Question

Twelve charges of charge $q$ are situated at the corners of the $12$ sided polygon of side $a$ what is net force on the charge $Q$ at the centre?

Answer 1

Hint: It is a tricky question the student should think about the concept of force between the charges. Remember that the given charges are positively charged. The given conductor is a polygon which is symmetric; it also affects the net force exerted on the test charge in the centre.

Complete step by step answer:
The force between two charges was studied by the scientist coulomb.
Coulomb’s law states that the force of attraction or repulsion between two point charges is directly proportional to the square of the distance between them. The direction of this force is along the electric lines of force which are nothing but imaginary lines.
The total number of electric lines of force, crossing through a particular area is known as the electric flux.
The electric lines of force are imaginary lines of force along which the point charges tend to move.
The law relates the flux through any closed surface and the net charge enclosed within the surface is known as the gauss law. The total flux of the electric field E over any closed surface is equal to $\dfrac{1}{{{\varepsilon _ \circ }}}$times the net charge enclosed by the surface.
$ \Rightarrow \phi = \dfrac{q}{{{\varepsilon _ \circ }}}$
Where,
$q$ , is the charge.
$\phi $ , is the electric charge
${\varepsilon _ \circ }$ , is the permittivity of free space
The closed imaginary surface is known as the Gaussian surface. Gauss’s law tells us that the electric flux of electric field through a closed surface depends only on the value of net charge inside the closed surface and not on the location of the charges. Charges outside the surface will not take part in the action. The charges on the outside are left out.
It is given that,
Twelve charges of charge $q$ are situated at the corners of the $12$ sided polygon.
And, a test charge $Q$ is placed inside the polygon in the centre
The magnitude of the charges are the same. Given charges are positive charges it will repel each other. Due to symmetry there will be no force acting in the centre of the polygon since the test charge inside the polygon is also a positive charge. So the net force exerted on the charge $Q$ by the other charges is zero.

Note: Basically the electric field inside a conductor is zero when the charges flow outside the conductor. Here in this problem since the polygon has twelve sides and it is symmetrical, there will be no electric field inside the polygon as the charges on the side repel and the direction of electric field will be outside since the test charge inside the polygon is also a positive charge.