Q.What is the ratio of the charges $q_1/q_2$ for the following electric field line pattern (the number of field lines emerging from each charge is proportional to its magnitude)?

Equipotential surfaces and electric field lines always meet at right angles.

An equipotential surface is a surface on which the electric potential V has the same value at every point. If the electric field E had any component along the equipotential surface, moving a test charge along that surface would require work (since W = qE·dl along that component), and the potential would change — contradicting the definition of an equipotential surface.

Therefore the electric field (and hence the electric line of force, which is tangent to E at every point) can have no component along the surface; the field must be directed entirely along the normal to the surface. This means:

E ⊥ equipotential surface, i.e. the line of force is always perpendicular to the equipotential surface at the point where it crosses it.

This is exactly analogous to contour lines on a map (equal height/equal potential) always being crossed at right angles by the direction of steepest descent (the field direction).

Field lines of an isolated positive charge point away from it in every direction, since the force on a small positive test charge is repulsive (away from the source charge).

Electric field lines show the direction a positive test charge would move if placed at that point. For an isolated positive point charge, the force on a nearby positive test charge is repulsive, pushing it directly away from the charge along the line joining them.

So the field lines of a single positive charge are straight lines radiating outward from the charge in all directions (like spokes of a wheel), extending to infinity, with no lines terminating on the charge itself.

Electric field lines convey direction (tangent), strength (density of lines), and the nature of charge (start on +, end on −).

An electric field line carries several pieces of information:

• Direction: the tangent at any point gives the direction of E there.
• Field strength: where lines are crowded (dense) the field is strong, and where they are sparse it is weak.
• Nature of charge: lines originate from positive charges and terminate on negative charges, so their start/end reveals the sign of the charge.

Hence field lines provide all of these.

For a single isolated positive point charge, the electric field lines are straight lines pointing radially outward from the charge in every direction, uniformly spread in 3-D (spherically symmetric).

Since $\vec E = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2}\hat r$ for a point charge, the field vector at every point in space is directed along the radius vector away from the charge (for $q>0$). So the field-line diagram is: the charge $q$ sits at the centre, and an evenly-spaced set of straight lines emerge from it and extend outward to infinity in all directions - denser near the charge (stronger field) and spreading apart farther away (weaker field), but never crossing each other, never forming closed loops, and never terminating except at infinity (since there is no negative charge nearby to end on).

True. If two field lines intersected, the field at that point would have two different directions, which is impossible since the field at any point is unique.

Electric field lines represent the direction of the electric field at every point in space — the tangent to a field line at any point gives the direction of $\vec{E}$ there.

If two field lines were to intersect at a point, then at that single point there would be two tangents, meaning two different directions for the electric field simultaneously. But the electric field at any given point has one unique magnitude and direction (it is the vector sum of contributions from all the charges present, which always gives a single resultant). Two directions at one point is a contradiction, so field lines never cross.

A field line's tangent gives the direction of $\vec E$. Two crossing lines would give two directions at one point, which is impossible — so they cannot intersect.

The tangent to an electric field line at any point gives the direction of the electric field there. If two lines intersected, we could draw two tangents at the point of intersection, implying the electric field points in two directions at once. Since the electric field at a point can have only one direction (and one magnitude), field lines can never cross one another.