In my book it has been given that
In a charge-free region, electric field lines can be taken to be continuous curves without any breaks
what does actually the word charge free region means? And what does this statement means?
I know this question has been answered previously but i am not convinced by previous solutions kindly help me.
Best Answer
I would take it to mean that, for the Electric field $\overrightarrow{E}$.
$$\overrightarrow{\nabla}\cdot\overrightarrow{E}=0$$
Everywhere within your charge-free region (lets call it M). Intuitively you can think of a point charge as a place where all of the electric field lines begin or end. picture the field as several arrows, they will point toward or away from the charge, this is what the divergence in the above equation measures.
You could also consider (applying Gauss's Law to the above) to be:
$$\int_{\partial M}\overrightarrow{E}\cdot d\overrightarrow{a}=0$$ where $\partial M$ denotes the boundary of the charge free region. Though this is less helpful as it really means the total charge is zero in M (so you could have charges here so long as they sum to zero).