Eight solid plastic cubes, each 3.00 cm on each edge, are glued together to form each one of the

Eight solid plastic cubes, each 3.00 cm on each edge, are...

Eight solid plastic cubes, each 3.00 $\mathrm{cm}$ on each edge, are glued together to form each one of the objects (i, ii, in, and iv) shown in Figure $\mathrm{P} 23.37$ . (a) Assuming each object carries charge with uniform density 400 $\mathrm{nC} / \mathrm{m}^{3}$ throughout its volume, find the charge of each object. (b) Assuming each object carries charge with uniform density 15.0 $\mathrm{nC} / \mathrm{m}^{2}$ everywhere on its exposed surface, find the charge on each object. (c) Assuming charge is placed only on the edges where perpendicular surfaces meet, with uniform density $80.0 \mathrm{pC} / \mathrm{m},$ find the charge of each object.

Key Concepts

Composite Geometric Structures

Composite geometric structures involve objects made by combining several simple shapes, such as multiple cubes joined together. Understanding the overall geometry is essential to correctly determine parameters such as volume, surface area, or edge length, which are needed when applying density concepts (volume, surface, or linear) to calculate the total charge. This concept emphasizes the importance of breaking down complex shapes into simpler parts to facilitate integration and accurate analysis.

Linear Charge Density

Linear charge density describes the amount of charge per unit length along a line, such as along the edges of an object. In scenarios where charge is confined to the edges where surfaces meet, the total charge is calculated by multiplying the linear charge density by the total length of those edges. This concept applies to cases such as charged filaments or edges of composite geometrical objects.

Surface Charge Density

Surface charge density is the measure of charge per unit area on the surface of an object. When an object has charge only on its exposed surfaces, the total charge is found by multiplying the surface charge density by the effective surface area of the object. This concept is particularly important in understanding phenomena related to conductors, capacitors, and other systems where charge resides on the surface.

Volume Charge Density

Volume charge density refers to the amount of charge per unit volume distributed uniformly throughout a material. When dealing with objects of various shapes, the total charge is obtained by multiplying the volume charge density by the volume over which the charge is distributed. This concept is fundamental in determining charge distribution when the entire volume of an object carries a uniform charge, as often encountered in problems involving dielectric materials or uniformly charged bodies.

Transcript

00:01 For this problem on the topic of electric fields, we are given eight solid plastic cubes, each with the edge length of 3 centimeters, which are glued together to form one of the objects as shown in the figure.

00:12 Firstly, if each of the object carries a charge with uniform density 400 nanocolums per cubic meter throughout its volume, we want to find the charge of each of these objects.

00:22 Secondly, the charge of uniform density 15 nanoculums per square meter is provided to each object on its object.

00:31 Exposed surface and we again want to find the charge on each object.

00:35 And lastly we are told that if charge is only placed on the edges where the perpendicular surfaces meet with the uniform density of 80 picocouloms per meter, we want to again find the charge of each object.

00:48 So firstly, in a every object has the same volume and this volume v is equal to 8 into 0 .03 meters cubed, which gives a total volume of 2 .16 times 10 to the minus 4 cubic meters.

01:17 Now for each, they have the same charge q, which is the charge density row times the volume.

01:27 And this is equal to 400 times 10 to the minus 9 koums per cubic meter times the total volume of 2 .16 times 10 to the minus 4 cubic meters which gives the charge on each of the objects to be 8 .64 times 10 to the minus 11 columns.

02:03 Next we are given the charge density for the area and we must count the nine square centimeter squares painted with the charge.

02:16 And so firstly we have six times four, which is 24 squares.

02:24 And so the total charge queue is the surface charge density sigma times the area a.

02:34 And this is equal to 15 times 10 to the minus 9 couloms per square meter times 24 times the area 9 times 10 to the minus 4 square meters.

02:59 This gives the total charge for the first object to be 3 .24 times 10 to the minus 10 culloms.

03:17 For the second object we have 34 squares exposed and so q again is the surface charge density times the area and this is equal to 15 times 10 to the minus 9 couloms per square meter times multiplied by 34 exposed squares times the area of 9.

03:55 Times 10 to the minus 4 meter squared...

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