A physics student uses induction to give two small spheres...

A physics student uses induction to give two small spheres the same charge $q$. He then proceeds to measure the magnitude of the repulsive force between the spheres as a function of their separation. The findings are given in the table. Make a linearized plot of the data, in which force is on the $y$ axis. Using a "best fit" to the data, determine the charge on each of the spheres.

Key Concepts

Linearization Technique

Linearization involves transforming a nonlinear relationship into a linear one by a suitable change of variables, which simplifies analysis and interpretation. In the context of Coulomb's law, by plotting the force (F) against the inverse square of the separation distance (1/r^2), the data is expected to lie on a straight line. This transformation allows one to use standard linear regression techniques to extract the parameters of the original nonlinear relationship.

Linear Regression and Best Fit Analysis

Linear regression is a statistical method used to determine the best-fit line through a set of data points by minimizing the sum of the squared differences between the observed and predicted values. In experiments involving Coulomb's law, applying a best-fit analysis to the linearized data permits the accurate estimation of physical parameters, such as the charge on the spheres. The slope of the best-fit line in the plot of F versus 1/r^2 is directly related to the product of Coulomb's constant and the square of the charge, enabling the determination of the charge value.

Coulomb's Law

Coulomb's law governs the electrostatic interaction between charged particles, stating that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance separating them. This relationship is expressed mathematically as F = k * (q1 * q2) / r^2, where k is Coulomb's constant. It is fundamental in understanding how charged bodies interact in an electric field and forms the basis for analyzing experimental data involving charged objects.

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