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 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...

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...

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 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...

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) 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 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) 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) 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...

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...

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...

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...

(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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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 ....

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...

Use the orthogonality properties of the spherical harmonics to prove the following identities for a function (r) which satisfies Laplaces equation in and on an origin-centered spherical surface S of...

Four identical positive point charges sit at (a, a), (a, a), (a,a), and (a,a) in the z = 0 plane. Very near the origin, the electrostatic potential can be written in the form (a) Deduce the non-zero...

The z-axis runs down the center of an infinitely long heating duct with a square cross section. For a real metal duct (not a perfect conductor), the electrostatic potential (x, y) varies linearly...

The Poisson integral formula gives the potential at any point r inside a sphere if we specify the potential (r S ) at every point on the surface of the sphere. Derive this formula by summing the...

Let n be the normal to an equipotential surface at a point P. The principal radii of curvature of the surface at P are R 1 and R 2 . A formula due to George Green relates normal derivatives (/n n )...

For a short time, the 250-kg roller-coaster car with passengers is traveling along the spiral track at a constant speed such that its position measured from the top of the track has components r = 10...

Draw the shear and moment diagrams for the compound beam. The beam is pin connected at E and F. A -L B LILIL W -7 D

Gear A is held fixed, and arm DE rotates clockwise with an angular velocity of DE = 6 rad/s and an angular acceleration of DE = 3 rad/s. Determine the angular acceleration of gear B at the instant...

Pulley A has a weight of 30 lb and a centroidal radius of gyration k B = 0.6 ft. Determine the speed of the 20-lb crate C at the instant s = 10 ft. Initially, the crate is released from rest when s =...

The electric fan is mounted on a swivel support such that the fan rotates about the z axis at a constant rate of z = 1 rad/s and the fan blade is spinning at a constant rate s = 60 rad/s. If at the...

An AISI 1040 cold-drawn steel tube has an outside diameter of 50 mm and an inside diameter of 42 mm. The tube is 150 mm long, and is capped on both ends. An internal pressure of 40 MPa is applied....

A 20-mm-diameter steel shaft, made of AISI 1035 HR steel, transmits power while rotating at 400 rev/min. Assume any bending moments in the shaft to be relatively small compared to the torque....

Write (r r') = (r r')(z z') and use direct integration to derive Weyls formula for the free-space Green function in three dimensions, Go(r, r') = 1 20 S dk__ik(r-r' ) __-k|z-z'l k (27)

Find the free-space Green functionG (d) 0 (r, r') in d = 1, 2, 3 space dimensions by the method of eigenfunction expansion. For d = 2, you will need (i) an integral representation of J 0 (x); (ii)...

For Problem 1217 a satisfactory design is Double the size of the bearing dimensions and quadruple the load to 3600 lbf. Data in Problem 1217 Design a central annular-groove pressure-fed bearing with...

An empty beer can is bounded by the surfaces z = 0, z = h, and = R. By slamming it against his forehead, a frustrated football fan dents the can into the shape shown below. Our interest is the...

Maintain the plane z = 0 at potential V and introduce a grounded conductor somewhere into the space z > 0. Use the magic rule for the Dirichlet Green function to find the charge density (x, y)...

The plane z = 0 is grounded except for an finite area S 0 which is held at potential 0 . Show that the electrostatic potential away from the plane is p(x, y, z) = Polz| 2 So dr' r-r'*

A point electric dipole with moment p sits at the center of a grounded, conducting, spherical shell of radius R. Use the method of images to show that the electric field inside the shell is the sum...

A point charge q is placed at a distance 2R from the center of an isolated, conducting sphere of radius R. The force on q is observed to be zero at this position. Now move the charge to a distance 3R...

The free-space Green function in two dimensions (potential of a line charge) is: Use the method of direct integration to reduce the two-dimensional equation to a one-dimensional equation and...

Suppose that a collection of image point charges q 1 , q 2 , . . . , q N is used to find the force on a point charge q at position r q due to the presence of a conductor held at potential C . Let U...

An infinite slab with dielectric constant = / 0 lies between z = a and z = b = a + c. A point charge q sits at the origin of coordinates. Let = ( 1)/( + 1) and use solutions of Laplaces equation...

