Gauss' Law (Rec 4)

Gauss’ Law
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This activity is based on the following concepts:
Electric flux through a surface is defined in the following way:
o Divide the surface into small patches
o Multiply the area of a patch by the component of the electric field in the direction
perpendicular to the patch
o Add the electric field contributions from all patches on the surface — that sum is the
flux!
o In practice, we will consider the patches to be infinitesimally small, and hence the
sum is replaced by an integral.
Gauss' Law is a fundamental law of nature that says:
o The electric flux through any closed surface is equal to the total charge enclosed by
that surface divided by ε 0 .
o We write this mathematically as:
Q
E dA enc
The electric field inside a metal (in electrostatic equilibrium) is zero!
Some math that you must review:
o What is the surface area of a sphere of radius R?
o What is the volume of a sphere of radius R?
o What is the surface area of a cylinder of length L and radius R?
o What is the volume of a cylinder of length L and radius R?
0

Exercise 1: Understanding Electric Flux
The figures below show different surfaces in the presence of different electric fields. Determine
the electric flux in each case, briefly showing your reasoning.
15°
E E
Q1. Rectangle of area A with a uniform electric
field E oriented at an angle of 15° with respect
to the normal to the surface of the rectangle.
Flux:
Q2. Rectangle of area A with a uniform electric
field E oriented parallel to the surface of the
rectangle.
Flux:
E
Q3. Spherical surface of radius R with an
electric field pointing radially outward from the
center of the sphere. The magnitude of the
electric field at the surface of the sphere is
everywhere E.
Flux:

Exercise 2: Applying Gauss' Law
The figure at right shows a 2D cross-section of a charge
arrangement containing a point charge +Q surrounded by an
uncharged spherical metallic shell with inner radius R 1 and outer
radius R 2 . Your goal is to use Gauss' Law to find out the magnitude
of the electric field at the points A and B that are located at
distances R A and R B from the center of the sphere as shown in the
figure. You will also use Gauss' Law to determine the charge
induced on the inner surface of the spherical shell. Remember to
use the spherical symmetry of the problem in your arguments.
B
A
R 1
R 2
+Q
Q4. To determine the field at point A, draw a spherical Gaussian surface of radius R A , concentric
with the point charge and metal shell.
In terms of the electric field strength at A (E A ), and the radius R A , what is the electric flux
through this surface? Use the definition of electric flux instead of Gauss’ Law.
How much charge is enclosed by this Gaussian sphere?
Using Gauss’ Law, and your expression for the electric flux at A, determine the electric
field at point A.
Q5. Another student in this class makes the following two claims:
The charge on the inside surface of the spherical shell has to be −Q, and the
charge on the outer surface of the shell is zero.
Is this student correct in making both of these statements? You may want to use concepts such as
(i) Gauss’ Law, (ii) the knowledge that E = 0 inside a metal in electrostatic equilibrium, (iii) the
spherical symmetry of the problem, and (iv) the principle that charge must be conserved to
convince us of what the correct answer should be. Be sure to address both statements.

Q6. Finally, determine the electric field at point B by imagining a spherical Gaussian surface of
radius R B .
In terms of the electric field at B (E B ) and the radius R B , what is the electric flux through
this surface? Use the definition of electric flux instead of Gauss’ Law.
What is the total charge enclosed by this surface?
Using Gauss’ Law, determine the electric field at point B.