A conical surface (an empty ice-cream cone) carries a uniform surface charge σ. The height of the cone is h, as is the radius of the top. Find the potential difference between points a (the vertex) and b (the center of the top).
> The purpose of this problem is to explain what causes the dramatic white cloud pattern generated in the flow field over the F/A-18C Hornet shown on the cover of this book. This problem is both a tutorial and a quantitative calculation involving the reade
> Consider a Mach 3 flow at 1 atm pressure initially moving over a flat horizontal surface. The flow then encounters a 20 degree expansion corner, followed by a 20 degree compression corner that turns the flow back to the horizontal. Calculate the pressure
> Consider an oblique shock wave with a wave angle of 30◦ in a Mach 4 flow. The upstream pressure and temperature are 2.65*104 N/m2 and 223.3 K, respectively (corresponding to a standard altitude of 10,000 m). Calculate the pressure, temperature, Mach numb
> Repeat Problem 9.18, except with θ=30◦. Again, we will use these results to compare with a quasi-one-dimensional calculation in Problem 10.16. The reason for repeating this calculation is to examine the effect of the much more highly two-dimensional flow
> Consider a two-dimensional duct with a straight horizontal lower wall, and a straight upper wall inclined upward through the angle θ=3◦. The height of the duct entrance is 0.3 m. A uniform horizontal flow at Mach 2 enters the duct and goes through a Pran
> Consider the supersonic flow over a flat plate at an angle of attack, as sketched in Figure 9.35. As stated in Section 9.7, the flow direction downstream of the trailing edge of the plate, behind the trailing edge shock and expansion waves, is not precis
> Consider a circular cylinder (oriented with its axis perpendicular to the flow) and a symmetric diamond-wedge airfoil with a half-angle of 5◦ at zero angle of attack; both bodies are in the same Mach 5 freestream. The thickness of the airfoil and the dia
> Consider sonic flow. Calculate the maximum deflection angle through which this flow can be expanded via a centered expansion wave.
> Consider the Space Shuttle during its atmospheric entry at the end of a mission in space. At the altitude where the Shuttle has slowed to Mach 9, the local heat transfer at a given point on the lower surface of the wing is 0.03 MW/m2. Calculate the norma
> Consider a diamond-wedge airfoil such as shown in Figure 9.36, with a half-angle ε=10◦. The airfoil is at an angle of attack α=15◦ to a Mach 3 freestream. Calculate the lift and wave-drag coefficients for the airfoil.
> Consider an infinitely thin flat plate at an angle of attack α in a Mach 2.6 flow. Calculate the lift and wave-drag coefficients for (a) α=5◦ (b) α=15◦ (c) α=30◦ (Note: Save the results of this problem for use in Chapter 12.)
> A supersonic flow at M1=3, T1=285 K, and p1=1 atm is deflected upward through a compression corner with θ=30.6◦ and then is subsequently expanded around a corner of the same angle such that the flow direction is the same as its original direction. Calcul
> A supersonic flow at M1=1.58 and p1=1 atm expands around a sharp corner. If the pressure downstream of the corner is 0.1306 atm, calculate the deflection angle of the corner.
> Consider the supersonic flow over an expansion corner, such as given in Figure 9.25. The deflection angle θ=23.38◦. If the flow upstream of the corner is given by M1=2, p1=0.7 atm, T1=630◦R, calculate M2, p2, T2, ρ2, p0,2, and T0,2 downstream of the corn
> A slender missile is flying at Mach 1.5 at low altitude. Assume the wave generated by the nose of the missile is a Mach wave. This wave intersects the ground 559 ft behind the nose. At what altitude is the missile flying?
> The entropy increase across a normal shock wave is 199.5 J/(kg. K). What is the upstream Mach number?
