I was reading some information about the constant of electromagnetism $\epsilon_0$ $[\frac{C^2}{N\cdot M^2}]$ and according to my understanding it is the amount how much electric field is permitted in the space (vacuum). I do not understand this idea so much and I would like for an example in real life when the $\epsilon_0$ plays role.
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It doesn’t have any physical significance. It is nothing more than an ugly artifact of SI units. In other unit systems, such as Gaussian units, it doesn’t exist and “vacuum permittivity” isn’t even a meaningful concept.
More generally, no physical constant with dimensions has physical meaning, because its value depends on arbitrarily chosen units. As a simple example, what is significant about the speed of light is not its particular value but that it isn’t zero, isn’t infinite, and is much larger than other speeds we typically observe. Dimensionless ratios of other speeds to the speed of light are what have physical significance.
For more information, see the question Dimensionless constants in physics.
G. Smith
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Imagine a positive charge placed in vacuum, we can understand its lines of forces are directed outward (because it is positive charge) with equal distances between them, now the amount of how easily the electric field lines go through this space is determined by the value of the $\epsilon_0$
Sagigever
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