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...

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...

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 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...

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 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...

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...

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 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) =...

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....

(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...

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...

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...

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 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 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 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...

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...

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...

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...

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).

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...

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...

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...

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...

(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...

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...

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....

The half-space z > 0 has uniform magnetization M = Mz. The half-space z < 0 has uniform magnetization M = + Mz. Find the magnetic field B at every point in space using (a) The method of magnetization...

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...

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...

Consider a pistoncylinder assembly containing 0.125 kg of wet steam with a quality of 0.45 at 300 kPa (state 1). Energy is added to the steam at constant temperature by a heat interaction until the...

Consider a pistoncylinder assembly containing 0.85 kg of air initially at 400 K and 620 kPa (state 1). The piston such that the final pressure of the air is 300 kPa. The process occurs constant...

Consider a pistoncylinder assembly containing 0.10 kg of N 2 initially at 425 K and 150 kPa (state 1). The piston moves such that the final pressure of the air is 350 kPa. The process occurs at...

Two identical, current-carrying rectangular loops are oriented at right angles, one in the vertical x-y plane, one in the horizontal x-z plane. The horizontal loop moves infinitesimally slowly from z...

Let (r) be an arbitrary scalar function. A magnetic field which satisfies B = B is called force-free because the Lorentz force density j(r) B(r) vanishes everywhere. There is some evidence that...

The chemical diagnostic tool of nuclear magnetic resonance uses a static magnetic field B 0 = B 0 z and a small-amplitude radio-frequency magnetic field B 1 (t) to orient and manipulate nuclear spins...

The diagram shows three small magnetic dipoles at the vertices of an equilateral triangle. Moments m B and m C point permanently along the internal angle bisectors. Moment m A is free to rotate in...

A superconducting sphere (radius R) placed in a uniform magnetic field B spontaneously generates currents on its surface which produce a dipole magnetic field. The field is equivalent to that...

Consider a pistoncylinder assembly containing 0.09 kg of air initially at 600 K and 350 kPa (state 1). The piston moves such that 18 kJ of work is done by the air on the surroundings. The process...

Consider a pistoncylinder assembly containing 0.08 kg of air initially at 950 K and 400 kPa (state 1). The piston moves such that 30 kJ of work is done by the air on the surroundings. The process...

Consider a pistoncylinder assembly containing 0.14 kg of air initially at 500 K and 250 kPa (state 1). The piston moves such that 5 kJ of work is done on the air by the surroundings. The process...

The Levitron TM is a toy in which a spinning magnetic top floats stably in air above a magnetized base. The potential energy of this system is V B (r) = Mgz m B(r) if we model the top as a point...

A filamentary wire carries a current along the positive z-axis in the presence of a constant magnetic field B = B 0 x + B'z. (a) Find the straight line in space along which the magnitude of the total...

(a) Use an energy method to determine how much current must be passed through the coil if it is to remain unstretched when a mass m is hung from its bottom end. Neglect stray fields at the ends of...

Consider a collection of current loops where Using the notation of the text for variables that are to be held constant in partial derivatives, prove that the magnetic force exerted on the ith loop...

A circular loop of radius R carries a current I 1 . A straight wire with a current I 2 in the plane of the loop passes a distance d from the loop center. (a) Use an energy method to show that the...

Consider a pistoncylinder assembly containing 0.25 kg of air initially at 750 K and 200 kPa (state 1). The piston moves such that 20 kJ of work is done on the air by the surroundings. The process...

Consider a pistoncylinder assembly containing 0.10 kg of air initially at 300 K and 300 kPa (state 1). The piston moves such that 35.9 kJ of work is done on the air by the surroundings. The process...

A straight wire carries a current I 1 down the symmetry axis of a toroidal solenoid with a rectangular cross section of area ab. The solenoid has inner radius R and is composed of N turns of a wire...

The figure shows two square, current-carrying loops with side length a and center-to-center separation c. The currents I 1 and I 2 circulate in the same direction. (a) Evaluate Neumanns integral and...

(a) Show that the Helmholtz theorem representation of the magnetization M(r) is equivalent to the equation B M = 0 (H M + M). (b) The figure shows a thin film (infinite in the x and y directions)...

The stepped rod shown in Figure P573 is subjected to a direct tensile force that varies from 8500 to 16 000 lb. If the rod is made from SAE 1340 OQT 700 steel, compute the resulting design factor....

The right leg of an athlete on a rowing machine can be modeled as a linkage as shown, where A represents the ankle (which is stationary), K the knee, and H the hip. At the instant when = 75, the...

Gear A has a mass of 1 kg and a radius of gyration of 30 mm; gear B has a mass of 4 kg and a radius of gyration of 75 mm; gear C has a mass of 9 kg and a radius of gyration of 100 mm. The system is...

The pendulum consists of two rods: AB is pin supported at A and swings only in the y-z plane, whereas a bearing at B allows the attached rod BD to spin about rod AB. At a given instant, the rods have...

The floor of a light storage warehouse is made of 6-in.-thick cinder concrete. If the floor is a slab having a length of 10 ft and width of 8 ft, determine the resultant force caused by the dead load...

A soaked sponge is experiencing dry air flowing over its surface. The air is at 1 atm and zero relative humidity. Determine the difference in the air temperature and the surface temperature of the...

Ethyl alcohol is spilt and spread over a flat table into a layer of approximately 1 m in diameter wide. Dry air at 25C and 1 atm flows over the ethyl alcohol surface with a free stream velocity of 1...

Steam at 5 bar, 300 C is condensed to saturated liquid in at 5 bar in a heat exchanger at a mass flow rate of 1 kg/s. Cooling is provided by water, which enters the heat exchanger at 20 C 1 bar and...

The decomposition of nitrous oxide is found to proceed as follows: What is the order of this reaction with respect to N,O, and overall? +21/20 NO N + k[NO] -N0 = 1+ k[NO]

A 1100-K n-nonane thermally cracks (breaks down into smaller molecules) 20 times as rapidly as at 1000 K. Find the activation energy for this decomposition.

The pyrolysis of ethane proceeds with an activation energy of about 300 kJ/mol. How much faster is the decomposition at 650C than at 500C?

For a gas reaction at 400 K the rate is reported as (a) What are the units of the rate constant? (b) What is the value of the rate constant for this reaction if the rate equation is expressed as dp A...

A certain reaction has a rate given by If the concentration is to be expressed in mollliter and time in hours, what would be the value and units of the rate constant? -TA = 0.005 C, mol/cm. min

For the enzyme-substrate reaction , the rate of disappearance of substrate is given by What are the units of the two constants? -TA 1760[A][E] 6+ CA mol/m.s

Snake-Eyes Magoo is a man of habit. For instance, his Friday evenings are all alike-into the joint with his week's salary of $180, steady gambling at "2-up" for two hours, then home to his family...

The following four problems consider an isothermal single-phase flow reactor operating at steady-state and constant pressure. Given a gaseous feed, C A0 = C B 0 = 100, A + 2B R , C B = 20. Find X A...

In an isothermal batch reactor 70% of a liquid reactant is converted in 13 min. What space-time and space-velocity are needed to effect this conversion in a plug flow reactor and in a mixed flow...