A point particle with charge q and mass m is fixed at the origin. An identical particle is released from rest at x = d. Find the asymptotic (x ) speed of the released particle.

If is a real constant, the continuity equation is satisfied by the charge and current distributions. The given j represents current flowing in toward the origin of coordinates. But the given is...

An infinitely long cylindrical solenoid carries a spatially uniform but time-dependent surface current density K(t) = K 0 (t/). K 0 and are constants. Find the electric and magnetic fields...

In 1942, Boris Podolsky proposed a generalization of electrostatics that eliminates the divergence of the Coulomb field for a point charge. His theory retains E = 0 but replaces Gauss law by (a)...

A surface current density K(r S , t) flows in the z = 0 plane which separates region 1 (z > 0) from region 2 (z < 0). Each region contains arbitrary, time-dependent distributions of charge and...

Let be a parameter and define new electric and magnetic field vectors as linear combinations of the usual electric and magnetic field vectors: E' = E' cos + cBsin cB' = E sin + cB cos . (a) Show...

Consider a collection of point particles fixed in space with charge density is : Suppose that : E(r, t = 0) = B(r, t = 0) = 0 and (a) Construct a simple current density which satisfies the continuity...

A particle with charge q is confined to the x-y plane and sits at rest somewhere away from the origin until t = 0. At that moment, a magnetic field B(x, y) = (x)(y) turns on with a value of which...

The magnetostatic equation B = 0 j is not consistent with conservation of charge for a general time-dependent charge density. Show that consistency can be achieved using B = 0 j + j D and a...

The electric and magnetic fields for time-independent distributions of charge and current which go to zero at infinity are (a) Calculate E and E. (b) Calculate B and B. The curl calculation...

Let F 1 and F 2 be the instantaneous forces that act on a particle with charge q when it moves through a magnetic field B(r) with velocities 1 and 2 , respectively. Without choosing a coordinate...

Let r 1 (r 2 ) point to a line element ds 1 (ds 2 ) of a closed loop C 1 (C 2 ) which carries a current I 1 (I 2 ). Experiment shows that the force exerted on I 1 by I 2 is (a) Show that (b) Use (a)...

Let S be the surface that bounds a volume V . Show that (a) s dS = 0; (b) 1/3 s dS r = V.

Compute the unit normal vector n to the ellipsoidal surfaces defined by constant values of Check that you get the expected answer when a = b = c. (x, y, z) = V + 6 2

The buckets on the conveyor travel with a speed of 15 ft/s. Each bucket contains a block which falls out of the bucket when = 120. Determine the distance s to where the block strikes the conveyor....

Particles A and B are traveling around a circular track at a speed of 8 m/s at the instant shown. If the speed of B is increasing by (a t ) B = 4 m/s 2 , and at the same instant A has an increase in...

The boat is originally traveling at a speed of 8 m/s when it is subjected to the acceleration shown in the graph. Determine the boats maximum speed and the time t when it stops. - t(s) 1 24 9 + 1//3...

Point charges q 1 , q 2 , . . ., q N are embedded in a body with permittivity in . The latter is itself embedded in a body with permittivity out . Find the total polarization charge Q pol induced...

The text proved that the force on an isolated dielectric is Where E(r) is the total field at an interior point r and E avg (r S ) is the average of the total field just inside and just outside the...

Write the Helmholtz theorem expression for D(r) and eliminate D itself from the integrals you write down. How does this formula simplify (if at all) for simple dielectric matter?

A metal ball with charge Q sits at the center of a thin, spherical, conducting shell. The shell has charge Q' and the space between the shell and the ball is filled with matter with dielectric...

A dielectric body with permittivity in is embedded in an infinite volume of dielectric matter with permittivity out . The entire system is polarized by an external electric field E ext . If is the...

The parallel-plate capacitor shown below is made of two identical conducting plates of area A carrying charges q. The capacitor is filled with a compressible dielectric solid with permittivity and...

An origin-centered sphere with permittivity and radius a is placed in a uniform external electric field E 0 . What radius b < a should an origin-centered, perfectly conducting sphere similarly...

Application 6.3 modeled a meson (a quark-antiquark pair) as a finite dipole placed at the center of a spherical cavity with radius R and unit dielectric constant scooped out of an infinite medium...

