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 )...

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

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...

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)...

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'*

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...

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...

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...

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...

The diagram below shows a rod of length L and net charge Q (distributed uniformly over its length) oriented parallel to a grounded infinite conducting plane at the distance d from the plane. (a)...

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...

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) Let (,) be a solution of Laplaces equation in a cylindrical region < R. Show that the function (,) = (R 2 /,) is a solution of Laplaces equation in the region > R. (b) Show that a suitable...

(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...

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...

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...

Let b be the perpendicular distance between an infinite line with uniform charge per unit length and the center of an infinite conducting cylinder with radius R = b/2. (a) Show that the charge...

(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 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

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...

(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...

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...

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...

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...

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

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 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...

The annulus shown below is cut from a planar metal sheet with thickness t and conductivity . (a) Let V be the voltage between the edge CD and the edge FA. Solve Laplaces equation to find the...

The diagram below shows an ohmic film with conductivity , thickness d, infinite length, and semi-infinite width. A total current I enters the film at the point A through a line contact (modeled as a...

(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) 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...

A circular loop with radius R and current I lies in the x-y plane centered on the z-axis. The magnetic field on the symmetry axis is R B(z) = (R + z)/2. z In cylindrical coordinates, B,(p, z) =...

The diagram below illustrates a reciprocal principle satisfied by an ohmic sample of any shape. The principle asserts that if the impressed currents satisfy I A = I B , the measured voltages satisfy...

A spherical shell with radius a has conductivity in the angular range 1 < < 2 . Otherwise, the shell is perfectly conducting and a potential difference V is maintained between = 0 and = . (a)...

Current flows on the surface of a spherical shell with radius R and conductivity . The potential is specified on two rings as ( = ) = V cos n and ( = ) = V cos n. Show that the rate at which Joule...

(a) Consider a semi-infinite and tightlywound solenoid with a circular cross section. Prove that the magnetic flux which passes out through the open end of the solenoid is exactly one-half the flux...

(a) Two rings of radius R, coaxial with the z-axis, are separated by a distance 2b and carry a current I in the same direction. Make explicit use of the formula for the magnetic field of a single...

Find the surface current density K(,) on the surface of sphere of radius a which will produce a magnetic field inside the sphere of B

A charge Q is uniformly distributed over the surface of a sphere of radius R. The sphere spins at a constant angular frequency with = z. Use B = to find the magnetic field everywhere.

The figure below shows an infinitely long current filament wound in the form of a circular helix with radius R and pitch l , i.e., is the distance along the z-axis occupied by one wind of the helix....

Use the solid angle representation of the magnetic scalar potential (r) to find B(r) everywhere for an infinite, straight line of current I . State carefully the surface you have chosen to cut the...

The conductivity of the Earths atmosphere increases with height due to ionization by solar radiation. At an altitude of about H = 50 km, the atmosphere can be considered practically an ideal...

A current I starts at z = and flows up the z-axis as a linear filament until its hits an origin-centered sphere of radius R. The current spreads out uniformly over the surface of the sphere and flows...

Let B(x, z) be the magnetic field produced by a surface current density K(y, z) = K(z) y confined to the x = x 0 plane. (a) Show that the Biot-Savart law for this situation reduces to a one...

Show that the normal derivative of the Coulomb gauge vector potential suffers a jump discontinuity at a surface endowed with a current density K(r S ).

Biot and Savart derived their eponymous formula using a currentcarrying wire bent as shown below. Find B(r) in the plane of the wire at a distance d from the bend along the axis of symmetry. P a a

The magnetic scalar potential in a volume V is (x, y, z) = (C/2) ln(x 2 + y 2 ). Find a vector potential A = A x x + A y y which produces the same magnetic field.

Consider a charge distribution (r) in rigid, uniform motion with velocity . (a) Show that the magnetic field produced by this system is B(r) = (/c 2 ) E, where E(r) is the electric field produced by...

A cylindrical solenoid with length L and cross sectional area A = R 2 is formed by wrapping n turns per unit length of a wire that carries a current I . Estimate the magnitude of the magnetic field...

(a) Show by direct calculation that the Coulomb gauge condition A = 0 applies to (b) Find the choice of gauge where a valid representation of the vector potential is A(r): Ho 4 dr' j(r) |rr|

A current I 0 flows up the z-axis from z = z 1 to z = z 2 as shown below. (a) Use the Biot-Savart law to show that the magnetic field in the z = 0 plane is (b) Symmetry and the Coulomb gauge vector...

