A long coaxial cable consists of an inner cylindrical...

A long coaxial cable consists of an inner cylindrical conductor with radius $a$ and an outer coaxial cylinder with inner radius $b$ and outer radius $c$. The outer cylinder is mounted on insulating supports and has no net charge. The inner cylinder has a uniform positive charge per unit length $\lambda$. Calculate the electric field (a) at any point between the cylinders a distance $r$ from the axis and (b) at any point outside the outer cylinder. (c) Graph the magnitude of the electric field as a function of the distance $r$ from the axis of the cable, from $r = 0$ to $r = 2c$. (d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.

Key Concepts

Surface Charge Distribution

The surface charge distribution on conductors in electrostatic equilibrium is determined by the requirement of a zero electric field inside the conductor. This principle, when combined with Gauss's Law, allows for the calculation of the induced charges on surfaces that face charged regions. Distinguishing between the inner and outer surface charges in a coaxial cable involves applying these conditions at both surfaces.

Graphical Analysis of Field Distribution

Graphing the electric field as a function of radial distance illustrates how the field changes across different regions in the system. In the context of a coaxial cable, the plot shows distinct behaviors: a rising field due to the central line charge, subsequent changes where induced charges on the conductive surfaces alter the field, and the eventual decay of the field at distances beyond the outer conductor. This visualization aids in understanding the spatial variation of the field magnitude.

Boundary Conditions in Electrostatics

Boundary conditions in electrostatics involve the continuity of the electric field and potential across interfaces between different materials, such as conductors and insulators. In systems involving conductors, the electric field inside a conductor is zero, and the surface charge distributions adjust to satisfy this condition. These conditions are crucial for correctly determining the field in multi-layered structures like coaxial cables.

Cylindrical Symmetry

Cylindrical symmetry arises in systems where the charge distribution and geometry are invariant along an axis and in rotations about that axis. This symmetry implies that the electric field depends solely on the radial distance from the axis, which simplifies the application of Gauss's Law because the field has a uniform magnitude over a cylindrical Gaussian surface and is directed radially.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface. It simplifies the calculation of electric fields in systems exhibiting symmetry, such as cylindrical or spherical distributions, by allowing for the use of a Gaussian surface to directly equate the net flux to the charge inside divided by the vacuum permittivity.

Electric Field of a Line Charge

For an infinitely long, uniformly charged line (or thin cylinder approximated as a line charge), the electric field at a radial distance is given by a simple formula derived using Gauss's Law. This concept is fundamental to understanding how the field behaves in the region surrounding a charged cylindrical conductor and provides a basis for computing fields in more complex coaxial arrangements.

Transcript

00:01 Going to apply goss's law to this gossian cylinder here for part a we can say that a is going to be less than r which is less than b and we can say that the electric flux is then equaling the electric field times two pi r l and we know that the charge enclosed would be equalling the linear charge density lambda times l the charge this would be the charge this would be the charge this would be the charge charge on the length l of the inner conductor that is inside the gaussian surface.

00:39 And so we can say that then the electric flux equals the charge of the enclosed divided by the primitivity of free space.

00:58 This gives us that then the electric field multiplied by 2 pi r times l will be equaling pi l sorry lambda l divided by epsilon not of course this gives us then that the length is going to cancel out and we have that the electric field is going to be equaling the linear charge density lambda divided by two pi r times the electric permit the permittivity of free space or the permittivity of the therefore the direction of the electric field is we can say radially outward.

01:54 For part b, this would be the, this would be essentially the diagram.

02:01 So we're going to play goss's law where for part b, r is greater than some length c.

02:09 And we can say that the electric flux would be equal to the electric field times 2 pi r times l.

02:18 And again, the enclosed charge is equaling pi times l.

02:25 This is going to give us here.

02:30 As we can see, the enclosed charged, the magnetic flux all are giving us the exact same values.

02:39 So here, the electric charge is, again, the linear charge density divided by 2 pi r multiplied by epsilon not.

02:49 And here the enclosed charge is again positive.

02:52 So the direction of e would be again radially outward.

03:00 So we can now, for part c, we have that within a conductor, the electric field is equaling zero.

03:12 So we can say electric field equals zero within a conductor.

03:18 And we can then say thus the electric field is going to equal zero for radii.

03:26 Less than a.

03:27 And so we can say that e is equalling the linear charge density divided by 2 pi r epsilon not...

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