1 Million+ Step-by-step solutions

Using the definitions in Eqs. l. 1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive,

(a) When the three vectors are coplanar;

(b) In the general case.

Is the cross product associative? (A x B) x C = A x (B x C).

If so, prove it; if not, provide a counterexample.

Find the angle between the body diagonals of a cube.

Use the cross product to find the components of the unit vector n perpendicular to the plane shown in Fig. 1.11.

Prove the BAC-CAB role by writing out both sides in component form.

Prove that [A x (B x C)] + [B x (C x A)] + [C x (A x B)] = 0. Under what conditions does A x (B x C) = (A x B) x C?

Find the separation vector π from the source point (2, 8, 7) to the field point (4, 6, 8). Determine its magnitude (π), and construct the unit vector π.

(a) Prove that the two-dimensional rotation matrix (1.29) preserves dot products. (That is, show that Ay By + A z B z = Ay By + A z B z.)

(b) What constraints must the elements (Rij) of the three-dimensional rotation matrix (1.30) satisfy in order to preserve the length of A (for all vectors A)?

Find the transformation matrix R that describes a rotation by 120o 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) How do the components of a vector transform under a translation of coordinates (x = x, y = y – a, z = z, Fig. 1.16a)?

(b) How do the components of a vector transform under an inversion of coordinates (x = – x, Y = – y, z = –z, Fig. 1.16b)?

(c) How does the cross product (1.13) of two vectors transform under inversion? [The cross-product of two vectors is properly called a pseudo vector because of this "anomalous" behavior.] Is the cross product of two pseudo vectors a vector, or a pseudo vector? Name two pseudo vector quantities in classical mechanics.

(d) How does the scalar triple product of three vectors transform under inversions? (Such an object is called a pseudoscalar.)

Find the gradients of the following functions:

(a) f(x, y, z) = x2 + y3 + z4.

(b) f(x, y, z) = x2y3z4.

(c) f(x, y, z) = ex sin(y) ln (z).

The height of a certain hill (in feet) is given by h(x, y) = 10(2xy – 3x2 – 4y2 – 18x + 28y + 12), where y is the distance (in miles) north, x the distance east of South Hadley.

(a) Where is the top of the hill located?

(b) How high is the hill?

(c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point?

Let π, be the separation vector from a fixed point (x', y’, z') to the point (x, y, z), and let π be its length. Show that

(a) ∆ (π2) = 2 π

(b) ∆ (1/π) = – π/π2.

(c) What is the general formula for ∆(πn)?

Suppose that f is a function of two variables (y and z) only. Show that the gradient ∆ f = (∂f/∂y)y + (∂f/∂z)z transforms as a vector under rotations, Eq. 1.29.

Calculate the divergence of the following vector functions:

(a) va = x2 x + 3xz 2 y – 2xz z.

(b) vb = xy x + 2yz y + 3zx z.

(c) vc = y2x + (2xy + z2)y + 2yzz.

Sketch the vector function v = r/r2, and compute its divergence. The answer may surprise you... can you explain it?

In two dimensions, show that the divergence transforms as a scalar under rotations.

Calculate the cuds of the vector functions in Prob. 1.15.

Construct a vector function that has zero divergence and zero cud everywhere. (A constant will do the job, of course, but make it something a little more interesting than that!)

Prove product rules (i), (iv), and (v).

(a) If A and B are two vector functions, what does the expression (A ∙ ∆)B mean? (That is, what are its x, y, and z components in terms of the Cartesian components of A, B, and ∆?)

(b) Compute (r ∙ ∆)r, where r is the unit vector defined in Eq. 1.21.

(c) For the functions in Prob. 1.15, evaluate (va ∙ ∆)vb.

(For masochists only.) Prove product rules (ii) and (vi). Refer to Prob. 1.21 for the definition of (A ∙ ∆)B.

Derive the three quotient rules.

(a) Check product rule (iv) (by calculating each term separately) for the functions

A = x x + 2 y y + 3z z; B = 3y x – 2xy.

(b) Do the same for product rule (ii).