> The pressure upstream of a normal shock wave is 1 atm. The pressure and temperature downstream of the wave are 10.33 atm and 1390 ◦R, respectively. Calculate the Mach number and temperature upstream of the wave and the total temperature and total pressur
> The flow just upstream of a normal shock wave is given by p1=1 atm, T1=288 K, and M1=2.6. Calculate the following properties just downstream of the shock: p2, T2, ρ2, M2, p0,2, T0,2, and the change in entropy across the shock.
> Consider the isentropic flow over an airfoil. The freestream conditions correspond to a standard altitude of 10,000 ft and M∞=0.82. At a given point on the airfoil, M=1.0. Calculate p and T at this point. (Note: You will have to use the standard atmosphe
> Consider a flat plate at zero angle of attack in a hypersonic flow at Mach 10 at standard sea level conditions. At a point 0.5 m downstream from the leading edge, the local shear stress at the wall is 282 N/m2. The gas temperature at the wall is equal to
> Consider the isentropic flow through a supersonic nozzle. If the test-section conditions are given by p=1 atm, T=230 K, and M=2, calculate the reservoir pressure and temperature.
> At a given point in a flow, T=700 ◦R, p=1.6 atm, and V=2983 ft/s. At this point, calculate the corresponding values of p0, T0, p∗, T∗, and M∗.
> At a given point in a flow, T=300 K, p=1.2 atm, and V=250 m/s. At this point, calculate the corresponding values of p0, T0, p∗, T ∗, and M ∗.
> The temperature in the reservoir of a supersonic wind tunnel is 519 ◦R. In the test section, the flow velocity is 1385 ft/s. Calculate the test-section Mach number. Assume the tunnel flow is adiabatic.
> Prove that the total pressure is constant throughout an isentropic flow.
> The stagnation temperature on the Apollo vehicle at Mach 36 as it entered the atmosphere was 11,000 K, a much different value than predicted in Problem 8.17 for the case of a calorically perfect gas with a ratio of specific heats equal to 1.4. The differ
> When the Apollo command module returned to earth from the moon, it entered the earth’s atmosphere at a Mach number of 36. Using the results from the present chapter for a calorically perfect gas with the ratio of specific heats equal to 1.4, predict the
> In the test section of a supersonic wind tunnel, a Pitot tube in the flow reads a pressure of 1.13 atm. A static pressure measurement (from a pressure tap on the sidewall of the test section) yields 0.1 atm. Calculate the Mach number of the flow in the t
> On March 16, 1990, an Air Force SR-71 set a new continental speed record, averaging a velocity of 2112 mi/h at an altitude of 80,000 ft. Calculate the temperature (in degrees Fahrenheit) at a stagnation point on the vehicle.
> Derive the Rayleigh Pitot tube formula, Equation (8.80).
> Consider a light, single-engine, propeller-driven airplane similar to a Cessna Skylane. The airplane weight is 2950 lb and the wing reference area is 174 ft2. The drag coefficient of the airplane CD is a function of the lift coefficient CL for reasons th
> Consider an NACA 2412 airfoil (the meaning of the number designations for standard NACA airfoil shapes is discussed in Chapter 4). The following is a tabulation of the lift, drag, and moment coefficients about the quarter chord for this airfoil, as a fun
> (a) A point charge q is inside a cavity in an uncharged conductor (Fig. 2.45). Is the force on q necessarily zero?11 (b) Is the force between a point charge and a nearby uncharged conductor always attractive?12
> Find the electric field a distance z above the center of a square loop (side a) carrying uniform line charge λ (Fig. 2.8). [Hint: Use the result of Ex. 2.2.]
> Two spherical cavities, of radii a and b, are hollowed out from the interior of a (neutral) conducting sphere of radius R (Fig. 2.49). At the center of each cavity a point charge is placed—call these charges qa and qb. (a) Find the surf
> A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b, as in Fig. 2.48). The shell carries no net charge. (a) Find the surface charge density σ at R, at a, and at b. (b)
> Find the interaction energy / for two point charges, q1 and q2, a distance a apart. [Hint: Put q1 at the origin and q2 on the z axis; use spherical coordinates, and do the r integral first.]
