Figure 29-52 shows, in cross section, four thin wires that are parallel, straight, and very long

Figure 29-52 shows, in cross section, four thin wires that...

Key Concepts

Vector Addition and Field Rotation

Vector addition explains how the individual magnetic field contributions combine to yield a net field whose final direction and magnitude depend on the relative positions and distances of the sources. Adjusting the positions of current-carrying wires alters the magnitude and direction of their respective fields, thus rotating the net field direction. Understanding this concept is key to solving problems involving adjustments in geometry to achieve a desired field orientation.

Right-Hand Rule

The right-hand rule is used to determine the direction of the magnetic field around a current-carrying wire. Pointing the thumb in the direction of the current, the curl of the fingers shows the circular direction of the magnetic field. This concept is essential for analyzing the directions of the magnetic fields produced by each wire in the system.

Magnetic Field from a Long Straight Current-Carrying Wire

A long straight current-carrying wire produces a magnetic field that circles the wire. Its magnitude at a distance r from the wire is given by the formula B = (??I)/(2?r), where ?? is the permeability of free space and I is the current. This concept is crucial for understanding how each wire contributes to the net magnetic field in the system.

Superposition Principle in Magnetism

The net magnetic field at any point in space is the vector sum of the magnetic fields contributed by all current sources. Each wire’s magnetic field, calculated individually using the magnetic field formula for a straight wire, must be added vectorially to determine the overall field direction and magnitude.

Transcript

00:01 So for this problem we have four wires that are parallel and are very long and they carried identical currents that we are going to call i and they also are at a distance deep of 15 centimeters from the center or the origin of the coordinate system where they create a magnetic field in this point they create a magnetic field b so before and trying to solve the questions for this problem we need to see how this net magnetic field is going to be and from the magnetic fields produced by each of these wires.

01:20 So to do that, we first need to use the right -hand rule to obtain the direction of each of these magnetic fields.

01:32 In here, we can see in this figure that we use the right -hand rule to obtain the direction of the magnetic field.

01:44 As we know, the magnetic field produced by a wire are circular, concentriced, and circular lines that depends on the radios of the point of measure.

02:05 Since we are measuring here at a distance deep from each wire, we will have the same magnitude for all of the.

02:15 This magnetic field so we will have the magnetic field one the magnitude of the magnetic field one that is equal to the magnitude of the magnetic field two three and four and it is equal to mu of zero the current that and because they have the same current i and the distance to the origin of this coordinate system which is d.

02:51 So in here we can see that for example for the wire one if we use the right -hand rule we will have lines the point in the center of the wire indicates that the current is outward meanwhile when it is across like in wire 2, the current is inward.

03:27 So for the case of the wire 1, we could see that the magnetic field at the center.

03:40 The magnetic field, in this case, is counter -clodwild.

03:45 So it will produce at the center a vector like this.

03:49 So this is the vector b1.

03:53 Now for the wire 3, we could see that the direction will be again counter cloudwise, but because it is in this position, it will create a vector downward.

04:18 So this is b3.

04:20 Now for the wire 2 we'll have that it is inward, so we put our thumb in that direction and make circles.

04:39 So that will indicate that in this case it is cloudwise.

04:46 So it will produce at the center of a bedward like this.

04:55 And in the case of the current 4, because it is outward, we will have the same direction, a battery in the same direction of b2, so this is b4.

05:13 From here, when we sum all of these magnetic fields to produce a net magnetic field, we will have 1, 2, 3, and 4.

05:31 The figure we could see that b3 and b1 will cancel each other because they have an opposite direction and the same magnitude so they will counsel and we will have two times the magnitude that we obtain here so that will be in the resulting net bettort in this case should be just in the x component because as i said the contributions of e1 and b2 cancel is shorter and those corresponds to the only y components for this net magnetic field.

06:32 So the first problem the first question for this problem is to determine what what is the value of ets, where ets represent how much we move the wire one along the ets assets, so that the net magnetic field bed tour rotates counter -cladwise by 30 degrees.

07:01 So for that, we use this formula.

07:04 We note that the tangent of an angle of the angle teta is equal to to the y component over the x component.

07:15 So in this case, we know that the angle that we want is 30 degrees, so we will have tangent of 30 degrees.

07:26 We will set what that value of the y company is going to be, but we know the value for the x component.

07:37 We can see in here that it is this value in here.

07:48 And so that we will have that the white campaign should be the product of this, the tangent of 30 degrees.

08:03 Now, we know that the tangent of 30 degrees is one over, it's one over the square root of 3.

08:18 So now from the figure we said that the only contributions to the white component of this net magnetic field are the magnetic field produced by the current one and three as you can see in the picture those are the only ones that had a white component.

08:47 So in this case, and in fact they only have a y component...

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