A uniform electric field E 0 exists throughout a homogeneous dielectric with permittivity . What is the electric field inside a vacuum cavity cut out of the interior of the dielectric in the shape of...

(a) The entire volume between two concentric spherical shells is filled with a material with uniform polarization P. Find E(r) everywhere. (b) The entire volume inside a sphere of radius R is filled...

A polarizable sphere of radius R is filled with free charge with uniform density c . The dielectric constant of the sphere is . (a) Find the polarization P(r). (b) Confirm explicitly that the total...

The electrostatic polarization inside an origin-centered sphere is P(r) = P(r). (a) Show that (r) outside the sphere is equal to the potential of a point electric dipole at the origin with a moment...

A cube is polarized uniformly parallel to one of its edges. Show that the electric field at the center of the cube is E(0) = P/3 0 . Compare with E(0) for a uniformly polarized sphere.

The polarization in all of space has the form P = P(r R)r, where P and R are constants. Find the polarization charge density and the electric field everywhere.

Students are often told that E = F q /q defines the electric field at a point if F q is the measured force on a tiny charge q placed at that point. More careful instructors let q 0 to avoid the...

A semiconductor with permittivity occupies the space z 0. One dopes such a semiconductor by implanting neutral, foreign atoms with uniform density N D in the near-surface region 0 z d. Assume...

An air-gap capacitor with parallel-plate area A discharges by the electrical breakdown of the air between its parallel plates (separation d) when the voltage between its plates exceeds V 0 . Lay a...

Find the total electrostatic energy of a ball with radius R and uniform polarization P.

Find a polarization P(r) which produces a polarization charge density in the form of an origin-centered sphere with radius R and uniform volume charge density P .

Two spheres with radius R have uniform but equal and opposite charge densities . The centers of the two spheres fail to coincide by an infinitesimal displacement vector . Show by direct superposition...

A conducting shell of radius R has total charge Q. If sawed in half, the two halves of the shell will fly apart. This can be prevented by placing a point charge Q' at the center of the shell. (a)...

(a) A spherical metal shell is charged to an electrostatic potential V . Cut this shell in half and pull the halves infinitesimally apart. Find the force with which one hemisphere of the shell repels...

A battery maintains the potential difference V between the spheres of a spherical capacitor with capacitance C. Move the center of the inner sphere away from the center of the outer sphere by an...

A long, straight wire has length L and a circular cross section with area a 2 . Arrange two such wires so they are parallel and separated by a distance d. You may assume that L >> d >> a and ignore...

Two pyramid-shaped conductors each carry a net charge Q. (a) Transfer charge Q from pyramid 2 to pyramid 1. Derive a condition on the coefficients of potential P ij which guarantees that this charge...

A non-conducting square has a fixed surface charge distribution. Make a rectangle with the same area and total charge by cutting off a slice from one side of the square and gluing it onto an adjacent...

A point electric dipole with moment p is placed at the center of a hollow spherical cavity scooped out of an infinite conducting medium. (a) Find the surface charge density induced on the surface of...

Let d be the separation between two infinite, parallel, perfectly conducting plates. The lower surface of the upper plate has charge per unit area 1 > 0. The upper surface of the lower plate has...

A point dipole p is placed at r = r 0 outside a grounded conducting sphere of radius R. Use Greens reciprocity (and a comparison system with zero volume charge density) to find the charge drawn up...

A grounded metal plate is partially inserted into a parallel-plate capacitor with potential difference 2 1 > 0 as shown in the diagram below. Find the elements of the capacitance matrix. Assume...

A conducting disk of radius R held at potential V sits in the x-y plane centered on the z-axis. (a) Use the charge density for this system calculated in the text to find the potential everywhere on...

The square region defined by a x a and a y a in the plane z = 0 is a conductor held at potential = V. The rest of the plane z = 0 is a conductor held at potential = 0. The plane z = d is also a...

Three concentric spherical metallic shells with radiic > b > a have charges e c , e b , and e a , respectively. Find the change in potential of the outermost shell when the innermost shell is...