> Consider two concentric spherical shells, of radii a and b. Suppose the inner one carries a charge q, and the outer one a charge -q (both of them uniformly distributed over the surface). Calculate the energy of this configuration, (a) using Eq. 2.45, and
> Here is a fourth way of computing the energy of a uniformly charged solid sphere: Assemble it like a snowball, layer by layer, each time bringing in an infinitesimal charge dq from far away and smearing it uniformly over the surface, thereby increasing th
> Find the energy stored in a uniformly charged solid sphere of radius R and charge q. Do it three different ways: (a) Use Eq. 2.43. You found the potential in Prob. 2.21. (b) Use Eq. 2.45. Don’t forget to integrate over all space. (c) Use Eq. 2.44. Take a
> Consider an infinite chain of point charges, ±q (with alternating signs), strung out along the x axis, each a distance from its nearest neighbors. Find the work per particle required to assemble this system. [Partial Answer: / for some dimensionless nu
> Two positive point charges, qA and qB (masses m A and m B) are at rest, held together by a massless string of length a. Now the string is cut, and the particles fly off in opposite directions. How fast is each one going, when they are far apart?
> Find the transformation matrix R that describes a rotation by 120◦ about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin.
> (a) Three charges are situated at the corners of a square (side a), as shown in Fig. 2.41. How much work does it take to bring in another charge, +q, from far away and place it in the fourth corner? (b) How much work does it take to assemble the whole co
> (a) Check that the results of Exs. 2.5 and 2.6, and Prob. 2.11, are consistent with Eq. 2.33. (b) Use Gauss’s law to find the field inside and outside a long hollow cylindrical tube, which carries a uniform surface charge σ. Check that your result is consi
> Find the electric field a distance z above one end of a straight line segment of length L (Fig. 2.7) that carries a uniform line charge λ. Check that your formula is consistent with what you would expect for the case z (( L.
> Check that Eq. 2.29 satisfies Poisson’s equation, by applying the Laplacian and using Eq. 1.102.
> Use Eq. 2.29 to calculate the potential inside a uniformly charged solid sphere of radius R and total charge q. Compare your answer to Prob. 2.21.
> Find the potential on the axis of a uniformly charged solid cylinder, a distance z from the center. The length of the cylinder is L, its radius is R, and the charge density is ρ. Use your result to calculate the electric field at this point. (Assume that
> Using Eqs. 2.27 and 2.30, find the potential at a distance z above the center of the charge distributions in Fig. 2.34. In each case, compute E=-∇V, and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectiv
> For the configuration of Prob. 2.16, find the potential difference between a point on the axis and a point on the outer cylinder. Note that it is not necessary to commit yourself to a particular reference point, if you use Eq. 2.22.
> For the charge configuration of Prob. 2.15, find the potential at the center, using infinity as your reference point.
> (a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot products. (That is, show that (b) What constraints must the elements (Rij ) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of A (fo
> Find the potential a distance s from an infinitely long straight wire that carries a uniform line charge λ. Compute the gradient of your potential, and check that it yields the correct field.
> Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V (r).
> One of these is an impossible electrostatic field. Which one? (a) E = k[xy xˆ + 2yz yˆ + 3xz zˆ]; (b) E = k[y2 xˆ + (2xy + z2) yˆ + 2yz zˆ]. Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your
> Find the electric field (magnitude and direction) a distance z above the midpoint between equal and opposite charges (±q), a distance d apart (same as Example 2.1, except that the charge at x = +d/2 is −q).
> Calculate ∇×E directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.63 if you get stuck.
