It took me a long time to understand that current is constant throughout the circuit. And now I am stuck at one last thing: The speed of charge. Here is what I am thinking:
Current is defined as quantity of charge passing through a point at a given time, and from this we can deduce that current is constant throughout a circuit. But, if this is true, then wouldn't the speed of charge through a resistor be increased? Let us think about an analogy, water flowing through a pipe. Suppose water is flowing through a broader pipe(analogous to a less-resistance wire), and a greater quantity of water can flow past a point because of larger cross-sectional area. Now, suddenly water encounters a smaller pipe(analogous to the resistor). So, less quantity of charge can flow through it because of less cross-sectional area. But, if the volume of water flowing in a time $T$(analogous to current) is same, won't the rate of flow of water in the narrower pipe(the resistor) be more?
I am asking this question because while seeing some answers for the question that "Why is current constant?" , I saw some people saying that speed of charge slows down a bit through a resistor, which seems incorrect to me.
Answer
The answer to your question is "yes".
The relevant relationship is $I= nqAv_d$ where $I$ is the current, $q$ is the charge on a single charge carrier $n$ is the number of charge carriers per unit volume and $v_d$ is the magnitude of drift velocity.
In a series circuit the current is constant as a consequence of the law of conservation of charge.
This means that the amount of charge per second entering a conductor at one end must equal the amount of charge per second leaving the conductor at the other end.
Charge is not created or destroyed within the conductor.
To simplify matter let’s look at the variables one at a time and assume that the current is the same in a complete circuit or part of a circuit.
Keeping the area and the charge on a charge carrier the same means that if a conductor has fewer charge carriers per unit volume the drift speed must be larger than the drift speed in a conductor with more charge carrier per unit volume.
So how is this increased drift speed achieved?
It is achieved by having a larger voltage (per unit length of the conductor) across the conductor which has the fewer charge carriers per unit volume.
So if you had a piece of copper and piece of iron (a worse electrical conductor that copper) which had the same dimensions as the copper in series and you passed a current though them the voltage across the iron would be larger than the voltage across the copper.
Put another way the resistance of the sample of iron is larger than that of copper.
This is your resistor and copper wire situation.
Suppose now there were two copper wires connected in series of the same length but one piece of wire had twice the cross sectional area of the other.
To transport the same amount of charge per second through both wires the drift velocity in the thinner wire must be twice that of the thicker wire.
You could say that there were in total fewer charge carrier in the thinner wire than in the thicker wire and so to convey the same charge per second the charge carriers would have to move twice as fast.
How is this extra speed achieved, again by having a larger voltage across the thinner wire that the thicker wire. The resistance of the thinner wire is greater than that of the thicker wire.
You can do a similar analysis comparing the charge on the charge carriers if they are not the same.
All this you can convert to the analogy of a flow of water with the pressure difference being the analogy of voltage and the length and cross-sectional area of pipes being analogous to the length and the cross-sectional area of the conductor.
Usually it is the volume flow per second which is cited as the analogue of current but that means that there is a problem with finding an equivalent to the number of charge carries per unit volume in the electrical case.
So it is better to say that the mass transported per second is the analogue of current.
How this is done in practice I do not know. Perhaps it has to be shown as two separate circuits?
Anyway is you have two pipes of the same dimensions and need to transport the same amount of mass per second through the pipes then the liquid with the smaller density (number of charge carriers per unit volume) would have to travel faster (drift velocity) and so there has to be larger pressure difference (voltage) across the pipe with the lower density liquid flowing through it.
An finally as a caveat to my initial answer to your question it is theoretically possible by juggling the area of the specimens (resistor and wire) to make the charge carriers in a resistor move slower than that in a wire but in practice this is not possible because the charge carrier density in copper is so much greater than that of the substances used in resistors.
No comments:
Post a Comment