges, and inversely proportional to the square of the distance between the centers of the two charges. In mathematical vector notation Coulomb’s Law is expressed as

where Fr is the force on particle 1 due to particle 2 in Newtons, q is the charge on 12 1 particle 1 in Coulombs, q2 is the charge on particle 2 in Coulombs, rˆ12 is a unit vector originating at the center of particle 1 and pointing directly away from particle 2, r is the distance between the centers of the two particles in meters, and k is a constant, given by

Introduction Coulomb formulated his law and tested it using a device he called the torsion balance. One sphere suspeded from a fiber has the same charge as a second sphere. These spheres are charged in the same manner so they repel each other. This causes the the fiber supporting one of the spheres to twist. To measure this repulsive force, Coulomb counter-acted the repulsive force by twisting the suspension head through the angle theta needed to keep the two spheres a certain distance apart. The force of repulsion was balanced by the force resulting from the twisting of the suspension head. Thus, the angle theta provided a relative measure of the force of repulsion between the two spheres. Coulomb performed a similar experiment to test the force of attraction between two spheres of opposite charges. We used the video analysis of a different set up to investigate this law. Our setup consisted of a charged ball suspended by two strings, and a charged ball mounted on a lucite prod. This is shown in Fig. 1. The prod has the same charge as the suspended sphere. Since like charges repel, the ball moves away from the prod as it is brought closer. The distance that the suspended ball moves can be used as a measure of the force of repulsion. Equations To calculate the repulsive force using our method, a look at the forces acting on the sphere and how they are related mathematically is helpful. In Figure 1, the charged ball has been displaced from equilibrium by a force, Fr = Fr . This force is due the prod with a charge c 12q2 on it which is placed on the same horizontal line as q1. If q1 is at rest then it is in equilibrium, and thus the net force on it,

Using xˆ , yˆ notation for these vectors, the tension on the string, Fr , the force of gravity, Fg , and the force due to the prod, Fc can be defined as follows:

Adding these vectors gives

The zero vector is defined as 0 xˆ + 0 yˆ , and by the properties of vector equivalence, two equal vectors must have equivalent components, so

Solving eq. 2 for |Fr | and using the fact that |Fg| = ma , where m is the mass of the ball gg and ag is the acceleration of gravity,

Substituting this result into eq. 1, and solving for

Substituting this result into eq. 1, and solving for

which can be rewritten using the definition of the tangent function as

In Figure 1, we see that

Equation 4 is the definition of tangent, and equation 5 is the Pythagorean Theorem. The effective length, L, is the distance from the ball to the point of suspension and not the length of the string, which can be seen in Figure 2. Solving eq. 5 for y and substituting this result into eq. 4,

Using this result equation 3 can be written as

Methods and Materials In this investigation two ping-pong balls that were covered with conducting paint were stroked with a fur-charged rubber rod and touched together to equalize their charges. One of the negativelycharged balls is hung from a long bifilar pendulum and the other, which serves as a prod, was attached to an insulated rod, as shown in Figure 1.

Figure 1: A charged ping-pong ball is repelled from an equally charged prod. At equilibrium, the vector sum of the gravitational force, the tension in the string, and the Coulomb force on...
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