A very long, solid insulating cylinder has radius R; bored...

A very long, solid insulating cylinder has radius $R$; bored along its entire length is a cylindrical hole with radius $a$. The axis of the hole is a distance $b$ from the axis of the cylinder, where $a<b<R$ (Fig. P22.54). The solid material of the cylinder has a uniform volume charge density $\rho .$ Find the magnitude and direction of the electric field $\overrightarrow{\boldsymbol{E}}$ inside the hole, and show that $\overrightarrow{\boldsymbol{E}}$ is uniform over the entire hole. (Hint: See Problem 22.53.)

Key Concepts

Gauss's Law

Gauss's Law relates the net electric flux through a closed surface to the charge enclosed by that surface, allowing one to compute electric fields for systems with high symmetry. In particular, it is an essential tool for finding the field due to uniformly charged bodies, as it simplifies the integration required by taking advantage of symmetry to determine the contribution of each charge element.

Superposition Principle

The superposition principle states that the total electric field due to multiple charge distributions is the vector sum of the individual fields produced by each of those charges. This concept is crucial when dealing with cavities or cutouts in a charged object, as the field can be calculated by considering the cavity as a region with an 'opposite' charge distribution whose field cancels part of the field produced by the original charged body.

Uniform Charge Distribution

A uniform charge distribution implies that the charge density is constant throughout the object. This characteristic leads to predictable and symmetric electric field patterns, which simplifies the calculation process for electric fields in and around the material. Specifically, for objects like cylinders or spheres, it enables the use of Gauss's Law effectively to derive the field as a function of position.

Method of Virtual Charges (Negative Distribution)

The method of virtual charges involves conceptually removing the effect of a portion of the charged body by introducing an equivalent charge distribution with opposite sign. In the context of cavities, this method models the absence of material by superimposing a negative version of the original charge distribution, which helps in analytically deriving the electric field within the cavity by utilizing the superposition principle.

Uniformity of the Electric Field in a Cavity

Under certain configurations, such as a cavity within a uniformly charged object created by a uniform offset of the cavity relative to the center, the resulting electric field inside the cavity can be shown to be constant throughout its volume. This uniformity arises from the linearity of the resulting potential distribution when the effects of the rest of the charge distribution and the virtual negative distribution are superimposed, making the electric field independent of the specific location within the cavity.

Transcript

00:01 Welcome to the problem number 22 .58.

00:05 This is chapter 22 goes low.

00:07 This is a very interesting problem so in this question we have given that there is a long cylinder and has contained some charge density draw so the charge density and within this there is a whole passing to the whole cylinder which has radius a and is at distance b from the center of the cylinder so this is this is a distance be from the center of the cylinder so this is is another hole which is in cylindrical shape passing through this hole so if you want to expand this one so this is our hole and this is an insulating material so this is a hole so this can in this condition we are given that the hole is not coaxial that means which has been given that this hole is at distance b and has radius a is smaller than b and which is smaller than capital so in this condition we have this much hole and outside this region we have rochard density so we need to find out the electric field inside that hole there are many ways to do this question so the one i am using it's very intuitive and very easy so let's take a small cylindrical shape gaussian surface so this is our hole these are same and this is a cylindrical shape don't think that this cylinder has something to do with the cylinder so let me try to project this one here so this cylinder is exactly seems like a small cylinder here passing through this one so looking at this one you have this slender and this is a surface which has which has containing row volume or row chart density and and whatever the volume we don't need to care about that so let's apply the gaussian low or the ghost low for this surface so we have e .d .s.

02:37 Equals to q and closed over epsilon.

02:41 Let's take that the hole has distance or the point inside that hole is x at distance x from the surface of this hole and we are digging also the same distance x x side that which has containing charge row.

02:59 So if you want to project this one, so let me make it more clear.

03:03 So this is our hole and this is our sphere.

03:12 This is our sphere and this is our cylinder.

03:17 And this one is the face of the cylinder...

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