Flux, Gauss' law

Flux

Problem:

A disk with radius r = 0.10 m is oriented with its normal unit vector n at an angle of 30o to a uniform electric field E with magnitude 2.0*103 N/C.
(a)  What is the electric flux through the disk?
(b)  What is the flux through the disk if it is turned so that its normal is perpendicular to E?
(c)  What is the flux through the disk if its normal is parallel to E?

Solution:


Gauss' law, spherical symmetry

Problem:

A solid conducting sphere of radius 2 cm has a charge of 8 microCoulomb.  A conducting spherical shell of inner radius 4 cm and outer radius 5 cm is concentric with the solid sphere and has a charge of -4 microCoulomb.
(a)  What is the magnitude and direction of the electric field at r = 1 cm?
(b)  What is the magnitude and direction of the electric field at r = 3 cm?
(c)  What is the magnitude and direction of the electric field at r = 4.5 cm?
(d)  What is the magnitude and direction of the electric field at r = 7 cm?

Similar problem:

Problem:

Inside a sphere of radius R and uniformly charged with the volume charge density ρ, there is a neutral spherical cavity of radius R1 with its center a distance a from the center of the charged sphere.  If (R1 + a) < R, find the electric field inside the cavity.

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Solution:

Problem:

A charge distribution produces an electric field E = A(1 - exp(-βr))(r/r3) where A and β are constants.  Find the net charge within the radius r = 1/β.

Solution:

Problem:

Determine the charge distribution that will give rise to the potential V(r) = kq exp(-mr)/r, with m a positive constants.  Calculate the total charge in the distribution.

Solution:

Problem:

(a)  Consider a non-conducting sphere with radius a.  This sphere carries a net charge Q, assumed to be uniformly distributed.  Find the electric field inside and outside the sphere.  Sketch the result.
(b)  Now consider a conducting sphere with radius a carrying a net charge Q.  Find the electric field inside and outside the sphere.

Solution:

Problem:

A sphere of radius R has volume charge density ρ = Krn, for some constants K and n. The region r > R is filled with a conductor (all the way to infinity).
(a)  Find the volume charge density ρ in the region r > R, inside the conductor, and the surface charge density at r = R.
(b)  Find the electric field E everywhere, i.e. for r < R and for r > R.
(c)  Find the potential Φ everywhere, taking Φ to vanish at infinity.
(d)  How much energy is stored in this system?

Problem:

Two metallic spheres of the same radius r are immersed in a homogeneous liquid with resistivity ρ.  What is the total resistance between two spheres?  Assume that the distance between two spheres is much larger than the sphere radius.

Solution:

Problem:

A charge Q is uniformly distributed through the volume of a sphere of radius R.  Calculate the electrostatic energy stored in the resulting electric field.

Solution:

Problem:

A model of the hydrogen atom was proposed before the advent of quantum mechanics, which consists of a single electron of mass m and an immobile uniform spherical distribution of positive charge with radius R.  Assume that the positive charge interacts with the electron via the usual Coulomb interaction but otherwise does not offer any resistance to the motion of the electron.
(a)  Explain why the electron’s equilibrium position is at the center of the positive charge.
(b)  Show that the electron will undergo simple harmonic motion if it is displaced a distance d < R away from the center of the positive charge.  Calculate its frequency of oscillation.
(c)  How big would the atom need to be in order to emit red light with a frequency of 4.57*1014 Hz?  Compare your answer with the radius of the hydrogen atom.
(d)  If the electron is displaced a distance d > R from the center, will it oscillate in position?  Will it undergo simple harmonic motion?  Explain!

Solution:

Problem:

A spherical charge distribution is given by
ρ = ρ0(1 - r/a),    r < a,
ρ = 0,    r > a.
(a)  Calculate the total charge Q.
(b)  Find the electric field and potential for r > a.
(c)  Find the electric field and potential for r < a.
(d)  Find the electrostatic energy of this charge distribution.

Solution:

Similar problem


Gauss' law, cylindrical symmetry

Problem:

A 500 m length of high-voltage cable is undergoing electrical testing.  The cable consists of two coaxial conductors, the inner of 5 mm diameter and the outer of 9 mm internal diameter.  The space between the conductors is filled with polythene which has a relative permittivity of 2 and which can withstand electric field strength of 60 MVm-1.
(a)  Find the maximum voltage which can be applied between the conductors and the energy stored in the cable at this voltage.
(b)  If the cable is to be discharged to a safe level of 50 V in 1 minute, what value of resistance must be connected across it?  What is the maximum power and the total energy dissipated in the resistance?

Solution:

Problem:

A plasma is generated inside a long hollow cylinder of radius R.  It has the charge distribution
ρ(r) = ρ0/(1 + (r/a)2)2,
where r is the distance to the center, and ρ0 and a are constants.
(a)  What is the electric field inside and outside the cylinder?
(b)  Setting V(r=0) = 0, find the potential at all points r < R.
(c)  What are the equilibrium positions of a particle with charge q placed inside the cylinder, assuming the charge does not alter ρ(r).  What is the force acting on the particle if it is displaced by a distance ε << a from an equilibrium position.  Are the equilibrium positions stable? 

Solution:


Gauss' law, planar symmetry

Problem:

Consider an infinite plane with a uniform charge density σ located at z = 0.
(a)  Using Gauss' law, find the electric field created by this plane.
(b)  Find the potential Φ(z).
(c)  Locate another plane with charge density -σ at z = d.  Find the potential Φ(z) everywhere.
What is the magnitude of the potential jump across the dipolar layer configurations of the two planes?
(d)  Find the force per unit area between the planes.

Solution:

Problem:

A charge Q is placed a distance D from an infinite slab of non-conducting material with charge density ρ and thickness d.  What is the force on the charge?

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Solution: