The Coaxial Cable. A long coaxial cable consists of an inner...

The Coaxial Cable. 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 $r=2 c$ . (d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.

Key Concepts

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.

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.

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.

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.

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.

Transcript

00:01 Hi everyone this is the problem based on gosfiram to calculate the electric fuel intensity at any point.

00:09 Here it is given cylindrical conductor having inner radius a and outer cylinder.

00:23 Inner radius of outer cylinder is b and outer radius of outer cylinder is c.

00:31 So it is forming the coaxial cable.

00:39 Linear charge density of inner cylinder is lambda column per meter.

00:45 We have to find electric field at first part r greater than a but less than b.

00:57 We have to find a form r to be greater than c.

01:02 We have to find the graph between electric field and distance for r is called to zero to r is called to two c and we have to find linear charge density of inner surface of outer conductor let us see here first part first we have to consider a question surface to solve this problem so suppose this is the gussian surface as a cylinder of radius r which is greater than a but less than b for first case this distance so for first part we have to apply the cost syrup flux will be equal to q and closed upon a epsilon not so flux will be e into 2 5 or l.

02:28 If lambda is length of cylindrical gusian surface, then charge and close will be lambda into l.

02:36 So electric field you will get lambda upon 2 pi epsilon not r.

02:58 This is the answer of part a.

03:00 This is for r to be greater than but less than b.

03:07 Second part.

03:11 For second part, we have to consider the gussian surface...

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