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Let d and s be two unequal lengths. Assume that charge is distributed on the z = 0 plane with a surface density (a) Integrate to find the total charge Q on the plane. (b) Show that the potential (z)... Suppose that the electrostatic potential produced by a point charge was not Coulombic, but instead varied with distance in a manner determined by a specified scalar function f (r): (a) Calculate the... The figure below shows a cube filled uniformly with charge. Determine the ratio 0 / 1 of the potential at the center of the cube to the potential at the corner of the cube. -% S A two-dimensional disk of radius R carries a uniform charge per unit area > 0. (a) Calculate the potential at any point on the symmetry axis of the disk. (b) Calculate the potential at any point on... (a) Use Greens second identity, V d 3 r (f 2 g g 2 f ) = S dS (f g g f), to prove that the potential (0) at the center of a charge-free spherical volume V is equal to the average of (r) over the... (a) Show that the electric field produced by a uniform charge density confined to the volume V enclosed by a surface S can be written (b) Show that the electric field due to an arbitrary but... Suppose the electrostatic potential of a point charge were (r) = (1/4 0 )r (1+) rather than the usual Coulomb formula. (a) Find the potential (r) at a point at a distance r from the center of a... The figure below shows a circular hole of radius b (white) bored through a spherical shell (gray) with radius R and uniform charge per unit area . (a) Show that E(P) = (/2 0 )[1 sin( 0 /2)]r, where... A charge distribution (r) with total charge Q occupies a finite volume V somewhere in the half-space z < 0. If the integration surface is z = 0, prove that z=0 dS 2. E = ala . 20 (a) Find E(r) if (x, y, z) = 0 (x) + 0(x) 0(x b). (b) Show by explicit calculation that (x, y, z) does not exert a net force on itself. The z-axis coincides with the symmetry axis of a flat disk of radius a in the x-y plane. The disk carries a uniform charge per unit area < 0. The rim of the disk carries an additional uniform charge... Show that the torque exerted on a charge distribution (r) by a distinct charge distribution ' (r') is N = 1 471060 [ar [d farfar rxr r'/0 (r)p'(r'). Draw the electric field line pattern for a line of five equally spaced charges with equal magnitude but alternating algebraic signs, as sketched below. Be sure to choose the scale of your drawing and... The Cartesian components of the electric field in a charge-free region of space are E k = C k + D jk r j , where C k and D jk are constants. (a) Prove that D jk is symmetric (D jk = D kj ) and... (a) A point charge q > 0 with total energy E travels through a region of constant potential V1 and enters a region of potential V2 < V1. Show that the trajectory bends so that the angles 1 and 2 in... Use Gauss law to find the electric field when the charge density is: (a) Expresses the answer in Cartesian coordinates. (b) Express the answer in cylindrical coordinates. (c) Express the answer in... If the photon had a mass m, Gauss law with E = changes from 2 = / 0 to an equation which includes a lenght L = /mc: Experimental searches for m use a geometry first employed by Cavendish where a... 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... 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... 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... 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)... 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 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 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 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.... 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 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... 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... 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... 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... 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 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. 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... 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. 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 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) 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 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) 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. 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... 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 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 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... 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... 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 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.... 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... 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... 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... 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...
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