A disk with radius R and uniform positive charge density σlies horizontally on a tabletop. A sma

step 1 Here for the solution for the part A, the expression for the electric field on the axis of disk

A disk with radius R and uniform positive charge density...

A disk with radius $R$ and uniform positive charge density $\sigma$ lies horizontally on a tabletop. A small plastic sphere with mass $M$ and positive charge $Q$ hovers motionless above the center of the disk, suspended by the Coulomb repulsion due to the charged disk. (a) What is the magnitude of the net upward force on the sphere as a function of the height $z$ above the disk? (b) At what height $h$ does the sphere hover? Express your answer in terms of the dimensionless constant $v \equiv 2 \epsilon_{0} M g /(Q \sigma) .$ (c) If $M=100 \mathrm{~g}, Q=1 \mu \mathrm{C}, R=5 \mathrm{~cm},$ and $\sigma=10 \mathrm{nC} / \mathrm{cm}^{2},$ what is $h ?$

A disk with radius $R$ and uniform positive charge density $\sigma$ lies horizontally on a tabletop. A small plastic sphere with mass $M$ and positive charge $Q$ hovers motionless above the center of the disk, suspended by the Coulomb repulsion due to the charged disk.
(a) What is the magnitude of the net upward force on the sphere as a function of the height $z$ above the disk? (b) At what height $h$ does the sphere hover? Express your answer in terms of the dimensionless constant $v \equiv 2 \epsilon_{0} M g /(Q \sigma) .$ (c) If $M=100 \mathrm{~g}, Q=1 \mu \mathrm{C}, R=5 \mathrm{~cm},$ and $\sigma=10 \mathrm{nC} / \mathrm{cm}^{2},$ what is $h ?$

Key Concepts

Coulomb’s Law

Coulomb’s Law describes the electrostatic force between two point charges. The force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This fundamental principle is used to determine the interaction between charged objects, whether point charges or elements of a charge distribution, by applying the superposition principle.

Electric Field due to a Uniformly Charged Disk

A uniformly charged disk produces an electric field that varies with the distance along the axis perpendicular to the disk. Determining this field involves integrating contributions from infinitesimal charge elements over the entire disk. The resulting field expression, which includes the radius of the disk and the distance from the disk, is essential for analyzing the forces on charged particles located along the symmetry axis.

Force on a Charged Particle in an Electric Field

The force that an electric field exerts on a charged particle is given by the product of the charge and the electric field vector at its location. This concept is crucial when calculating the net force acting on a charged sphere suspended in an externally generated electric field, and it forms the basis for determining how the sphere’s motion or equilibrium position is influenced by the field.

Equilibrium of Forces

In many electrostatic problems, particularly those involving levitation or static equilibrium, the electric force is balanced by another force, such as gravity. Establishing the condition for equilibrium involves setting the magnitude of the upward electrostatic force equal to the downward gravitational force. This balance is used to derive relationships between the system’s parameters and to determine the stable position of the charged particle.

Non-dimensional Parameters and Scaling

Non-dimensional parameters are used to simplify and generalize the analysis of physical systems by combining relevant physical constants and variables into dimensionless groups. In the context of electrostatic levitation problems, introducing a dimensionless parameter can help in expressing the equilibrium conditions in a compact form, making it easier to study how varying physical parameters such as mass, charge, and geometrical dimensions affect the system.

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