> Two spheres, each of radius R and carrying uniform volume charge densities +ρ and -ρ, respectively, are placed so that they partially overlap (Fig. 2.28). Call the vector from the positive center to the negative center d. Show that
> An infinite plane slab, of thickness 2d, carries a uniform volume charge density ρ (Fig. 2.27). Find the electric field, as a function of y, where y = 0 at the center. Plot E versus y, calling E positive when it p
> A long coaxial cable (Fig. 2.26) carries a uniform volume charge density ρ on the inner cylinder (radius a), and a uniform surface charge density on the outer cylindrical shell (radius b). This surface charge is negative and is of just the rig
> A thick spherical shell carries charge density / (Fig. 2.25). Find the electric field in the three regions: (i) r (ii) a (iii) r > b. Plot |E| as a function of r , for the case b = 2a.
> Find the electric field inside a sphere that carries a charge density proportional to the distance from the origin, ρ=kr , for some constant k. [Hint: This charge density is not uniform, and you must integrate to get the enclosed charge.]
> Find the separation vector r from the source point (2,8,7) to the field point (4,6,8). Determine its magnitude (r), and construct the unit vector rˆ.
> Find the electric field a distance s from an infinitely long straight wire that carries a uniform line charge λ. Compare Eq. 2.9.
> Use Gauss’s law to find the electric field inside a uniformly charged solid sphere (charge density ρ). Compare your answer to Prob. 2.8.
> Use Gauss’s law to find the electric field inside and outside a spherical shell of radius R that carries a uniform surface charge density σ. Compare your answer to Prob. 2.7.
> A charge q sits at the back corner of a cube, as shown in Fig. 2.17. What is the flux of E through the shaded side?
> (a) Twelve equal charges, q, are situated at the corners of a regular 12-sided polygon (for instance, one on each numeral of a clock face). What is the net force on a test charge Q at the center? (b) Suppose one of the 12 q’s is removed (the one at “6 o’
> In case you’re not persuaded that ∇2(1/r) = −4πδ3(r) (Eq. 1.102 with rr = 0 for simplicity), try replacing r by and watching what happens Specially, let / To demonstrate th
> (a) Find the divergence of the function First compute it directly, as in Eq. 1.84. Test your result using the divergence theorem, as in Eq. 1.85. Is there a delta function at the origin, as there was for / What is the general formula for the divergence
> The integral / is sometimes called the vector area of the surface S. If S happens to be flat, then |a| is the ordinary (scalar) area, obviously. (a) Find the vector area of a hemispherical bowl of radius R. (b) Show that a = 0 for any clo
> Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that: (a) / where c is a constant, in the divergence theorem; use the product
> Here are two cute checks of the fundamental theorems: (a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem (v=∇T, in this case). Show that the result is consistent with what you already knew about second derivatives. (b) Combine Corollary
> Prove that [A × (B × C)]+ [B × (C × A)]+ [C × (A × B)]= 0. Under what conditions does A × (B × C) = (A × B) × C?
> Check the divergence theorem for the function v = r 2 sin θ rˆ + 4r 2 cos θ θˆ + r 2 tan θ φˆ, using the volume of the “ice-cream coneâ
> A Lincoln Continental is twice as long as a VW Beetle, when they are at rest. As the Continental overtakes the VW, going through a speed trap, a (stationary) policeman observes that they both have the same length. The VW is going at half the speed of lig
> A rocket ship leaves earth at a speed of / When a clock on the rocket says 1 hour has elapsed, the rocket ship sends a light signal back to earth. (a) According to earth clocks, when was the signal sent? (b) According to earth clocks, how long after
> Generalize the laws of relativistic electrodynamics (Eqs. 12.127 and 12.128) to include magnetic charge. [Refer to Sect. 7.3.4.]
> The natural relativistic generalization of the Abraham-Lorentz formula (Eq. 11.80) would seem to be / This is certainly a 4-vector, and it reduces to the Abraham-Lorentz formula in the nonrelativistic limit v
> Use the Larmor formula (Eq. 11.70) and special relativity to derive the Liénard formula (Eq. 11.73).
