Repeat Exercise 28.39 for the case in which the current in...

Repeat Exercise 28.39 for the case in which the current in the central, solid conductor is $I_{1}$, the current in the tube is $I_{2}$, and these currents are in the same direction rather than in opposite directions.

Key Concepts

Ampere's Law

Ampere's Law relates the line integral of the magnetic field around a closed path to the current passing through that path. This fundamental principle in magnetostatics is used to calculate magnetic fields in systems with high symmetry, thereby simplifying the evaluation of magnetic fields in various current configurations.

Superposition Principle

The superposition principle states that the net magnetic field created by multiple currents is the vector sum of the magnetic fields produced by each current individually. This principle is critical when dealing with composite current systems, as it allows for the separation of contributions from different current-carrying elements and the subsequent combination to determine the overall field.

Symmetry Considerations

Utilizing symmetry in magnetic field calculations helps in choosing appropriate integration paths and simplifies the evaluation of the magnetic field. By identifying axes or planes of symmetry within the system, one can reduce the complexity of the problem, often allowing the magnetic field to be expressed as a function of radial distance from a central axis or other symmetric variables.

Current Distribution in Conductors

Understanding the distribution of electric current within conductors, whether in solid wires or tubular structures, is crucial for accurately determining the resultant magnetic field. This involves knowing how the current density is distributed in different regions, particularly when the geometry of the conductors leads to different levels of enclosed current for a given Amperian loop.

Transcript

00:01 Okay, this is question 28 .44.

00:03 And we have the same situation in the question right before.

00:08 So we have a coaxial cable.

00:09 We have a wire with current i1 inside.

00:14 And then another thick cylinder on the outside with current i2.

00:18 And now what we want to do is consider the case where these two currents are going to be in the same direction.

00:27 And we're assuming that they're different.

00:28 So there's i1 inside and i2 outside.

00:32 And in the first part of the question, we want to consider a point where the radius or the point is in between these two between radius a and radius b.

00:47 So some point maybe here.

00:49 And we're going to call that that's r that we're considering.

00:54 So we want to look at this point r here.

00:58 This is going to be some distance again r from the center of this first wire.

01:09 So to calculate this we're going to want to use the fact that the integral of the magnetic field is going to be equal to mu not times the current enclosed.

01:19 So this i here represents the current enclosed by this line integral.

01:26 That's what this is.

01:27 There's really a little circle here to indicate that this is a line integral indicating a circle...

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