How can an electric field have a direction in space although the electric potential has no direction?

Question

How can an electric field have a direction in space although the electric potential has no direction?

Answer 1

The potential is the amount of work needed to bring a unit positive charge from infinity to that point. Whatever the method you use to bring it to that point the work is the same irrespective of direction.

Once it is brought to that point, it will not remain there and it will tend to go to infinity unless some external force prevents it from going. That is why say that the point has some potential to do some work.

Thus this is some scalar quantity and no direction is involved in potential.
But field is the force acting on the unit charge.

Since the charge tends to go away from the position where there is potential, we say some force is acting on the charge, the force per unit charge is field. Naturally force is a vector and has direction involved in it. The direction in which the charge moves is the direction of the field. Thus field has direction, whereas the potential has no direction.

Answer 2

Think of it this way: the potential is like asking how high up a hill you are, the field is like asking about how something will roll downhill from where you are. How high is just a number, but how something goes downhill is a matter not just of amount but direction.

Now, if you know the height of every point on a hill, you can use that information to figure out which way things will roll, just like having an equation for electric potential (which has no direction) will let you find the electric field (which does have a direction).

Answer 3

Forces and fields (which are the carriers of force) have direction for teh simple reason that you can observe the interaction of the force with matter - matter moves in the direction of the net forces acting on it. Potential energies, on the other hand, are relative to an arbitrary definition of where zero potential energy is. For instance, if I define the zero of potential energy for a roller coaster car to be at the highest point of the roller coaster, and you define it to be at the bottom, when we work out the speed of the car at the bottom, we get the same result. But if potential had direction, then where we placed the zero point would matter. We'd get different answers for the problem above but since we know total energy is conserved, the couldn't be correct.

So basically, energy is scalar quantity since it has no direction and cna be arbitrarily changed in value by moving the zero point of potential energy.