Figure 29-50 a shows, in cross section, two long, parallel...

Figure $29-50 a$ shows, in cross section, two long, parallel wires carrying current and separated by distance $L$. The ratio $i_{1} / i_{2}$ of their currents is $4.00 ;$ the directions of the currents are not indicated. Figure $29-50 b$ shows the $y$ component $B_{y}$ of their net magnetic field along the $x$ axis to the right of wire $2 .$ The vertical scale is set by $B_{y s}=4.0 \mathrm{nT},$ and the horizontal scale is set by $x_{s}=20.0 \mathrm{~cm}$ (a) At what value of $x>0$ is $B_{y}$ maximum? (b) If $i_{2}=3 \mathrm{~mA},$ what is the value of that maximum? What is the direction (into or out of the page) of (c) $i_{1}$ and (d) $i_{2}$ ?

Figure $29-50 a$ shows, in cross section, two long, parallel wires carrying current and separated by distance $L$. The ratio $i_{1} / i_{2}$ of their currents is $4.00 ;$ the directions of the currents are not indicated. Figure $29-50 b$ shows the $y$ component $B_{y}$ of their net magnetic field along the $x$ axis to the right of wire $2 .$ The vertical scale is set by $B_{y s}=4.0 \mathrm{nT},$ and the horizontal scale is set by $x_{s}=20.0 \mathrm{~cm}$
(a) At what value of $x>0$ is $B_{y}$ maximum? (b) If $i_{2}=3 \mathrm{~mA},$ what is the value of that maximum? What is the direction (into or out of the page) of (c) $i_{1}$ and (d) $i_{2}$ ?

Key Concepts

Magnetic Field from a Long Straight Current-Carrying Wire

This concept refers to the magnetic field generated by an infinite or long straight conductor carrying an electric current. The magnitude of this magnetic field is given by the formula B = (??/2?)(I/r), where I is the current, r is the radial distance from the wire, and ?? is the permeability of free space. Understanding this principle is crucial because it provides the building block for analyzing more complex systems involving multiple wires.

Right-Hand Rule for Magnetic Fields

The right?hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By pointing the thumb of the right hand in the direction of the current, the curl of the fingers shows the circular direction of the magnetic field. This concept is essential in predicting the orientation of magnetic fields produced by multiple currents.

Superposition Principle in Magnetic Fields

According to the superposition principle, the net magnetic field resulting from multiple sources is the vector sum of the fields produced by each individual source. This concept is important when analyzing systems with more than one current-carrying conductor because it allows the determination of the overall magnetic field by considering each wire’s contribution independently.

Optimization of Magnetic Field Strength

In problems involving the net magnetic field from multiple sources, there is often a spatial point where the resultant field strength is maximized. Finding this point involves understanding how the individual magnetic fields vary with distance and setting up the conditions for their constructive interference or cancellation. This concept underpins the analysis required to determine the point of maximum net magnetic field in a system of parallel current-carrying wires.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that this figure shows in a cross -ception two long parallel wires carrying current and separated by the distance held.

00:11 The ratio between the currents of the current of the wire 1 and 2 of their currents is equal to 4 and the directions of the currents are not indicated.

00:29 So the figure 20 and the other figure b, this one right here, it shows the white component of the magnetic field, of their net magnetic field along the axis to the right of wire 2.

00:45 So the vertical scale is set that b, y, s, this value right here and this value, is equal to 4 nanostela.

01:01 And the horizontal scale is said that ets, this value right here, is equal to 20 centimeters.

01:18 So for part a of this problem, we are asked at what value of x greater than zero.

01:31 The value of the white component of the magnetic field is a maximum, is maximum.

01:42 Is maximum.

01:49 Now, the fact that the y, the fact that the y component of the magnetic field is 0 at this position where x is equal to 10 centimeters because it is half of this value, as you can see, half of the x value.

02:14 So that implies that the currents are in terms, opposite directions.

02:20 Thus, we can obtain from there that the white component of the magnetic field is equal to mu -sub -0 times the current 1 divided by 2 times pi times the separation distance that in this case is l plus x minus mu -sup 0 times the current 2 divided by 2 times pi times x.

02:47 So in here we will find we can write this in the following form.

02:54 We can write this as misoob zero times the current 2 divided by 2 times 4.

03:03 In here we have used the fat, this ratio right here, this ratio that we are given for the currents, this one right here.

03:11 We have used that in here just to put this in terms of only the current 2.

03:18 And this times 4 divided by the distance l plus x minus 1 over x.

03:26 So to get the maximum, we take the derivative with respect to x and set equal to 0.

03:36 So if we do that with this expression, this led us to the following equation.

03:43 This will lead us to 3 times x squared minus 2 times x.

03:50 Lx minus l squared equals to zero.

03:54 Which factors and becomes, we can factorize this, and we will have that this is three lt plus lt times l's minus lt is equal to zero, which has the physically asset of solution, we can have that s should be equal to lt.

04:19 And this produces a maximum, producing the maximum so that when we substitute that into the equation for the white component of the manly field, we obtained that this is mu -sub -0 times the current 2 divided by 2 times pi times the distance alt.

04:37 To proceed further, we must determine the value of l...

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