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