Definition of current density J?

[Researcher720]

I am confused by the definition of current density in Maxwell electrodynamics. Perhaps someone can help me out?

As I understand it, the current density function can be written as
$$ \vec{J} = \rho \vec{v}_S$$
where ρ is the charge density function and v_S is the continuous source charge velocity function. What I am confused about is why there isn't another part involving the test charge (or detector, or observation point) velocity? For example
$$\vec{J} = \rho ( \vec{v}_S - \vec{v}_T)$$
where v_T is the test charge (or detector or observation point) velocity in the arbitrary coordinate system chosen.

If you have a test charge, and source charges, since you can't tell if the source charges are moving with a constant velocity versus the test charges moving in the opposite direction at constant velocity, it seems that the current density J should involve the difference of these two independently moving objects (test and sources). What am I missing?

 

[Dale]

The velocity is defined with respect to some specified inertial frame, not with respect to a test charge. The fields can be (but do not need to be) defined with respect to hypothetical test charges, but the current density is not.

 

[Researcher720]

No, I can't agree with that without a lot more info. Yes, velocity is defined with respect to some specified inertia frame. Why do you say that the current density does not need to be defined with respect to the test charge?

For example, consider a test charge "T" and a source charge "S" in the chosen inertial frame of reference. We take "T" to be our detector. Let us put T and S on the x-axis and any motion will be on the x-axis, for this example. If T is at rest and S is moving in the +x direction at a constant speed v_S, the detector T will record some amount of force over time. Now, reset the situation. Have S at rest and T move in the -x direction at a constant speed v_T = -v_S. Then the detector should record the same force over time as in the first case. So, with many S charges making up an (approximate) current density, we need to include both v_S and v_T.

For example, v_S and v_T could have the same value, both S and T moving in the same direction at the same speed. In this case, the detector should see not changing force over time because the S and T remain stationary with respect to one another, but not with respect to the frame of reference. If
$$ \vec{J} = \rho \vec{v}_S $$
then this case would indicate that there is a current even when S is not moving with respect to T. But with the definition
$$ \vec{J} = \rho (\vec{v}_S - \vec{v}_T) = \vec{0} $$
because there is no relative motion between S and T. No relative motion means no current. And the detector reads a constant value.

You need to include both T and S velocities because the physics says it doesn't matter which one is stationary and which one is in motion, because you can go to another inertial frame of reference in which both are in motion in the new inertial frame of reference.

 

[Dale]

Why do you say that the current density does not need to be defined with respect to the test charge?

Why would you say that it does? Do you have any reference which defines current density using a test charge? The MIT textbook does not, nor has any other textbook I have seen.

 

[Dale]

For example, consider a test charge "T" and a source charge "S" in the chosen inertial frame of reference. We take "T" to be our detector. Let us put T and S on the x-axis and any motion will be on the x-axis, for this example. If T is at rest and S is moving in the +x direction at a constant speed v_S, the detector T will record some amount of force over time.

The force on T is mediated by the fields at T’s position. The concept of a test charge can be introduced for the fields. It does not need to be introduced twice, and in fact doing so would be problematic as you would now have two sets of test charges, one for the fields and another for the sources, and with no particular requirement that they be the same.

No, the idea of test charges for defining current density is not only unnecessary for your above example, it is a fundamentally bad idea and inconsistent with the literature.