In The Case Of Solid Charged Sphere

Electrostatics is one of the essential branches in Physics, dealing with a charged particle in the static position. The electrostatic force was first accidentally discovered and investigated by the Dutch physicist Pieter Van Musschenbroek in 1976, from the University of Leiden. Following this, Charles Augustin de Coulomb, a French physicist, put forth the famous Coulomb Law, regarding electrostatics.

DEFINITION FOR ELECTROSTATICS:

Electrostatic force: The electrostatic force between particles due to electric charges is the attractive and repulsive force between particles. Electrostatics is a study of electromagnetic phenomena, present only when particles are static or unmoving; a phenomenon established only after the system attains a state of equilibrium.

This phenomenon is labelled as electrostatic because the charged particles will always be stationary or in rest.  That is why the current-carrying charged particles are known as electrostatics.

There are three basic laws in electrostatics. They state that similar charges repel each other, opposite charges attract each other, and charged objects attract neutral objects.

IN THE CASE OF SOLID CHARGED SPHERES:

To explain the special case of solid charged spheres the article will go on to explain important laws pertaining to the topic. These are Coulomb’s Law and Gauss’ Theorem.

Coulomb’s Law:

As proposed by Charles Augustin de Coulomb, this law states that the force of attraction between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between the charges. Thus the corresponding formula is as follows:

Force = Product of the charges / distance² 

1 ⁄ 4πε_{0 *} q₁  q₂  ⁄ r² 

where,  1 ⁄ 4πε₀ = Constant

F = Force

q = charge

r = distance between the charges

This is also known as Coulomb’s constant, the S.I unit of which is  N m² C^-2

Note that a uniformly charged spherical shell interacts with external fields similarly to a point charge (as long as it stays uniformly charged). The material of the shell can be overlooked as long as the charge is spread in a thin, spherically symmetric shell.

The electric field of distributed charges, such as those produced by a uniformly charged spherical shell, cylinder, or plate, can be measured using Gauss’s law, which is as follows –

Gauss’s Theorem:

Gauss’s Theorem is one of the most important theorems used to calculate electric flux.

The total flux through a closed surface is equal to the  1 ⁄ ε times of the amount of charge enclosed by the surface.

The mathematical form of Gauss’s law is

Φ = Q ⁄ ε

Where,

Φ = electric flux

Q = total charge enclosed by the surface

ε= permittivity of the medium

The unit of permittivity is m^-3 kg^-1 S⁴ A² 

Gauss’ theorem assumes the surface to always be spherical.

The following are a few examples of solid charged spheres:

Assuming some quantities,

Φ = electric flux

Q = charge enclosed by the Gaussian surface

E = electric field

A = surface area

r = radius of the sphere

R = distance between the response point and the centre of the sphere

1 ⁄ 4πε₀= electric field constant

The electric field of a uniformly conducting sphere:

If, r > R which means electric field outside the solid sphere

Φ = E. A

Φ = E . (4πr²) 

From Gauss’s theorem, we know that

Φ = Q ⁄ ε₀

Comparing both the values of , Φ we get

E (4πr²) = Q ⁄ ε₀

E= Q ⁄ 4πε₀r² 

If, r < R which means electric field inside the solid sphere.

E= Qr ⁄ 4πε₀R³ 

When r < R the Gaussian surface encloses a lesser charge than the total charge in the sphere. So, the electric field will be less than the one outside the sphere.

Conclusion:

Electrostatics is defined as a study of the electromagnetic phenomena that occurs only if there are no moving charges. From Coulomb’s Law, it is proved that the force of attraction between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between the charges. According to Gauss’ theorem, the total flux through a closed surface is equal to the times of the amount of charge enclosed by the surface.

In the case of solid charged sphere examples, the electric field inside the solid sphere is discussed.