ssm Suppose in Figure 21.27a that I1=I2=25 A and that the separation between the wires is 0.016

Ssm Suppose in Figure 21.27a that I1=I2=25 A and that the...

Key Concepts

Magnetic Field Produced by a Current-Carrying Conductor

A current-carrying conductor produces a magnetic field whose magnitude and direction depend on the current and distance from the wire. This field is given by an expression involving the permeability of free space, current magnitude, and distance (commonly ?0I/(2?r) for a long straight conductor), and its direction is determined using the right?hand rule. This concept is essential for predicting interactions between current-carrying wires.

Right-Hand Rule

The right-hand rule is a fundamental tool in electromagnetism used to determine the direction of the magnetic field around a current-carrying conductor or the direction of force on a conductor in a magnetic field. By pointing the thumb of the right hand in the direction of the conventional current, the curling fingers indicate the direction of the magnetic field lines, which is critical for analyzing force directions in such systems.

Lorentz Force on a Current Element

The Lorentz force law describes the force experienced by a current element in a magnetic field, given by F = I(L × B), where I is the current, L is the vector length of the conductor, and B is the magnetic field. This force is crucial in understanding how current-carrying wires interact with both their own and external magnetic fields, leading to attractive or repulsive forces.

Superposition Principle in Magnetic Fields

The superposition principle states that the net magnetic field at a point is the vector sum of individual magnetic fields from different sources. This principle is employed to cancel out unwanted magnetic forces by carefully applying an external field that counteracts the magnetic field produced by another current source, thereby achieving force cancellation.

Transcript

00:01 In this problem, we have two very long current carrying wires that are separated by a particular distance, and they're carrying currents in the opposite direction.

00:10 So what does that look like? we have, let's say, we'll call it wire one, carrying current one in this direction, and then we have wire two, say, carrying current two, two a little better, current two in this direction, and these wires are going to be separated.

00:36 By a particular distance, we'll call it r.

00:39 Now, ultimately the question is asking about how do we cancel out the repulsion that is happening between these two wires, right? so what is happening here? wire one, because it's carrying a current i, is creating a magnetic field.

00:56 Now, the magnetic field varies as we move away from the wire, but the magnitude of it is given by the equation, and i'm going to write it here for a particular the magnetic field from wire 1 at the location of wire 2 is going to be given by a particular equation mu not the permability of free space multiplied by the current through wire 1 divided by 2 pi times how far away wire is 1 is from wire 2 following the right hand rule if we put our thumb along the direction of current 1 then our fingers are going to wrap around going into the page to the left, to the right of wire one, and out of the page to the left.

01:43 Sorry, the magnetic field is going to be going into the page on this side of wire one and out of the page on this side of wire two.

01:52 I try to use left and right there, but there's really not a left and right direction on this diagram.

01:57 So what does that mean? that means that at this location here, along wire one, sorry, along wire two, wire one creates a magnetic field, of this magnitude, and because of our right -hand rule, it's going to be out of the page.

02:13 We can do the exact same thing for wire 2, and we'll see that the magnetic field created here from wire 2 at the location of wire 1 is equal to mu -0 times the current in wire 2, divided by 2 pi times how far away wire 2 is from wire 1.

02:30 Following the right -hand rule, if we put a thumb along the direction of i -2, that means we know to the right of wire 2, we follow our fingers, we have a magnetic field going in on this side, and then to the left of wire 2 on this side here, we have the magnetic field going out.

02:48 So that means the magnetic field is also going out.

02:51 The magnetic field due to the current in wire 2 at the location of wire 1 is going out of the screen.

02:58 So the question is, is there any way that we can neutralize the magnetic fields that are creating these repulsive forces between our wires.

03:06 Ultimately, the magnetic field that wire two creates at the location of wire one pushes wire one out because it's the force of wire two on wire one, pushes the wire one out because of our lawrence forces or magnetic forces, and the same thing's going to happen over here.

03:25 Wire one is going to push wire two away.

03:28 Now, we can get rid of this repulsive force if we can just cancel out the magnetic fields that are causing it, so the magnetic fields that arise from the currents in each wire...

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