Figure 29-87 shows a cross section of a hollow cylindrical conductor of radii a and b, carrying

Figure 29-87 shows a cross section of a hollow cylindrical...

Key Concepts

Magnetic Field Continuity and Boundary Conditions

When analyzing the magnetic field across different regions of a conductor, applying appropriate boundary conditions is essential. These conditions ensure that the field varies continuously or takes appropriate limiting values at the interfaces, such as the field being zero at certain boundaries or matching known expressions at the surface of a long straight wire. This consistency check confirms the validity of the derived expressions for the magnetic field in different regions.

Graphical Analysis of Magnetic Field Profiles

Plotting the magnetic field as a function of the radial distance provides a visual representation of how the field changes within and around the conductor. Such graphical analysis helps in understanding the transition of the magnetic field from regions inside the current-carrying material to regions external to it, revealing the influence of geometric factors and the distribution of current on the overall field behavior.

Hollow Versus Solid Conductor Analysis

The analysis of magnetic fields in conductors can vary depending on whether the conductor is solid or hollow. In a hollow cylindrical conductor, the current is confined to an annular region, which alters the distribution of the enclosed current and thus the resulting magnetic field. Comparing the different cases, such as when the inner radius is zero (solid conductor) or nonzero (hollow conductor), helps in understanding how the geometry of the conductor influences the magnetic field profile.

Uniform Current Distribution

A uniform current distribution refers to the situation where the electric current is spread evenly across the cross-sectional area of a conductor. This simplifies the analysis because the current density is constant throughout the conductor, allowing one to calculate the fraction of the total current enclosed within any given radius by directly relating it to the area, which is crucial for applying Ampere's law in these scenarios.

Ampere's Law

Ampere's law is a fundamental principle in electromagnetism that relates the magnetic field circulating around a closed loop to the total current enclosed by that loop. In problems with high symmetry, such as those involving cylindrical conductors, this law simplifies the calculation of the magnetic field because the integral path can be chosen such that the magnetic field is constant along the path, leading directly to an expression for the field in terms of the enclosed current.

Cylindrical Symmetry

Cylindrical symmetry is a type of geometric symmetry where physical quantities depend only on the radial distance from the axis and not on the angular or longitudinal coordinates. This symmetry is exploited in the analysis of current-carrying wires and cylindrical conductors, allowing the use of circular Amperian loops to easily determine the magnetic field magnitude as a function of the radial distance.

Transcript

00:01 Ampiers law is typically used to find the magnetic field of a long straight wire, but it really works for any long straight conductor carrying a current, as long as the current just depends on the variable radially outward.

00:22 In other words, something with cylindrical symmetry.

00:26 And a reminder of what amper's law says.

00:30 It says that the magnetic field projected along a path integrated all the way around the path is equal to mu not times the enclosed current.

00:45 So here we're going to take a look at the magnetic field in a situation where we have a conductor of finite thickness and it is hollow inside, almost like a coaxial cable without the middle cable down it.

01:06 So just a tube.

01:09 And if we are inside the tube, so if we are in the gap with r less than b, what we know is the enclosed current is equal to zero.

01:23 And through the symmetry of the situation, we thus conclude that b is equal to zero.

01:33 And then when we get inside the conductor, so an empyrian loop is shown there in green.

01:45 What we have to take into account is the amount of current enclosed.

01:52 The left -hand side of ampiers law, you almost never have to integrate.

02:00 Sorry, we have the later picture showing, i will hide it.

02:05 But you almost never have to integrate that left -hand side because you assume that the magnetic field is constant along a circular loop.

02:15 So for the loop shown b times 2 pi, r is equal to mu not times i enclosed.

02:27 And in order to figure out the enclosed current, it's good to think about a current density j, which is the i divided by the cross -sectional area.

02:44 And in this case, we're assuming, yeah, we have a total of little i.

02:50 Let's just call it a little i.

02:54 And the cross -sectional area is the cross -sectional area of the blue conductor.

03:03 So it's the difference between the outer circle with radius a squared minus the area of the inner circle removing.

03:18 The gap, which has radius b.

03:23 But anyway, the ion closed is munaut times the current density times the area enclosed by the empyrian loop, which is the circle that's encircled by the green minus, again, subtract the gap, area and closed by the that you're finding the magnetic field on.

04:08 Okay, and putting this all together, it looks a little bit weird, but the magnetic field is equal to 1 over 2 pi times the radius of the empyrian loop.

04:25 Then we have mu not, and then this kind of weird -looking current density that takes into account just the cross -sectional area of the conductor, and then multiplied by the cross -sectional area of the conductor enclosed by that loop.

04:49 So that's an important distinction...

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