# Is current directly proportional to electron density?

If current $$i$$ is directly proportional to free charge density $$n$$, then this means that a greater charge density will lead to a greater current. But simultaneously, this will also lead to more collisions which should in turn will reduce the drift velocity $$v_d$$?

Consider the equation: $$i=v_d\ e\ n \ A$$ where $$e$$ is the charge of an electron and $$A$$ is area of cross section.

So if $$n$$ increases and $$v_d$$ decreases then there will be a possibility that current may remain same. So, is it wrong to say that current is directly proportional to $$n$$? If $$v_d$$ is inversely proportional to $$n$$ then $$i$$ will always remain the same.

Is this reasoning correct?

But simultaneously, this will also lead to more collisions which should in turn will reduce the drift velocity $$v_{\rm d}$$?

If the double $$n$$ then the number of collisions would double but there are twice as many charge carriers so the number of collisions per charge carrier would be unchanged as would the drift velocity.

• @Farcher....So more 'n' will never affect the drift velocity? Commented Oct 3, 2023 at 14:43
• And if drift velocity is never affected by increasing 'n' then current will always increase as resistance will decrease. Right? Commented Oct 3, 2023 at 14:45
• But what if we increase temperature then it will affect both 'n' and relaxation time...increasing temperature leads to more ionisation therefore more 'n' and even increases the Kinetic energy which leads to more collisions and therefore less relaxation time and more collisions will reduce the drift velocity but still we will have more charge carriers due to ionisation so in this case how current would be affected....will it increase, decrease or remain the same? Commented Oct 3, 2023 at 14:49
• You did not mention any temperature increase in your question as you did not mention the type of conductor you are asking about. Commented Oct 3, 2023 at 16:35
• The current depends on the voltage applied and the resistance of the substances. Resistance is temperature dependent. Commented Oct 4, 2023 at 8:24

In most cases, the statement "$$X$$ is proportional to $$Y$$" assumes that all other variable quantities are held constant. In your case, the statement that "current is proportional to free charge density" assumes that other quantities do not change. This is similar to saying "kinetic energy is proportional to mass." A heavier mass may be harder to accelerate and so won't go as fast, leading to less kinetic energy, but the only thing we care about with proportionality statements is if everything else is held constant.

is it wrong to say that current is directly proportional to $$n$$?

The answer depends on the experiment. Mark H said, "'$$X$$ is proportional to $$Y$$' assumes that all other variable quantities are held constant." So, can you define a meaningful experiment in which $$v_d$$, $$e$$, and $$A$$ all are held constant while $$n$$ is changed? $$e$$ is constant by definition, and keeping $$A$$ constant is trivially easy, but what about $$v_d$$? How could you even measure $$v_d$$?

On the other hand, if I looked at your same equation and said, "drift velocity is inversely proportional to $$n$$," that would feel more meaningful to me because measuring current in a circuit and forcing constant current in a circuit are easy things to do.