Discussion in 'Homework Help' started by Debdut, Mar 11, 2012.
Current has a direction but why it is not a vector?
In some contexts, current can be treated as a vector, but it is better to say that current has magnitude that can be positive or negative. This makes it a scalar quantity. It some sense, the sign of the current can be interpreted as direction, for example in a circuit, but this is different than a spatial direction.
It is a confusing and ambiguous thing. Often the world is not black and white and you need to think in shades of gray.
Rather than a direction, you can think of current as being associated with an oriented area. Think of taking a small rectangle. This rectangle might allow charges to flow through it at a certain rate. The current divided by the area of the square is the notion of current density.
If you have a filamentary current density traveling along a path, then one can treat this like a current vector mathematically. However, it is understood that the reality is that you have a highly confined current density.
Usually, current density is treated as a vector, in 3D space, under the mathematics of vector calculus in electromagnetism. Integration (or adding up) of current density over an area gives current, which makes it a scalar.
In differential geometry, current density is a little more complicated than that (it is a two-form in 3D space), but that is a subject for another day.
Can you tell then why we do scalar summation for current?You said that sometimes we consider it vector?But overall thanks.
One abstract way to answer that is to say that the mathematical object that we call current is a scalar by definition. In vector calculations we take the vector current density and integrate it over area via a dot product. The area has a normal vector associated with it and the dot product of two vectors is a scalar.
If you use the exterior calculus of differential forms, the mathematical objects are more complex, but the end result is still that current has a scalar nature to it.
Thinking more simply. Consider that current is the time rate of change of charge as it flows through some confined area. Charge itself is not really a vector, so the time rate of change of it should not be a vector.
But, I still caution that mathematics is quite flexible and one can still imagine current as a vector. For example, what if you have a stream of charged particles moving at a particular velocity. Velocity is a vector and you could argue that current I=qv. Since v is a vector, then a scalar q times v should be a vector. This could be a gray area in definitions, although a purist would say that the particle stream is better described by current density, which is a vector field.