Here's an example of how we can use this principle
of superposition with magnetic fields,
considering two different wires
carrying a different amounts of current in opposite directions.
Supposing these two wires are 1 centimeter apart,
one placed above the other
and the lower one is carrying twice as much current as the upper one.
We could ask, what is the strength of the magnetic field
1 centimeter above the upper wire
which would be 2 centimeters above the lower wire'
And we should remember here that the magnetic field
is proportional to the current divided by
the distance you are from each wire.
So using the symbol proportionality law and your right-hand rule,
see if you can find the magnitude of the magnetic field at this location.
What we have when we try to solve this, is this,
remembering to draw our two wires
with the currents flowing in a proper directions.
So for the upper one, for example
we could pick this direction and call it subcurrent, I.
And then for the lower one we have current in the opposite direction,
we could call that for example current 2I.
And now we have to practice our right-hand rule,
for this current flowing to the right in this example.
We would have the magnetic field coming out of the page
above this current so be sure to use your right-hand rule
and make sure that makes sense
and then magnetic field going into the page below the current.
For the lower wire, we're going to have the opposite.
We have this current as moving to the left
and again using our right-hand rule
we can see that the current will be going,
the magnetic field rather will be going into the page above the wire
and will be coming out of the page below the wire.
Now here's the key thing,
it's hard to see exactly what's going on
when these two are written together
so maybe we can write them separately very quickly
the upper one and the lower one.
So we already know other directions and magnitudes of the current in both of these.
The important thing is that the magnetic fields
above this upper wire extend very far away from that wire.
The straight of the magnetic field goes down and down and down
depending on how far away you are from this wire,
but it does extend very far away from the wire.
And this is the important idea,
so this lower wire,
we'll also have magnetic fields going into the page above it.
And these magnetic fields will sort of die after,
becoming weaker and weaker and weaker.
And again the important point
is that far away from this wire
we still have a magnetic fields even at such a distance.
So taking a look at what we're asked about in this problem
which is specifically the magnetic field 1 centimeter
above the wire remembering that this is 1 centimeter
and we're asked about 1 centimeter above the other wire.
What we would like to know is maybe at this position up here,
1 centimeter up, we have magnetic fields going out of the page
from the upper wire but we have magnetic fields going into the page
from the lower wire so they're competing with each other.
The question is how strong will it be?
All we have to do is remember that the magnetic field's strength
is proportional to the current divided by the distance we are away from our wire.
So let's call this wire, wire 1.
Let's call this wire, wire 2.
What we know is that the magnetic field from wire 1,
let's not say equal to because again we're not going to worry about the actual numbers
but it's proportional to the current going through wire 1,
divided by the distance we are from wire 1 which is only 1 centimeter.
Whereas the magnetic field from wire 2
will be proportional to the current flowing through wire 2
which is twice as strong divided by the distance away we are from wire 2.
There is one last thing we have to do here
which is taking to account the directions
and one of these cases we have current going into the page
and then the other case we have current going out of the page.
So this means is that these magnetic fields will be pointing in different directions
and it doesn't really matter which one we call positive or negative.
We could pick a convention.
So let's for example, say that into the page like this is our positive direction.
In this case, the magnetic field from the lower wire
is the one that has a positive magnetic field by our convention here.
And the one from the upper wire will be the negative current.
So now to find the total magnetic field
we just have to use the principle of superposition,
the total magnetic field now is equal to B1 plus B2
which will be proportional to minus I over 1 plus 2I over 2.
Cancelling these 2's, we see that, we simply get minus I plus I.
So the reason this is so easy to solve
is that the magnetic field at this particular point is exactly zero.
The reason again being that wire this lower wire
has twice as much current flowing through it,
it is also twice the distance away being 2 centimeters instead of 1 centimeter
away from the point that we're interested in.
So in this problem we could see exactly
how we would use our proportionality law for the current
as well as the right-hand rule in some of the conventions that we've defined
in terms of where the magnetic field is
and what its strength is to find some basic properties of magnetic field
at certain distances away from a wire with current flowing through it.