Lecture 2-Gauss_ Law

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Engineering Electromagnetics

LECTURE 2: Gauss’ Law

Instructor: Sir Tim

Summer Term

Gauss’ Law

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

ELECTRIC FLUX DENSITY

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Electric Flux

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Electric Flux Density (D)

Quantity similar to electric field butindependent of the medium.

𝐷 = ϵ 𝐸

Hence, each individual flux is given by:

Ψ = 𝐷 ⋅ 𝑑 𝑆

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Gauss’ Law

Gauss’ Law states that the total electric fluxpassing through any closed surface is equalto the total charge enclosed by that surface.

Ψ𝑡𝑜𝑡𝑎𝑙 = 𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 = 𝐷 ⋅ 𝑑 𝑆 = 𝜌𝑉𝑑𝑉

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 1

Three point charges, Q1 = 30 nC, Q2 = 150nC, and Q3 = -70 nC, are enclosed bysurface S. What net flux crosses S?

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 2

Charge in the form of a plane sheet withdensity ρs = 40 μC/m2 is located at z = -0.5m. A uniform line charge of ρl = -6 μC/mlies along the у axis. What net flux crossesthe surface of a cube 2 m on an edge,centered at the origin.

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 3

Apply Gauss’ Law to derive the electric fieldof standard charge distributions:

a) Infinite line charge

b) Infinite sheet charge

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 4

The volume in cylindrical coordinatesbetween ρ = 2 m and ρ = 4 m has a uniformvolume charge density, ρV = p C/m3 . UseGauss’ Law to find the electric fieldintensity in all regions.

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 5

Where does the capacitor store its energy inthe form of electric field? Assume that theupper plate is positive and the lower plate isnegative and the respective charges areequal in magnitude.

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 6

Three concentric spherical shells r = 1, r =2 and r = 3 m respectively, have chargedistributions 2, -4 and 5 μC/m2.

a) Calculate the total flux through r = 1.5m and r = 2.5 m.

b) Find E at r = 0.5, r = 2.5 and r = 3.5 m.

Gauss’ Law

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

DIVERGENCE THEOREM

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Differential Form of Gauss’ Law

Another form of the Gauss’ Law takesadvantage of the Divergence Theorem inVector Analysis.

𝐷 ⋅ 𝑑 𝑆 = 𝛻 ⋅ 𝐷

But since 𝐷 ⋅ 𝑑 𝑆 = 𝜌𝑉𝑑𝑉

𝜌𝑉 = 𝛻 ⋅ 𝐷

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Del Operator

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 1

The electric flux density of a rectangularparallelepiped formed by the planes x = 0and x = 1, y = 0 and y = 2 and z = 0 andz = 3 is D = 2xyax + x2ay C/m2. Determinethe net flux passing through the surface.

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Problem 2

A cube, 2 m on an edge, centered at theorigin and with edges parallel to thecoordinate axes has an electric field of E =10x3/3ε0 ax V/m. Determine the net chargeinside the cube.

Gauss’ Law

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

SEATWORK

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Seatwork

1. Given that

Determine D at all regions.

2. In free space, D = 2y2ax + 4xyay – azmC/m2. Find the total charge stored in theregion 1 < x < 2, 1 < y < 2, and – 1 < z < 4using closed surface integral and then the pointform of Gauss’ Law. Compare the results.

ECE Department, TUP Manila Engineering Electromagnetics Timothy M. Amado, ECE Summer

Thank you.

Lecture 2: Gauss’ Law

Engineering Electromagnetics

Timothy M. AmadoFaculty, Electronics Engineering Department

Technological University of the Philippines – Manila

tim.amado08@gmail.com