In Fig. 29-57, four long straight wires are perpendicular to the page, and their cross sections

In Fig. 29-57, four long straight wires are perpendicular to...

Key Concepts

Vector Representation Using Unit Vectors

In vector analysis, unit vectors (typically i, j, and k) are used to represent directional components of physical quantities such as force. Expressing forces in unit vector notation allows for clear and concise communication of both magnitude and direction. In problems involving forces on current-carrying wires, unit-vector notation helps decompose the force into its components, making it easier to perform vector addition and analyze the overall system dynamics.

Right-Hand Rule

The right-hand rule is a systematic method used to determine the direction of a magnetic field generated by a current-carrying wire, and also to deduce the direction of the force on a current-carrying conductor in an external magnetic field. By aligning the thumb of the right hand with the current direction, the curl of the fingers indicates the circular direction of the magnetic field, which then helps in correctly determining the vector components of forces induced by magnetic interactions.

Lorentz Force on a Current-Carrying Conductor

When a conductor carrying an electric current is placed within an external magnetic field, it experiences a force described by the Lorentz force law, F = I(L x B), where I is the current, L is the length vector in the direction of the current, and B is the magnetic field. This vector relationship shows that the force is perpendicular to both the current direction and the magnetic field, and its calculation requires the use of cross products and the right-hand rule to determine the proper direction.

Magnetic Field of a Long Straight Current-Carrying Wire

A long straight wire carrying an electric current produces a magnetic field that circulates around the wire. The magnitude of the field at a distance r from the wire is given by the expression B = ??I/(2?r), where ?? is the permeability of free space and I is the current. This fundamental relationship is crucial when determining the magnetic fields produced by multiple wires in an arrangement, as it allows one to calculate the contribution at any point in space from an individual wire.

Superposition Principle in Electromagnetism

The superposition principle states that when multiple magnetic fields are present, the net magnetic field at any given point is the vector sum of the individual fields produced by each current element or wire. This principle is essential when dealing with systems of several current-carrying wires, as it allows one to analyze the cumulative effects on a particular wire by calculating and then summing the contributions from all other wires.

Transcript

00:01 Hi there.

00:03 So for this problem, we need to calculate the net force that is produced in the wire four because of the other three wires.

00:16 In this case, we are going to use the magnetic force that is in this point for and then zoom all of the components.

00:30 So what i'm going to do is to draw all the forces that the wire 4 fills because of the wire 1, 2, and 3.

00:44 So to do that, we can see, we know that when two wires have the same direction in current, they attract each other.

00:56 So in that sense, the force that feels the wire four because of one is in this direction to that curve, to that wire.

01:13 Now, the force that feels, the wire, four because of three, goes to the left because it feels, it feels an opportunity.

01:29 An opposite force because they have an opposite direction in their current.

01:34 So this is 4 and 4 3.

01:41 And the other one goes in the line that connect them.

01:47 And this is also in the opposite direction because they have different directions in current.

01:56 So this is 4 -2.

02:00 Um, so this is 4 -2.

02:02 Um, so the net force that it is filled in 4 that we call it like this, it's just the superposition of all of these forces that we will have for 1, 4 2, 4 3.

02:26 And this angle in here, we can easily obtain it because we know that this, this, this, these wires form a square, sorry, form a square, and each edge has a length of 30 .5 centimeters.

02:55 So in that sense, this in here forms 45 degrees because it is half of 90 degrees that is what it forms the x axis with the y axis.

03:13 So we have that angle that we are called teta 45 degrees.

03:21 So we start by calculating each company of these f4 force.

03:32 So the companies that we are going to call it components.

03:40 In this sense we have the fours company.

03:44 So we are going to take into account only the forces that are acting on the ad acids or at least have some component on it.

03:55 So we have minus f43 because it is only in the component of that.

04:11 So this is just that.

04:14 And we also have minus 4 -2.

04:18 And we can see that this angle in here corresponds to this angle in here.

04:25 So it is also minus, but to obtain the x component, we need to multiply this by cousin of teta.

04:34 So having all of this, we can apply the definition for the magnetic force.

04:45 We know that that is given by the following.

04:50 Nuisance 0, the current i squared 2 pi a for this case, because from 0 .4 to 0 .3, this distance is a, which is this, but we are going to call it a.

05:14 So that's why we put this in here, minus the other one, epsilon 0, y, casino 40 degrees, this value in here, and over the distance pi a times the distance to, oh no, okay, i'm pretty like this.

05:43 The distance from the point four to the point two, we can calculate it by just using the hypotenuse.

05:54 We know that this in here is a, and we know that this in here is a.

06:00 So the hypotenuse is the square root of two times a, and that's the thing that we are going to put in here.

06:16 So that is 2 pi a the square root of 2.

06:38 So this is what we have for the s component of the force that is acting on the wire.

06:48 Now we do the white component.

06:54 So the white component in this case, we write it like this or want because it only has a vertical component, so it is simply that minus f2, and in this case, the component is given by the sign of theta, because we want the vertical component.

07:36 So it's sign of theta and it is minus because it is pointing downwards.

07:41 So it is minus.

07:43 And here we again apply the definition for the magnetic force.

07:50 So in this first case is mucus 0.

07:58 And the distance to the wire 1 is also a, so it's 2 pi a minus the one from the and this is here.

08:26 So we sum this...

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