Video Transcript
What is the reason behind making ohmic resistors from double-coiled wires? (A) To decrease the resistance of the wire. (B) To increase the resistance of the wire. (C) To avoid self-induction. Or (D) double-coiled wires are easier to manufacture.
Let’s start by talking a bit about how ohmic resistors are made. An ohmic resistor is one that obeys Ohm’s law. This means that the potential difference 𝑉 across an ohmic resistor is equal to the
current through the resistor 𝐼 multiplied by its resistance 𝑅. Because several metals obey Ohm’s law to a good degree, we can make ohmic resistors
out of metal wires.
Now, we can often forget this when we’re looking at circuit diagrams, but all normal
wires have resistance. So any regular wire can be considered a resistor. We can increase the resistance of a wire resistor simply by decreasing its diameter
or by increasing its length. This means that wire resistors are often made out of long pieces of wire.
A good way of getting a long wire resistor into a small space is to coil it up. And in fact, we usually wrap the wire around a core of nonconductive material as
well, which gives us something that looks a bit like this. Now, what we have here does actually function perfectly well as a resistor. However, when wire resistors are manufactured, it’s often done by first winding the
wire in one direction like this then looping it round and winding back down the
other way so that the wire is effectively wrapped in both directions around the
coil. Such a resistor can be said to be bifilar, or double coiled.
Now, this question simply asks us, why do this? Well, options (A) and (B) suggest that winding the wire this way changes the
resistance. Option (A) says that we double-coil the wire in order to decrease its resistance,
while option (B) says it’s done to increase the resistance. However, the shape that a wire is bent into doesn’t affect its resistance.
We can recall that the resistance 𝑅 of a wire with a uniform cross section is given
by 𝑅 equals 𝜌𝐿 over 𝐴, where 𝜌 is the resistivity of the metal, 𝐿 is the
length of the wire, and 𝐴 is its cross-sectional area. Under normal conditions, the resistivity of a material is constant. And the length and area of the wire aren’t affected by the winding process. This means that the way a wire is wound doesn’t affect its resistance. So we can rule out options (A) and (B).
Looking at the other answer options, we can also see that there’s an issue with
option (D). This says that double-coiled wires are easier to manufacture. But as we’ve seen, double-coiling a wire involves winding it in both directions, as
opposed to just winding it in one direction. Now, in practice, both single-coiled wires and double-coiled wires can be produced by
machines. So it may not necessarily be much harder. But it’s certainly not easier to double-coil a wire than it is to just coil it in one
direction. So we can eliminate option (D) as well.
This leaves us with just option (C), which says we make resistors from double-coiled
wires in order to avoid self-induction. Now, in order to understand what this means, we need to understand what
self-induction is.
Let’s start by considering a single coil of wire, also known as a solenoid, which is
carrying a current 𝐼. Now, because a current-carrying wire produces a magnetic field around itself, we can
use the right-hand grip rule to figure out the direction of the magnetic field
produced. And in this case, the net result is a magnetic field 𝐵 which goes from right to left
inside the coil.
Alternatively, we could say that a certain amount of magnetic flux, represented by
the symbol 𝜙 m, is created within the coil. The amount of magnetic flux that the coil produces per unit of current is known as
the inductance of the coil. This is expressed by the equation 𝐿 equals 𝜙 m over 𝐼, where 𝐿 is the inductance
of the solenoid.
So far so good, but let’s consider what happens when the magnetic field through the
solenoid changes, for example, when the current through the solenoid is turned on or
off. We know that a changing magnetic field will induce an emf in a conductor. This is expressed by Faraday’s law, which says that the emf induced in a coil is
proportional to the change in magnetic flux over time. So this means that if we set up a current in a solenoid, the resulting change in
magnetic field will itself induce an emf in the solenoid. This phenomenon is known as self-induction. The coil is inducing an emf in itself.
Now, it turns out that the emf induced in the solenoid will actually oppose the
existing current. This important rule is summed up by Lenz’s law, which tells us that the direction of
the emf induced in a conductor will oppose the change that created it. What this means for a solenoid is that when we start passing a current through it,
the sudden production of a magnetic field will induce an opposing current within the
wire loops. In this way, a single coil of wire will naturally oppose changes in current due to
self-induction. This behavior is not something we want from resistors. So single coils of wire are not good for making resistors.
It turns out that a much better way of making a resistor is to double-coil the wire
as we talked about earlier. Because alternating loops in a double-coiled resistor carry current in opposite
directions, this means that the magnetic fields produced by the loops act against
each other, and for the most part they cancel each other out. The overall magnetic field produced by a double-coiled wire is therefore much, much
smaller than that which is produced by a single-coiled wire. This means that the effects of self-induction are virtually eliminated. And so a double-coiled wire doesn’t exhibit the same resistance to changes in current
that a single-coiled wire would.
Hence, we can see why option (C) is the correct answer. Ohmic resistors are made from double-coiled wires in order to avoid
self-induction.
