Direction of electrical force in coloumb's law

[ankities]

is there any theoretical proof why force between the two charges act along the line joining them (acc to coloumb' s law)

 

[ReachingFoul]

I think because it's the shortest distance which can have the lowest energy consumption.

 

[chill_factor]

The Coulomb interaction is a conservative force which can be obtained from a scalar potential. The work needed to move a charged particle from an initial position to a final position is the same regardless of path. Every other path is the same as a straight line, shortest distance path.

OP:

From Maxwell's laws we have ∇ x E = dB/dt. In our case, we have a stationary charge and a charged test particle that doesn't influence the electric field, so we can say that dB/dt = 0 and ∇ x E = 0. The electric field is thus irrotational - it must have only a radial component. The radial component will be a straight line connecting the point charge with the test particle.

 

[K^2]

is there any theoretical proof why force between the two charges act along the line joining them (acc to coloumb' s law)

Symmetry is the simplest demonstration. If you rotate the world around the line connecting the two charges, nothing in the problem changes. That means, you should get exactly the same solution. The only direction of force that doesn't change if you rotate the whole problem is along the same line.

 

[andrien]

it holds only when they are stationary.if they are moving,the direction refers to the retarded position.

 

[Rap]

Symmetry is the simplest demonstration. If you rotate the world around the line connecting the two charges, nothing in the problem changes. That means, you should get exactly the same solution. The only direction of force that doesn't change if you rotate the whole problem is along the same line.

I agree with K^2. You cannot "prove" that its true, but there are deep physical principles, one of which is that, if you have an isolated system, the physics doesn't change if you rotate it. Another way of saying that is that there is no special or preferred direction. But we know charge is a scalar, it has no direction, so if the force were not along the a line connecting the two charges, then it would define a special direction in space, in violation of that principle. A direct result of that principle is that angular momentum is conserved. If the force were not central, the angular momentum of the two charges would constantly be increasing, again, a violation of the principle.

There are three similar principles. The physics for an isolated system doesn't change if you rotate it, which gives conservation of angular momentum. The physics for an isolated system doesn't change if you move it somewhere else, which gives conservation of linear momentum. And lastly, the physics for an isolated system doesn't change if you set it up now, or later (i.e. "move it in time"), which gives conservation of energy.