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Ssm Two circular loops of wire, each containing a single...

Key Concepts

Superposition Principle of Magnetic Fields

Magnetic fields obey the superposition principle, which means that the net magnetic field produced by multiple sources is the vector sum of the individual fields. This concept is essential when dealing with multiple current-carrying loops or other configurations because it allows for the analysis of each field separately before combining them to find the overall effect.

Vector Addition of Perpendicular Magnetic Fields

When magnetic fields from different sources are oriented in perpendicular directions, the net field is determined by applying vector addition using the Pythagorean theorem. This mathematical approach is important for finding the magnitude of the resultant field when individual components act at right angles to one another.

Magnetic Field from a Circular Current Loop

This concept involves using the Biot–Savart law to calculate the magnetic field at the center of a loop. For a single-turn circular loop carrying a current, the magnetic field at its center is given by a simple formula that depends on the current and the radius of the loop. This foundational idea is central to understanding how current distributions create magnetic fields in more complex systems.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field generated by a current-carrying conductor. It helps in establishing the orientation of the magnetic field with respect to the direction of the current flow, a critical aspect in analyzing and predicting the behavior of magnetic fields in various geometries.

Transcript

00:01 Hello, in this problem, we're asked to find the magnitude of the magnetic field located at the center of a little bit of a strange geometrical combination of two loops of current carrying wire.

00:16 Basically, what we have from a top -down view is two current carrying wires.

00:23 This is the top view, say, from a ring of wire like this, where we're going to take us two concentric.

00:30 Rings like this, but then put them kind of perpendicular inside of each other like this.

00:38 Basically, what's happening here, right, is two individual loops of current carrying wire.

00:47 Say something like this.

00:49 It doesn't matter which way the currents are going in general, so we'll just pick some.

00:56 So let's say the current is heading out of the page on this side of the wire and into the on this side of the wire, because right we have some circular loop going on.

01:05 If this is the case, then if we're using the right hand rule, if we wrap our fingers around the loop of current and try to follow that, then our thumb is going to point, in this case, for this loop, to the right, and therefore we know the magnetic field created by this loop of wire is to the right.

01:20 Well, this combination here on the left is just a combination of a loop like this, right? so maybe if this is the green loop, this would be the green loop here.

01:30 And then we can say maybe have a red loop here that's creating a magnetic field as well.

01:43 We have a red loop of wire that is heading this way.

01:47 And just for example purposes, like i said, doesn't quite matter.

01:51 But let's say that the current is heading out of the page here on the left side of this loop and into the page here on the right.

01:57 And if it's doing that, then that means we have a magnetic field from this loop that is heading up like this.

02:06 So this problem really hinges on the idea of the superposition principle.

02:09 So we have this strange combination of two loops on the left here, but really it's just the addition of the two situations, two individual situations we have on the right here, the two individual loops.

02:22 And we'll see, right, if we want the net magnetic field at the center of these loops, i just have to add, we'll call this one and two, i just have to add the two magnetic fields that i would get at the center of each of the individual loops together.

02:37 And but remember we are adding vectors.

02:41 So this vector b1 is to the right, this vector b2 is up.

02:45 What matters is that they're always going to be perpendicular to each other, right? so that if we have our b1 vector here and our b2 vector here, then the net resultant one is going to be the vector addition of those two.

03:00 So b net, the net magnetic field at the center is the addition of the two individual magnetic fields.

03:05 Now if we look at the trigonometry here, that means that if we want the magnitude of the net magnetic field at the center, which would be the magnitude of the addition of these two, we just have to take b1 squared magnitude of b1 plus the magnitude of b2, square them, add them together in square root...

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