> (a) Construct a tensor Dμν (analogous to Fμν ) out of D and H. Use it to express Maxwell’s equations inside matter in terms of the free current density / / (b) Construct the dual tensor Hμν (analogous to Gμν ). [Answer: H 01 ≡ Hx , H 12 ≡−cDz , etc.] (c
> In a laboratory experiment, a muon is observed to travel 800 m before disintegrating. A graduate student looks up the lifetime of a muon / and concludes that its speed was / Faster than light! Identify the student’s error, and find the actual speed of th
> A charge q is released from rest at the origin, in the presence of a uniform electric field /and a uniform magnetic field /Determine the trajectory of the particle by transforming to a system in which E=0, finding the path in that system and then transformi
> Check Stokes’ theorem for the function / using the triangular surface shown in Fig. 1.51. [Answer: a2]
> Derive” the Lorentz force law, as follows: Let charge q be at rest in S¯, so F¯ = qE¯ , and let S¯ move with velocity v = v xˆ with respect to S. Use the transformation rules (Eqs. 12.67 and 12.109) to rewrite F¯ in terms of F, and E¯ in terms of E and
> Two charges (q approach the origin at constant velocity from opposite directions along the x axis. They collide and stick together, forming a neutral particle at rest. Sketch the electric field before and shortly after the collision (remember that electro
> In a certain inertial frame S, the electric field E and the magnetic field B are neither parallel nor perpendicular, at a particular space-time point. Show that in a different inertial system S¯, moving relative to S with velocity v given by / the fields E
> A stationary magnetic dipole, m = m zˆ, is situated above an infinite uniform surface current, K = K xˆ (Fig. 12.44). (a) Find the torque on the dipole, using Eq. 6.1. (b) Suppose that the surface current consists of
> An ideal magnetic dipole moment m is located at the origin of an inertial system S¯ that moves with speed v in the x direction with respect to inertial system S. In S¯ the vector potential is / (Eq. 5.85), and the scalar potentia
> An electric dipole consists of two point charges ((q), each of mass m, fixed to the ends of a (massless) rod of length d. (Do not assume d is small.) (a) Find the net self-force on the dipole when it undergoes hyperbolic motion (Eq. 12.61)
> Find x as a function of t for motion starting from rest at the origin under the influence of a constant Minkowski force in the x direction. Leave your answer in implicit form (t as, a function of x ). / /
> A particle of mass m collides elastically with an identical particle at rest. Classically, the outgoing trajectories always make an angle of 90◦. Calculate this angle relativistically, in terms of φ, the scattering angle, a
> Calculate the threshold (minimum) momentum the pion must have in order for the process π+ p (K+Σ to occur. The proton p is initially at rest. / (all in MeV). [Hint: To formulate the threshold condition, examine the collision in the center-of- momentu
> Every 2 years, more or less, The New York Times publishes an article in which some astronomer claims to have found an object traveling faster than the speed of light. Many of these reports result from a failure to distinguish what is seen from what is ob
> Compute the line integral of v = (r cos2 θ) rˆ − (r cos θ sin θ) θˆ + 3r φˆ around the path shown in Fig. 1.50 (the points are labeled by th
> Inertial system S¯ moves at constant velocity v = βc(cos φ xˆ + sin φ yˆ) with respect to S. Their axes are parallel to one another, and their origins coincide at t=
> Show that the Liénard-Wiechert potentials (Eqs. 10.46 and 10.47) can be expressed in relativistic notation as Where /
> Show that the potential representation (Eq. 12.133) automatically satisfies ∂Gμν /∂ xν = 0. [Suggestion: Use Prob. 12.54.]
> You may have noticed that the four-dimensional gradient operator ∂/∂xμ functions like a covariant 4-vector—in fact, it is often written ∂μ, for short. For instance, the continuity equation, ∂μ Jμ=0, has the form of an invariant product of two vectors. Th
