Four very long, currentcarrying wires in the same plane intersect to form a square 40.0 cm on e

step 1 Hey, this is question twenty eight point twenty six. We have four very long, uh, current carryin

Four very long, currentcarrying wires in the same plane...

Key Concepts

Magnetic Field from a Current-Carrying Wire

This concept refers to the magnetic field produced by a steady current flowing through a long, straight conductor. The field's magnitude at a point is determined by the distance from the wire and the current's magnitude, and it circles the wire according to the Biot–Savart law or Ampère’s law. Understanding this principle is essential for calculating the individual field contributions at a given point in space.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field lines produced by a current-carrying conductor. By pointing the thumb of the right hand in the direction of the current, the curled fingers indicate the circular direction of the magnetic field. This rule is critical for analyzing the vector directions of magnetic fields and predicting how they add or cancel at a point.

Superposition Principle

The superposition principle states that the net magnetic field created by multiple sources is the vector sum of the fields due to each individual source. In problems involving several current-carrying wires, this principle allows one to calculate and combine the contributions from each wire. It is a fundamental concept when determining conditions for the net field to be zero.

Symmetry Considerations

Symmetry is a powerful tool in electromagnetism, especially in configurations like a square arrangement of wires. Recognizing symmetry in the spatial arrangement and current directions can simplify the analysis by revealing equal magnitudes and opposing field directions, which may lead to cancellation. This concept helps in reducing the complexity of problems by exploiting geometric or current symmetries.

Transcript

00:01 Hey, this is question twenty eight point twenty six.

00:04 We have four very long, uh, current carrying wires that are intersecting to form a square.

00:11 And we're told that we want the magnetic field in the center of this square to be zero, and we know the magnitude and direction of thes other three wires, but we want to figure out of the current further another three wires.

00:24 But we want to figure out what the current and the fourth wire is.

00:27 So, um, we are told that means we have this first wire over on the left, and that's carrying a current of ten amps going up.

00:38 This first top horizontal wire has a current of eight amps, and that's going to the right.

00:45 And the bottom wire also has, ah, twenty amps going also to the right.

00:50 But we don't know what this fourth one is here.

00:53 So we're told that these wires are very long, so we can assume that they're basically infinite.

00:58 So the equation for the magnetic field from an infinitely long wire is shown here where our is the distance to the perpendicular distance closest to the wire.

01:12 So, um, this is if we want to calculate the contribution from one wire.

01:20 Here's what we're going to get.

01:21 So we have four wires total, so you know that we're gonna have four terms when we add all these up together.

01:32 But one important thing here is we want to know whether each of these terms is going toe add or subtract.

01:38 So let's just start with this.

01:40 Ah, with this wire all the way over to the left here with ten amps.

01:44 If we use the right hand rule pointing our thumb in the direction of the currents of pointing it up, our fingers or raina curl into the page at point are.

01:53 So that means that it's gonna make a magnetic field that is pointing into the page or at this point in the middle.

02:01 So let's define this to be positive, positive going into the page.

02:06 So let's just take the first term.

02:09 The term from chris first aa wire risk.

02:15 Current is ten amps, and we're going to say that it's positive.

02:19 Ah, because it's going into the page.

02:23 Now let's look at this top chorizo montel wire.

02:27 Over here, the current is eight amps, and it's going to the right so if we use the right hand rule pointing our thumb in the direction of the current to the right, our fingers are going to curl into the page at this point here.

02:44 So we again have.

02:46 We're going to say that this is positive because it's contributing in the same direction into the page...

Please give Ace some feedback