Video Transcript
The platinum-iridium alloy wire in a hot-wire ammeter expands when its temperature increases and contracts when its temperature decreases. The temperature of the wire is dependent on the current in the wire. A hot-wire ammeter using such a wire will give a constant reading for an alternating current that has a particular peak value. Which of the following most correctly explains how an alternating current with a frequency of 50 hertz in the wire can produce a constant hot-wire ammeter reading? (A) The wire heats a hot-wire ammeter’s other mechanical components. The expansion and contraction of these components are out of phase with each other, so the reading on the ammeter remains constant. (B) The wire expands when its temperature increases much faster than it contracts when its temperature decreases, so the wire never reduces in temperature for a sufficient time to contract noticeably. Or (C) the frequency at which the wire can undergo a cycle of expansion and contraction is much smaller than the frequency of the alternating current, so the expansion of the wire corresponds to the effective value of the current.
So this question is asking us to identify the correct reason that a hot-wire ammeter displays a constant reading in response to an alternating current. Let’s clear the answer options off the screen for now so we can have a closer look at how this works. To start with, let’s recall that an alternating current is a current whose direction and magnitude are constantly changing. We can draw a graph with current on the vertical axis and time on the horizontal axis that shows how an alternating current varies over time.
Initially, at time zero, there’s zero current. After this, we can see that current increases up to some maximum value before decreasing again back down to zero. After this, the current increases in the negative direction up to some maximum negative value. This represents current going in the opposite direction. The magnitude of this current then decreases to zero again. And this cycle then repeats over and over again, with the current increasing in one direction and then decreasing again, then increasing in the opposite direction and decreasing again, and so on.
In this question, we’re dealing with an alternating current with a frequency of 50 hertz. This means that the current undergoes one full cycle like this 50 times per second, meaning that the current changes direction 100 times every second. Now, attempting to measure the size of a current like this is much more challenging than measuring that of a direct current, which is one that stays at a constant level in one direction. To measure a direct current, it’s common to use a device known as a moving coil ammeter, which is based on a galvanometer. In a galvanometer, the magnetic field produced by a current causes a needle to defect across a dial by an amount proportional to the size of the current. This works great for direct currents. However, it doesn’t work so well for alternating currents. This is because a galvanometer effectively measures the magnetic field produced by a current.
And if the current is rapidly alternating, then the magnetic field will alternate at the same frequency. This means if we connect our galvanometer to an alternating current source, we find that the needle fluctuates rapidly from side to side, which makes it essentially impossible to obtain an accurate reading. In contrast, a hot-wire ammeter, like the one mentioned in this question, is an ammeter which is designed specifically to measure the size of alternating currents. The way it does this is rather than looking at the electromagnetic effects of a current like a galvanometer does. It measures the thermal effects of charge flowing in a wire.
A hot-wire ammeter can measure the alternating current in a circuit by allowing a fraction of this current to flow along a platinum-iridium alloy wire. Attached to this wire is a silk thread which passes over a pulley, which is then attached to a spring which keeps it under tension. The pulley then has a needle attached to it, which points to a dial. The way that this works is that platinum-iridium wire produces heat when charge flows through it due to resistive dissipation. Although some of this heat is given away to its surroundings through conduction and radiation, the temperature of the wire itself increases, causing it to undergo thermal expansion. As the temperature of the wire increases, the rate at which it transfers heat to its surroundings also increases, until eventually it equals the rate at which the wire dissipates electrical energy. At this point, the temperature of the wire stops increasing.
Now this is where things get interesting. As it mentions in the question, the temperature of the wire is dependent on the current in the wire. However, we know that the current in the wire is rapidly alternating. So why is it then that the temperature of this wire doesn’t fluctuate too? Well, simply put, the thermal expansion and contraction of this wire is much slower than the alternation of the current. In a given period of time, the heat produced by the wire 𝑄 is proportional to the current squared. So it is technically true that the wire produces more heat when the magnitude of the current is at its maximum, which corresponds to these positions on the graph.
However, the overall temperature of the wire, as well as its thermal contraction and expansion effects, change so slowly that there’s no time for the wire to cool down and contract between these two points of maximum current magnitude. This means that for a current with a given peak value, a hot-wire ammeter will give a constant reading. So now, if we look back at our answer options, we can see that this is best described by option (C). An alternating current with a frequency of 50 hertz can produce a constant hot-wire ammeter reading because the frequency at which the wire can undergo a full cycle of expansion and contraction is much smaller than the frequency of the alternating current. So the expansion of the wire corresponds to the effective value of the current.