A capacitor is formed from three very long, concentric, conducting, cylindrical shells with radii a < b < c. Find the capacitance per unit length of this structure if a fine wire connects the inner...

Let C be the capacitance of capacitor formed from two identical, flat conductor plates separated by a distance d. The plates have area A and arbitrary shape. When d < < A, we know that the...

(a) What is the self-capacitance (in farads) of the Earth? How much energy is required to add one electron to the (neutral) Earth? (b) What is the self-capacitance (in farads) of a conducting...

Four identical conducting balls are attached to insulating supports that sit on the floor as shown below. One ball has charge Q; its support is fixed in space. The other three balls are uncharged but...

The text derived Greens reciprocity theorem for a set of conductors as a special case of a more general result. For conductors with charges and potentials (q k , K ) and the same set of conductors...

A metal ball with radius R 1 has charge Q. A second metal ball with radius R 2 has zero charge. Now connect the balls together using a fine conducting wire. Assume that the balls are separated by a...

A spherical conducting shell with radius b is concentric with and encloses a conducting ball with radius a. Compute the capacitance C = Q/ when (a) the shell is grounded and the ball has charge Q....

A solid conductor has a vacuum cavity of arbitrary shape scooped out of its interior. Use Earnshaws theorem to prove that E = 0 inside the cavity.

Two infinite conducting planes are held at zero potential at z = d and z = d. An infinite sheet with uniform charge per unit area is interposed between them at an arbitrary point. (a) Find the...

A research paper published in the journal Applied Physics Letters describes experiments performed with three identical spherical conductors suspended from above by insulating wires so a (fictitious)...

A point charge q lies a distancer > R from the center of an uncharged, conducting sphere of radius R. Express the induced surface charge density in the form where is the polar angle measured from a...

(a) Evaluate the exterior spherical multipole moments for a shell of radius R which carries a surface charge density (, ) = 0 sin cos . (b) Write (r < R,,) in the form (x, y, z, r). (c) Evaluate...

A charge distribution is contained entirely inside a black box. Measurements of the electrostatic potential outside the box reveal that all of the exterior multipole moments for = 1, 2, . . . are...

How does the leading contribution to the electrostatic interaction energy between two nitrogen molecules depend on the distance R between them?

Six point charges form an ideal hexagon in the z = 0 plane as shown below. The absolute values of the charges are the same, but the signs of any two adjacent charges are opposite. (a) What is the...

A soap bubble (an insulating, spherical shell of radius R) is uniformly coated with polar molecules so that a dipole double layer with = r forms on its surface. Find the potential at every point in...

Show that dW = E(r) dp is the work increment required to assemble a point electric dipole with moment dp at the point r beginning with charge dispersed at infinity.

Place a point electric dipole p = p z at the origin and release a point charge q (initially at rest) from the point (x 0 , y 0 , 0 ) in the x-y plane away from the origin. Show that the particle...

(a) Show that the charge density of a point quadrupole is (r) = Q ij i j (r r 0 ). (b) Show that the force on a point quadrupole in a field E(r) is Q ij i j E(r 0 ). (c) Show that the torque on...

(a) Let (R, , ) be specified values of the electrostatic potential on the surface of a sphere. Show that the general form of an exterior, spherical multipole expansion implies that b) The eight...

The z-axis is the symmetry axis of a disk of radius R which lies in the x-y plane and carries a uniform charge per unit area . Let Q be the total charge on the disk. (a) Evaluate the exterior...

(a) Show that the potential due to a double-layer surface S with a dipole density (r S ) n is where d is the differential element of solid angle as viewed from r. (b) Use this result to derive the...

The low-energy Born approximation to the amplitude for electron scattering from a neutron is proportional to the volume integral of the potential energy of interaction between the electron and the...

The diagram shows two identical, charge-neutral, origin centered disks. One disk lies in the x-z plane. The other is tipped away from the first by an angle around the z-axis. The charge density of...

Two coplanar dipoles are oriented as shown in the figure below. Find the equilibrium value of the angle ' if the angle is fixed. P Zo 0 P '0'

Identical point electric dipoles are placed at the vertices of the regular polyhedra shown below. All the dipoles are parallel but the direction they point is arbitrary. Show that the electric field...

