Lecture 5 Charge Density & Differential Charge. Sections: 2.3, 2.4, 2.5 Homework: See homework file

Transcription

1 Lecture 5 Charge Density & Differential Charge Sections: 2.3, 2.4, 2.5 Homework: See homework file

2 Point Charge as an Approximation charge occupies a finite olume and may hae arying density a charged body can be always iewed as made of charges so small that they appear as points point charge features olume, which is considered infinitesimal (a point) relatie to the distance from its center to the obseration point point charge features homogeneous charge distribution (its charge density ρ is constant) Q 3 Q = ρ [C] ρ = [C/m ] far away LECTURE 5 slide 2

3 Uniform Charge Distribution: Example A sphere of radius 1 cm contains charge Q = C. (a) Find its charge density if the charge is uniformly distributed. (b) Find the number of electrons per unit olume (the number density of electrons). LECTURE 5 slide 3

4 Total Charge and Aerage Charge Density close up P Q P 1 Q ρ ( ρ) = ρ a d = ( ) the olume integral as a Riemann sum up close, the details in the charge distribution matter consider the total charge Q as a collection of point charges Q = ρ ( P) ( P) ( P ) i i i Q Q ρ = = i i i i i Q = ρ( x, y, z ) dx dy dz d homogeneous charge distribution is an approximation where an aeraged charge density is assumed or ( ρ ) a ρi i LECTURE 5 slide 4 = i i i

5 Aerage Charge Density: Example A cube of side 1 cm contains charge. In 1/5 th of its olume, the charge density is ρ 1 = C/m 3. In 3/5 ths of its olume, the charge density is ρ 2 = C/m 3. In the remaining olume, ρ 3 = C/m 3. (a) Find the total charge Q in the cube. (b) Find the aerage charge density (ρ ) a. LECTURE 5 slide 5

6 Volume and Surface Charge Densities olume charge density is a differential (point-wise, local) description of the charge ρ Q dq 3 = lim =, C/m Q = ρd, C 0 d the concept of surface charge is used when the charge is spread in a wide thin sheet approximations inoled with surface charge: (i) the sheet s thickness h is negligible compared to its length and width, (ii) the charge density does not ary with the height Q = d = ds = ds h/2 ρ ρdh ρs S h/2 S 2 ρ = ρ ρ h h s = ρs LECTURE 5 slide 6 [C/m ]

7 Surface Charge Density the surface charge distribution is described by surface charge density ρ s a function of position on the surface Q = ρsds Q dq 2 ρs = lim =, C/m S s 0 s ds + h + ds Metal strip of thickness 100 μm is charged with density ρ = 10 pc/m 3. What is its surface charge density ρ s? LECTURE 5 slide 7

8 Field of Surface Charge infinite-surface charge implies symmetry of the field with respect to their plane o field ectors are anti-symmetric o field is perpendicular to charge plane equipotential lines force lines LECTURE 5 slide 8

9 Linear Charge Density linear charges are a useful approximation for charges the olume of which has two of its dimensions negligibly small compared to the other dimension (the length) ariations of the charge distribution in the cross-section are neglected s Q = ρd = ρds dl Q = ρ ldl V l s l l d ρ = s ρ dq ρl = ρ s= s= dl s l dq dl, C/m linear charge distribution is described by the linear charge density ρ l the field has cylindrical symmetry LECTURE 5 slide 9

10 CAN YOU SOLVE IN YOUR MIND? How much charge Q is there on a charged surface of area A = 1 mm 2 if ρ s = 4 μc/m 2? Q = How long is a uniformly charged string whose total charge is Q = 1 μc and its linear charge density is ρ l = 10 6 C/m? L = LECTURE 5 slide 10

11 Differential Charge dq (Charge Element) infinitesimal (differential) charge elements they all behae as point charges dq = ρd, dq = ρsds and dq = ρldl point charge surface charge linear charge used to ealuate the field of complicated charge configurations through the principle of superposition (integration oer the olume of the charge distribution) the fields (potential and force) of the point, surface and linear charges are known and sere as building blocks in solutions of complicated problems the field of the point charge is of fundamental importance all other solutions follow from it LECTURE 5 slide 11

12 You hae learned about olume charge distribution, its density and its relation to the total charge Q surface and line charge distributions, their densities and their relation to the total charge Q the symmetry of the field of a planar charge and a line charge the concept of differential charge and how it relates to olume, surface and linear charge densities LECTURE 5 slide 12