# Direction of electric field lines and electrostatic force

Direction of electric field and electrostatic force should be same by the equation

$$\vec{F} = \frac{k q q_0}{r^2}$$

Electric Field $$\vec{E} = \frac{k q}{r^2}$$

Let us suppose that there is a positive sphere on a plane surface of charge $Q$ and a particle of charge $-Q$ charge at some distance. So if the positive sphere is fixed then the particle with $-Q$ charge will get attracted towards the positive sphere.

We know that electric field lines move from positive to negative. So the field line should start from positive sphere to negative particle.

The direction of field line is away from positive sphere. According to electric field line equation above the direction of force should be towards negative particle but it is towards positive sphere.

What is wrong with my reasoning?

What is wrong with my reasoning?

Opposite charges attract because one of the charges has a negative sign. The force on the negatively charged particle is thus

$$\vec F_- = \frac{kQ(-Q)}{r^2}\hat r = -Q\,\frac{kQ}{r^2}\hat r = -Q\,\vec E_+$$

The force on the negatively charged particle is opposite the direction of the field from the positively charged particle

The electric field lines show the direction of the electric force acting on a unit positive charge at a particular point in space. So, therefore, the force acting on a negative charge, as is in your question, will act in the opposite direction shown by the electric field lines.

I believe that the fact that field lines are defined in terms of a unit positive charge is purely conventional, like the direction of current.

As for the equations, when you plug the charge into the equation, include the negative sign for negative charge. This will also illustrate how the direction of electric force on a negative charge is the opposite of that for a positive.