The text describes a Helmholtz coil as two parallel, coaxial, and circular current loops of radius R separated by a distance R. Each loop carries a current I in the same direction. (a) Use the...

A compact disk with radius R and uniform surface charge density rotates with angular speed . Find the magnetic dipole moment m when the axis of rotation is (a) The symmetry axis of the disk; (b) Any...

A quantum particle with charge q, mass m, and momentum p in a magnetic field B(r) = A(r) has velocity (r) = p/m (q/m)A(r). This means that a charge distribution (r) generates a diamagnetic current...

Show that the first non-zero term in an interior Cartesian multipole expansion of the vector potential can be written in the form A(r) = ( 0 /4)G r where G is a constant vector. Show that the...

Find the magnetic moment of a planar spiral with inner radius a and outer radius b composed of N turns of a filamentary wire that carries a steady current I. a b

Use Greens second identity to prove that G D (r, r') = G D (r', r).

Let j (r) be an arbitrary current distribution. (a) Show that the components of the magnetic dipole moment m = 1/ 2 d 3 r r j are invariant to a rigid shift of the origin of coordinates. (b) Show...

(a) Find the vector potential inside and outside a solenoid that generates a magnetic field B = Bz inside an infinite cylinder of radius R. Work in the Coulomb gauge. (b) The A haronov-Bohm effect...

A semi-infinite solenoid (conce ntric with the negative z-axis) has cross sectional area A = R 2 , n turns per unit length of a wire with current I , and magnetic moment per unit length m = nIA. When...

A charge Q is uniformly distributed over the surface of a sphere of radius R. The sphere spins at a constant angular frequency . (a) Show that the current density of this configuration can be written...

The text produced a spherical multipole expansion for the magnetic scalar potential (r) based on the identity A Cartesian expansion for the scalar potential can be developed from the same starting...

A filamentary current loop traverses eight edges of a cube with side length 2b as shown below. (a) Find the magnetic dipole moment m of this structure. (b) Do you expect a negligible or a...

The text writes two expressions for the magnetic field of point magnetic dipole at the origin: 0 (r) is the magnetic scalar potential of a point dipole. Prove that the delta function content of...

A voltage difference V 0 causes a steady current to flow from the top conductor to the bottom conductor (in the sketch below) through an ohmic medium with conductivity . Find an approximate...

(a) Let I be the current carried by a wire bent into a planar loop. Place the origin of coordinates at an observation point P in the plane of the loop. Show that the magnitude of the magnetic field...

If such a thing existed, the magnetic field of a point particle with magnetic charge g at rest at the origin would be B mono (r) = ( 0 gr/(4r 3 ). Show that the magnetic field of a point magnetic...

A particle with mass m and charge q moves non-relativistically in static fields E(r) and B(r). Show that a re-scaling of the magnetic field and the time is sufficient for a particle with mass M and...

Two finite-length, concentric, cylindrical solenoids carry current in the same direction. The outer solenoid is very slightly longer and has a very slightly larger radius than the inner solenoid, but...

Two origin-centered circular rings have radii a and b aand I b . A narrow insulating rod coincident with their common diameter permits the smaller ring to rotate freely inside the larger ring. Show...

Show that the formula for the magnetic dipole moment derived in Example 11.1, is consistent with the spherical multipole expansion of the vector potential derived in Section 11.4, Example 11.1 Let...

An axial electric field E = Ez and a radial magnetic field B = Bp coexist in the volume V between two short cylindrical shells concentric with the z-axis. Suppose V is filled with xenon gas and, at a...

Two point dipoles m 1 and m 2 on the x-axis are separated by a distance R and misaligned from the positive x-axis by small angles and as shown below. A uniform agnetic field B points along the...

Two identical point dipoles m = mz sit rigidly at (a, 0, 0). A third point dipole M is free to rotate at its fixed position (0, y, z). Find the Cartesian components of M which correspond to stable...

Two infinite, straight, parallel wires, each carrying current I in the same direction, are coincident with the lines (1, 0, z) and (1, 0, z). In addition, there is a large external field B 0 = B 0 z....

Let a steady current I flow up the y-axis and let the initial position and velocity of a particle with mass m and charge q be r 0 = (x 0, 0, 0) and v 0 = (0, 0 , 0).(a) Show that the motion of the...

Two long, parallel wires of length L, separation d, and cross sectional radius a are connected by a U-turn at each end to form a closed circuit. Insert a battery with potential difference V at one...