Question

In some old text it is mentioned that $4\pi $ lines of force originate from each unit positive charge. Comment on the statement in view of the fact that $4\pi $ is not an integer.

Answer 1

Hint:Number of lines can never be in integer. In this context, $4\pi $ is used in reference to the area which the force lines are enclosed in, that is a solid angle.

Step by step solution:
Lines of force can be defined as the path followed by a free-to-move test particle in any force field. When measured at a particular point on these lines, the tangent gives the direction and their density gives the magnitude of the particular force field.
The force field can be of any type: gravitational, electrical or magnetic.

A unit test charge is a charge that has the value of one Coulomb $(C)$. For example, an electron is said to have a unit negative charge and a proton has a unit positive charge.
In regard to a unit positive charge, there can be a limitless number of force lines within the bounds of the electrical field it produces. Electrical force lines extend radially from a point charge. If there is another charge present, the lines will move radially toward or away from it depending upon the charge. Another main characteristic is that they are always in the direction of the field line.

Physically representing them is impossible therefore the number of force lines are drawn selectively; in high numbers where the field has a high magnitude and always showing the direction of the field tangentially. Such a method of representation, but in calculation and in words, is $4\pi $.
$4\pi $ is called a solid angle. A solid angle is any area on a sphere (three dimensional) which has an area equivalent to that of its radius’s square. And $4\pi $ is the radian equivalent of $360^\circ $, one cycle.

Therefore, when $4\pi $ lines of force are mentioned in regards to a unit test charge of any polarity, it means that the force field lines are extending radially outward from it in a three dimensional spherical space.

Note:In a system of charges, electrical force field lines do not always move toward one another. Their movement depends on the type of charge of the rest of the particles in the system. As we know, similar charges repel and opposite charges attract. Therefore, accordingly force field lines vary in direction and density in physical representation.

Additional information:Electrical and magnetic force field lines are real. They can store energy and also transfer momentum. On the other hand, gravitational force lines are imaginary.