Two semi-infinite and grounded conducting planes meet at a right angle as seen edge-on in the diagram. Find the charge induced on each plane when a point charge Q is introduced as shown. 20

The text showed that the attractive force F between an origin-centered, grounded, conducting sphere of radius R and a point charge located at a point s > R on the positive z-axis varies as 1/s 3 when...

A steady current is produced by a collection of moving charges confined to a volume V . Prove that the rate at which work is done on these moving charges by the electric field produced by a static...

A wire with conductivity carries a steady current I. Confirm the statement made in the text that a charge Q = 0 I/ accumulates on the wires surface in the immediate neighborhood of a 90...

Two highly conducting spheres with radii a 1 and a 2 are used to inject and extract current from points deep inside a tank of weakly conducting fluid. Show that the resistance between the spheres...

A battery maintains a potential difference V between the two halves of the cover of a tank (Lh) filled with salty water. Find the current density j(x, y, z) induced in the water. N -L/2 L/2- X h

In 1910, Debye suggested that the work function W of a metal could be computed as the work performed against the electrostatic image force when an electron is removed from the interior of a finite...

(a) Use completeness relations to represent (x x')(y y') and then the method of direct integration for the inhomogeneous differential equation which remains to find the interior Dirichlet Green...

An infinitely long cylindrical conductor carries a constant current with density jz(r). (a) Despite Ohms law, compute the radial electric field Er (r) that ensures that the radial component of the...

An infinite, two-dimensional network has a honeycomb structure with one hexagon edge removed. Otherwise, the resistance of every hexagon edge is r. Find the resistance of the network when a current I...

Consider the vacuum diode problem treated in the text with the space between the plates filled with a poor conductor with dielectric permittivity . For matter of this kind, v = uE, where the mobility...

Show that the lines of current density j obey a law of refraction at the flat boundary between two ohmic media with conductivities 1 and 2. Use the geometry shown below. 0 02 0 10

(a) Use the completeness relation, and the method of direct integration to show that (b) Show that G(r, r') above is identical to the image solution for this problem. YEM (F)YM (P) = lm lm lm 1 sin 0...

The Dirichlet Green function for any finite volume V can always be written in the form (a) Use the physical meaning of the Dirichlet Green function to prove that (b) Use Earnshaws theorem to prove...

(a) A long straight rod with cross sectional area A and conductivity accelerates parallel to its length with acceleration a. Write down the Drude-like equation of motion for the average velocity v...

(a) Derive an integral expression for the charge density (, z) induced on the outer surface of a conducting tube of radius R when a point charge q is placed at a perpendicular distance s > R from the...

Steady current flows in the x-direction in an infinite, two-dimensional strip defined by |y| < L. The current density j is constant everywhere in the strip and the conductivity varies in space as The...

A uniform surface current K = Kz confined to a strip of width b carries a total current I . Find the magnetic field at a point in the plane of the strip that lies a perpendicular distance a from the...

(a) Use the Biot-Savart law to find B(r) everywhere for a current sheet at x = 0 with K = Kz. (b) Check your answer to part (a) by superposing the magnetic field from an infinite number of straight...

The z-axis coincides with the symmetry axis of a flat disk of radius R in the x-y plane. Sketch and justify in words the pattern of currents that must flow in the disk to produce the magnetic field...

A thin membrane with conductivity and thickness separates two regions with conductivity . Assume uniform current flow in the z-direction in the figure above. When is small, it makes sense to seek...

The diagram shows a wire connected to the Earth (conductivity E ) through a perfectly conducting sphere of radius a which is half-buried in the Earth. The layer of earth immediately adjacent to the...

The electrodes of a spherical capacitor have radii a and b > a. The inner electrode is grounded; the outer electrode is held at potential V. In vacuum diode mode, the thermionic current which flows...

A current I flows up the z-axis and is intercepted by an origincentered sphere with radius R and conductivity . The current enters and exits the sphere through small conducting electrodes which...

A square plate of copper metal can be used as a crude variable resistor by making suitable choices of the places to attach leads that carry current to and from the plate. (a) Current enters at A and...

(a) Use superposition and the magnetic field on the symmetry axis of a current ring to find the magnetic field at the midpoint of the symmetry axis of a cylindrical solenoid. The solenoid has radius...