(c) The same for rule (vi).

Calculate the Laplacian of the following functions:

(a) Ta= x2 + 2xy + 3z + 4.

(b) Tb = sin x siny sin z.

(c) Tc = e–5x sin 4 y cos 3z.

(d) v = x2 x + 3xz2y – 2xzz.

Prove that the divergence of a cud is always zero. Check it for function va in Prob. 1.15.

Prove that the curl of a gradient is always zero. Check it for function (b) in Prob. 1.11.

Calculate the line integral of the function v = x2x + 2yz y + y2 z from the origin to the point (1, 1, 1) by three different routes:

(a) (0, 0, 0) → (1,0, 0) → (1, 1,0) → (1, 1, 1);

(b) (0, 0, 0) → (0, 0, 1) → (0, 1, 1) → (1, 1, l);

(c) The direct straight line.

(d) What is the line integral around the closed loop that goes out along path (a) and back along path (b)?

Calculate the surface integral of the function in Ex. 1.7, over the bottom of the box. For consistency, let "upward" be the positive direction. Does the surface integral depend only on the boundary line for this function? What is the total flux over the closed surface of the box (including the bottom)?

Calculate the volume integral of the function T = z2 over the tetrahedron with comers at (0,0,0), (1,0,0), (0,1,0), and (0,0,1).

Check the fundamental theorem for gradients, using T = x2 + 4xy + 2yz3, the points a = (0, 0, 0), b = (1, 1, 1), and the three paths in Fig. 1.28:

(a) (0,0.0) → (1.0, 0) → (1, 1,0) → (1, 1.1);

(b) (0,0.0) → (0,0, 1) → (0, 1, 1) → (1, 1, 1);

(c) the parabolic path z = x2; y = x.

Test the divergence theorem for the function v = (xy) x + (2yz) y + (3zx) z. Take as your volume the cube shown in Fig. 1.30, with sides of length 2.

Test Stokes' theorem for the function v = (xy) x + (2yz) y + (3zx) z, using the triangular shaded area of Fig. 1.34.

Check Corollary I by using the same function and boundary line as in Ex. 1.11, but integrating over the five sides of the cube in Fig. 1.35. The back of the cube is open.

Find formulas for r, θ, Ф in terms of x, y, z (the inverse, in other words, of Eq. 1.62 ).

Express the unit vectors r, θ, Ф in terms of x, y, z (that is, derive Eq. 1.64). Check your answers several ways (r ∙ r = 1, θ ∙ Ф = r x θ = Ф , .... ). Also work out the inverse formulas, giving x, y, z in terms of r, θ ∙ Ф (and θ, Ф).

(a) Check the divergence theorem for the function v1 = r2r, using as your volume the sphere of radius R, centered at the origin.

(b) Do the same for v2 = (1/r2)r. (If the answer surprises you, look back at Prob. 1.16.)

Compute the divergence of the function v = (r cos θ)r + (r sin θ) θ + (r sin θ cos Ф) Ф. Check the divergence theorem for this function, using as your volume the inverted hemispherical bowl of radius R, resting on the xy plane and centered at the origin (Fig. 1.40).

Compute the gradient and Laplacian of the function T = r (cos θ + sin θ cos Ф). Check the Laplacian by converting T to Cartesian coordinates and using Eq. 1.42. Test the gradient theorem for this function, using the path shown in Fig. 1.41, from (0, 0, 0) to (0, 0, 2).

Express the cylindrical unit vectors s, Ф, z in terms of x, y, z (that is, derive Eq. 1.75). "Invert" your formulas to get x, y, z. in terms of s, Ф, z (and Ф).

(a) Find the divergence of the function v = s(2 + sin2 Ф)s + s sin Ф cos Ф Ф + 3z z.

(b) Test the divergence theorem for this function, using the quarter-cylinder (radius 2, height 5) shown in Fig. 1.43.

(c) Find the curl of v

Evaluate the following integrals:

(a) f26(3x 2 – 2x – 1) δ (x – 3)dx.

