Two small spheres, each carrying a net positive charge, are...

Two small spheres, each carrying a net positive charge, are separated by $0.400 \mathrm{~m}$. You have been asked to perform measurements that will allow you to determine the charge on each sphere. You set up a coordinate system with one sphere (charge $q_{1}$ ) at the origin and the other sphere (charge $q_{2}$ ) at $x=+0.400 \mathrm{~m}$. Available to you are a third sphere with net charge $q_{3}=4.00 \times 10^{-6} \mathrm{C}$ and an apparatus that can accurately measure the location of this sphere and the net force on it. First you place the third sphere on the $x$ -axis at $x=0.200 \mathrm{~m} ;$ you measure the net force on it to be $4.50 \mathrm{~N}$ in the $+x$ -direction. Then you move the third sphere to $x=+0.600 \mathrm{~m}$ and measure the net force on it now to be $3.50 \mathrm{~N}$ in the $+x$ -direction. (a) Calculate $q_{1}$ and $q_{2}$. (b) What is the net force (magnitude and direction) on $q_{3}$ if it is placed on the $x$ -axis at $x=-0.200 \mathrm{~m} ?$ (c) At what value of $x$ (other than $x=\pm \infty$ ) could $q_{3}$ be placed so that the net force on it is zero?

Key Concepts

Vector Nature of Electrostatic Forces

Electrostatic forces are vectors, possessing both magnitude and direction. Recognizing the vector nature of these forces is crucial, as it means that both the magnitude and the direction must be considered when summing forces from multiple sources using the superposition principle.

System of Equations

When multiple measurements or interactions are involved, establishing a system of equations becomes necessary. By applying Coulomb’s Law to different scenarios and equating the results to the measured forces, one can create simultaneous equations that, when solved, yield the values of unknown quantities such as the charges involved.

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electrostatic force between charges is proportional to the product of the absolute values of the charges and inversely proportional to the square of the distance between them. This law forms the basic quantitative relationship for calculating forces in electrostatic interactions.

Superposition Principle

The superposition principle in electrostatics asserts that the net force on any charge due to several other charges is the vector sum of the individual forces exerted by each of the other charges. This principle allows complex problems to be broken down into simpler, individual interactions which can then be summed to find the overall effect.

Transcript

Please give Ace some feedback