(1I) A spherical cavity of radius 4.50 cm is at the center of a metal sphere of radius 18.0 cm .

step 1 Okay, so we're doing chapter 22, problem 21 here. So it says a spherical cavity of radius 4 .5 c

(1I) A spherical cavity of radius 4.50 cm is at the center...

(1I) A spherical cavity of radius 4.50 $\mathrm{cm}$ is at the center of a metal sphere of radius 18.0 $\mathrm{cm} .$ A point charge $Q=5.50 \mu \mathrm{C}$ rests at the very center of the cavity, whereas the metal conductor carries no net charge. Determine the electric field at a point $(a) 3.00 \mathrm{cm}$ from the center of the cavity, (b) 6.00 $\mathrm{cm}$ . the center.

Key Concepts

Induced Charge Distribution

When external or internal charges are present near or inside a conductor, they induce a redistribution of charges on the conductor's surfaces. This induced charge arrangement ensures that the electric field within the conductive material remains zero while producing an appropriate external field to maintain electrostatic equilibrium.

Electric Fields in Cavities

The presence of a cavity inside a conductor influences how the electric field is distributed. Charges placed within a cavity create their own electric field that is confined to the cavity, whereas the surrounding conductor responds by rearranging its surface charges so that the field inside the conducting material is nullified. This separation of regions allows for the analysis of fields both within the cavity and in the nearby external regions.

Gauss's Law

Gauss's Law states that the total electric flux through a closed surface is proportional to the net charge enclosed by that surface. This principle is a cornerstone in electrostatics and is used to calculate electric fields in systems with high symmetry by relating the geometry of the surface to the enclosed charge.

Electrostatic Equilibrium in Conductors

In a conductor at electrostatic equilibrium, the electric field within the conducting material is zero. This is because free charges within the conductor rearrange themselves in response to any internal electric field until they cancel it out, ensuring that no net electric field exists inside the conductor.

Transcript

00:01 Okay, so we're doing chapter 22, problem 21 here.

00:04 So it says a spherical cavity of radius 4 .5 centimeters is at the center of a metal sphere of radius 18 centimeters.

00:16 So let's go ahead and draw this out here, with this being r1.

00:23 Call that r1 and r2 here, and this being r2.

00:29 A point charge q of 5 .5 microculems rests at the very center of the cavity, whereas the metal conductor carries no net charge.

00:42 Determine the electric field at point at 3 centimeters from the center of the cavity, and then b and c are different points.

00:50 Okay, so i'll start with point a here.

00:54 So first we need to consider a gaussian surface with a radius.

01:00 3 centimeters, 3 .0 centimeters away, which encloses all of the charge.

01:16 We should know that this corresponds to an e -field of the gaussian surface of q over 4 pi epsilon not par squared.

01:44 How much charge is enclosed 3 centimeters are way? well, the point charge is at the center, so this just encloses the total charge.

01:52 So we can just calculate what the e -field is here.

01:55 And it's given as 5 .5 times 10 to the negative 6 coulums over 3 .0 centimeters squared times k, which was 8 .988 times 10 to the 9.

02:15 Let's actually change this to 0 .03 meters squared, because we need everything in si units here for this to work out.

02:24 And this comes out to be 5 .49 times 10 to the 7 newton per coulum.

02:34 And since this is positive, we know that this is directed radially.

02:41 Awesome...

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