(b) f05 cosx δ (x – π) dx.

(c) fo 3 x3 δ (x q- 1) dx.

(d) f∞_∞ ln(x + 3) δ (x + 2) dx.

Evaluate the following integrals:

(a) f2_2(2x + 3) δ (3x) dx.

(b) f02(x3 + 3x + 2) δ (1 – x)dx.

(c) f1– l 9x2 δ (3x + 1)dx.

(d) fa_∞ δ (x – b) dx.

(a) Write an expression for the electric charge density p (r) of a point charge q at r'. Make sure that the volume integral of p equals q.

(b) What is the charge density of an electric dipole, consisting of a point charge – q at the origin and a point charge + q at a?

(c) What is the charge density of a uniform, infinitesimally thin spherical shell of radius R and total charge Q, centered at the origin? [Beware: the integral over all space must equal Q.]

Evaluate the following integrals:

(a) f all space (r2 + r ∙ a + a2) δ3 (r – a) dτ, where a is a fixed vector and a is its magnitude.

(b) fv | r – b|2 δ3 (5r) dr, where V is a cube of side 2, centered on the origin, and b = 4y + 3z.

(c) fv (r4 + r2(r ∙ c) + c4) δ3 (r – c) dτ, where V is a sphere of radius 6 about the origin, c = 5 x + 3y + 2z, and c is its magnitude.

(d) fv r ∙ (d – r) δ3 (e – r) dτ, where d = (1, 2, 3), e = (3, 2, 1), and V is a sphere of radius 1.5 centered at (2, 2, 2).

Evaluate the integral (where v is a sphere of radius R, centered at the origin) by two different methods, as in Ex.1.16.

(a) Let F1 = x2 z and F2 = x x + y y + z z. Calculate the divergence and curl of F1 and F2. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential.

(b) Show that F3 = yzx + zxy + xyz can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.

For Theorem I show that (d) → (a), (a) → (c), (c) → (b), (b) → (c), and (c) → (a).

For Theorem 2 show that

(d) → (a),

(a) → (c),

(c) → (b),

(b) → (c), and (c) → (a).

(a) Which of the vectors in Problem 1.15 can be expressed as the gradient of a scalar? Find a scalar function that does the job.

(b) Which can be expressed as the curl of a vector? Find such a vector.

Check the divergence theorem for the function v = r2 cos θ r + r2 cos Ф θ – r2 cos θ sin Ф Ф, using as your volume one octant of the sphere of radius R (Fig. 1.48). Make sure you include the entire surface.

Check Stokes' theorem using the function v = ay x + bx y (a and b are ,constants) and the circular path of radius R, centered at the origin in the xy plane.

Compute the line integral of v = 6x + yz2y + (3y + z)z along the triangular path shown in Fig. 1.49. Check your answer using Stokes' theorem.

Compute the line integral of v = (r cos2 θ) r – (r cos θ sin θ) θ + 3r Ф around the path shown in Fig. 1.50 (the points are labeled by their Cartesian coordinates). Do it either in cylindrical or in spherical coordinates. Check your answer, using Stokes' theorem.

Check Stokes' theorem for the function v = yz, using the triangular surface shown in Fig.1.51.

Check the divergence theorem for the function v = r2 sin θ r + 4r2 cos θ = r2 tan θФ. using the volume of the "ice-cream cone" shown in Fig. 1.52 (the top surface is spherical, with radius R and centered at the origin).

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 2 to Stokes' theorem with the divergence theorem. Show that the result is consistent with what you already knew.

Although the gradient, divergence, and cud theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:

(a) fv (∆T)dτ = fS T da.

(b) fv (∆ x v) dτ = – fS v x da.

(c) fv [T∆2U – U∆2T) ∙ (∆U)] dτ = fS(T∆U) ∙ da.

(d) fv (T∆2U – U∆2T) dτ = fS (T∆U – U∆T) ∙ da. [Comment: This is known as Green's theorem; it follows from (c), which is sometimes called Green's identity.]

(e) fS ∆T x da = – fp T dl.