This problem exploits the ring and disk electric fields calculated Example 2.1. (a) Find E(r) inside and outside a uniformly charged spherical shell by superposing the electric fields produced by a...

A spherical charge distribution 1 (r) has total charge Q 1 and a second, non-overlapping spherical charge distribution 2 (r) has total charge Q 2 . The distance between the centers of the two...

The Potential of a Charged Line Segment The line segment from P to P' in the diagram below carries a uniform charge per unit length . The vector a is coincident with the segment. The vectors c and b...

A surprisingly realistic microscopic model for the charge density of a semi-infinite metal (with z = 0 as its macroscopic surface) consists of a positive charge distribution And a negative charge...

A model hydrogen atom is composed of a point nucleus with charge +|e| and an electron charge distribution Show that the ionization energy (the energy to remove the electronic charge and disperse it...

Give a physical realization of the electrostatic boundary value problem whose solution is provided by the complex potential. f(w) = i V + V 2 + V - V 2 R+iw In [R+ R-iw

Let V (z) be the potential on the axis of an axially symmetric electrostatic potential in vacuum. Show that the potential at any point in space is V(p, z) = 1 fd 0 de V(z+ip cos ().

Let the z-axis be the symmetry axis for an infinite number of identical rings, each with charge Q and radius R. There is one ring in each of the planes z = 0, z = b, z = 2b, etc. Exploit the Fourier...

The figure below is a cross section of an infinite, conducting cylindrical shell. Two infinitesimally thin strips of insulating material divide the cylinder into two segments. One segment is held at...

A spherical shell of radius R is divided into three conducting segments by two very thin air gaps located at latitudes 0 and 0 . The center segment is grounded. The upper and lower segments are...

The x > 0 half of a conducting plane at z = 0 is held at zero potential. The x < 0 half of the plane is held at potential V . A tiny gap at x = 0 prevents electrical contact between the two halves....

Two wedge-shaped dielectrics meet along the ray = 0. The opposite edge of each wedge is held at a fixed potential by a metal plate. The system is invariant to translations perpendicular to the...

Consider a parallel-plate capacitor with circular plates of radius a separated by a distance 2L. A paper published in 1983 proposed a solution for the potential for this situation of the form Where J...

A set of known constants n parameterizes the potential in a volume r < a as Let z point along = 0 and insert a solid conducting sphere of radius R < a at the origin. Show that the force exerted on...

A conducting sphere with radius R and charge Q sits at the origin of coordinates. The space outside the sphere above the z = 0 plane has dielectric constant 1 . The space outside the sphere below...

The figure below shows an infinitely long cylindrical shell from which a finite angular range has been removed. Let the shell be a conductor raised to a potential corresponding to a charge per unit...

Find the volume charge density and surface charge density which must be placed in and on a sphere of radius R to produce a field inside the sphere of: E = 18 + 1/8 (1 - 138 - 1/0 2. Vo Vo x ) R3...

The figure shows an infinitely long and deep slot formed by two grounded conductor plates at x = 0 and x = a and a conductor plate at z = 0 held at a potential 0 . Find the potential inside the slot...

Two semi-infinite, hollow cylinders of radius R are coaxial with the z-axis. Apart from an insulating ring of thickness d 0, the two cylinders abut one another at z = 0 and are held at potentials V...

A spherical conducting shell centered at the origin has radius R 1 and is maintained at potential V 1 . A second spherical conducting shell maintained at potential V 2 has radius R 2 > R 1 but is...

A capacitor is formed by the infinite grounded plane z = 0 and an infinite, solid, conducting cone with interior angle /4 held at potential V. A tiny insulating spot at the cone vertex (the origin of...

Confirm Poisson formula (derived in Section 6.3) for the case when the volume V is a rectangular slab which is infinite in the x and y directions and occupies the interval t z t otherwise. Keep the...

Two flat conductor plates (infinite in the x- and y-directions) occupy the planes z = d. The x > 0 portion of both plates is held at = + 0 . The x < 0 portion of both plates is held at = 0 ....

The square region defined bya x a anda y a in the z = 0 plane is a conductor held at potential = V . The rest of the z = 0 plane is a conductor held at potential = 0. The plane z = d is also a...