Download Tutorial on Gauss` law

Electricity and Magnetism: PHY-204
Fall Semester 2013
Tutorial on Gauss’ law
1. The dipole moment of the water molecule is 6.1 × 10−30 C.m. In an electric field,
a molecule with a dipole moment will tend to settle into an equilibrium orientation
such that the dipole moment is parallel to the electric field. If disturbed from this
equilibrium orientation, the molecule will oscillate like torsional pendulum. Calculate
the frequency of small oscillations of this kind for a water molecule about an axis
through center of mass (and perpendicular to the plane of the three atoms) when the
molecule is in an electric field of 2.0 × 106 N/C. The moment of inertia of the molecule
about this axis is 1.39 × 10−47 kg.m2 .
2. A uniform electric field points everywhere in the +x direction, so E = Ex i. The
magnitude of the field is independent of the z coordinate and varies with x and y
according to Ex = C(x − 2y), where x and y are in meters and C is a constant with
value C = 3.6 × 102 N/(C.m). For a cube of side 2.0 m at the same position as shown
in Fig. 24.7, what is the flux through the y − z face at x = 1.0 m (side 1)?
Date: 6 September, 2013
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Electricity and Magnetism: PHY-204
Fall Semester 2013
3. Consider a point charge q located at one vertex of a cubical Gaussian surface, just
inside the cube (see Fig. 2). What is the electric flux through each face of the cube?
Fig. 2: Point charge located just inside the
vertex of a cubical; Gaussian surface
4. According to the Thomson model, the atom of helium consists of a uniform spherical
cloud of positive charge within which sit two electrons. Assume that positive cloud is
a sphere of radius 5.0 × 10−11 m with a charge of 2e uniformly distributed over the
volume. The two electrons are symmetrically placed with respect to the center (see
Fig. 3). What is the equilibrium separation of the electrons?
Fig. 3: Thomson model of the helium atom
5. A thick insulating spherical shell of inner radius a and outer radius b has a nonuniform
volume charge density given by ρ(r) = C/r3 , where C is a known constant.
(a) What is the total charge Q in the shell?
(b) Determine the value of electric field for r ≤ a, a ≤ r ≤ b, and r ≥ b.
Date: 6 September, 2013
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Electricity and Magnetism: PHY-204
Fall Semester 2013
6. A long cylinder of radius R has a uniform charge density ρ distributed over its volume.
′
A cylindrical hole of radius R has been drilled parallel to the axis of the cylinder
along its full length, and charge on this hole has been discarded. The axis of the
′
hole is at a distance h > R from the axis of the cylinder (see Fig. 4). Find the
electric field as a function of the radial distance shown in Fig. 4. Consider the cases
′
′
′
′
r ≤ h − R , h − R ≤ r ≤ h + R , h + R ≤ r ≤ R, and r ≥ R separately.
Fig. 4: Cylinder with a cylinderical cavity
Date: 6 September, 2013
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