Two tiny conducting spheres are identical and carry charges of equal magnitude but opposite sign, embarking us on an electrifying journey to explore the captivating world of electrostatics. This intriguing scenario sets the stage for a comprehensive investigation into the electric field, potential, force, energy, and practical applications associated with these charged spheres.
The interplay between these spheres unveils fundamental principles governing the behavior of electric charges, providing a cornerstone for understanding a wide spectrum of phenomena in physics, engineering, and beyond.
Sphere Characteristics and Charges
Two tiny conducting spheres, each with a radius of r, are separated by a distance d. The spheres are identical and carry charges of +q and -q, respectively.
Electric charge is a fundamental property of matter. It is the ability of a particle to experience an electric force. The electric charge of an object can be either positive or negative. The charges on the two spheres are equal in magnitude but opposite in sign.
Electric Field and Potential
The electric field generated by the charged spheres is given by:
$$E = \frac14\pi\varepsilon_0\fracqr^2$$
where:
- E is the electric field
- q is the charge on each sphere
- r is the distance between the spheres
- $\varepsilon_0$ is the permittivity of free space
The electric potential at a point in space is given by:
$$V = \frac14\pi\varepsilon_0\fracqr$$
where:
- V is the electric potential
- q is the charge on each sphere
- r is the distance between the spheres
- $\varepsilon_0$ is the permittivity of free space
Force Interaction
The electrostatic force between the two spheres is given by Coulomb’s law:
$$F = \frac14\pi\varepsilon_0\fracq_1q_2d^2$$
where:
- F is the electrostatic force
- q_1 and q_2 are the charges on the spheres
- d is the distance between the spheres
- $\varepsilon_0$ is the permittivity of free space
The force between the spheres is attractive if the charges are opposite in sign and repulsive if the charges are the same sign.
Energy and Work: Two Tiny Conducting Spheres Are Identical And Carry Charges Of
The electrostatic potential energy stored in the system of charged spheres is given by:
$$U = \frac14\pi\varepsilon_0\fracq_1q_2d$$
where:
- U is the electrostatic potential energy
- q_1 and q_2 are the charges on the spheres
- d is the distance between the spheres
- $\varepsilon_0$ is the permittivity of free space
The work done in moving charges within the electric field is given by:
$$W = q\Delta V$$
where:
- W is the work done
- q is the charge
- $\Delta V$ is the change in electric potential
The work done in separating the spheres to a given distance is given by:
$$W = \frac14\pi\varepsilon_0\fracq^2d_f
\frac14\pi\varepsilon_0\fracq^2d_i$$
where:
- W is the work done
- q is the charge on each sphere
- d_f is the final distance between the spheres
- d_i is the initial distance between the spheres
- $\varepsilon_0$ is the permittivity of free space
Applications and Examples
The interaction between charged spheres is relevant in a wide range of applications, including:
- Electrostatic precipitators
- Laser printers
- Photocopiers
- Inkjet printers
- Electrostatic spray painting
An experiment that can be used to demonstrate the principles discussed in this analysis is the following:
Suspend two small conducting spheres from a common support. Charge one sphere positively and the other negatively. Bring the spheres close together and observe the force between them. The force will be attractive if the charges are opposite in sign and repulsive if the charges are the same sign.
FAQ Section
What is the significance of the charges being identical?
The identical charges ensure that the spheres have equal but opposite charges, leading to a symmetrical distribution of charge and simplifying the analysis of their interactions.
How does the distance between the spheres affect the electrostatic force?
The electrostatic force between the spheres is inversely proportional to the square of the distance between them, as described by Coulomb’s law.
What practical applications stem from the understanding of charged sphere interactions?
Applications include electrostatic spray painting, laser printers, and particle accelerators, where controlling the behavior of charged particles is crucial.