The Integral is sometimes called the vector area of the surface S. If S happens to be flat, then |a| is the ordinay (scalar) area, obviously.

(a) Find the vector area of a hemispherical bowl of radius R.

(b) Show that a = 0 for any closed surface.

(c) Show that a is the same for all surfaces sharing the same boundary.

(d) Show that where the integral is around the boundary line.

(e) Show that for any constant vector c.

(a) Find the divergence of the function v = r/r. 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 r/r2? What is the general formula for the divergence of rn r?

(b) Find the curl of rn r?. Test your conclusion using Prob. 1.60b.

(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'clock"). What is the force on Q? Explain your reasoning carefully.

(c) Now 13 equal charges, q, are placed at the corners of a regular 13-sided polygon. What is the force on a test charge Q at the center?

(d) If one of the 13 q's is removed, what is the force on Q? Explain your reasoning.

(a) Find the electric field (magnitude and direction) a distance z above the midpoint between two equal charges, q, a distance d apart (Fig. 2.4). Check that your result is consistent with what you'd expect when z >> d.

(b) Repeat part (a), only this time make the right-hand charge – q instead of+q.

Find the electric field a distance z above one end of a straight line segment of length L (Fig. 2.7), which carries a uniform line charge λ. Check that your formula is consistent with what you would expect for the case z >> L.

Find the electric field a distance z above the center of a square loop (side a) carrying uniform line charge λ (Fig. 2.8).

Find the electric field a distance z above the center of a circular loop of radius r (Fig. 2.9), which carries a uniform line charge λ.

Find the electric field a distance z above the center of a flat circular disk of radius R (Fig. 2.10), which cames a uniform surface charge σ. What does your formula give in the limit R → ∞? Also check the case z >> R.

Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11), which cames a uniform charge density a. Treat the case z < R (inside) as well as z > R (outside). Express your answers in terms of the total charge q on the sphere.

Use your result in Prob. 2.7 to find the field inside and outside a sphere of radius R, which carries a uniform volume charge density p. Express your answers in terms of the total charge of the sphere, q. Draw a graph of |E| as a function of the distance from the center.

Suppose the electric field in some region is found to be E = kr3r, in spherical coordinates (k is some constant).

(a) Find the charge density p.

(b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)

A charge q sits at the back comer of a cube, as shown in Fig. 2.17. What is the flux of E through the shaded side?

Use Gauss's law to find the electric field inside and outside a spherical shell of radius R, which carries a uniform surface charge density a. Compare your answer to Prob. 2.7.

Use Gauss's law to find the electric field inside a uniformly charged sphere (charge density p). Compare your answer to Prob. 2.8.

Find the electric field a distance s from an infinitely long straight wire, which carries a uniform line charge λ. Compare Eq. 2.9.

Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, p = kr, for some constant k.

Problem 2.15 A hollow spherical shell carries charge density p = k/r2 in the region a < r < b (Fig. 2.25). Find the electric field in the three regions: (i) r < a, (ii) a < r < b, (iii) r > b. Plot |E| as a function of r.

A long coaxial cable (Fig. 2.26) carries a uniform volume charge density p 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 of just the right magnitude so that the cable as a whole is electrically neutral. Find the electric field in each of the three regions: (i) inside the inner cylinder (s < a), (ii) between the cylinders (a < s < b), (iii) outside the cable (s > b). Plot |E| as a function of s.

An infinite plane slab, of thickness 2d, cames a uniform volume charge density p (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 points in the + y direction and negative when it points in the – y direction.

Two spheres, each of radius R and carrying uniform charge densities + p and – p, 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 the field in the region of overlap is constant, and find its value.

Calculate ∆ x E directly from Eq. 2.8, by the method of Sect. 2.2.2. Refer to Prob. 1.62 if you get stuck.

One of these is an impossible electrostatic field. Which one?

(a) E = k[xy + 2yz y + 3xz z];

(b) E = k[y2 x + (2xy + z2) y + 2yz z].

Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing V V.

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).

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.

For the charge configuration of Prob. 2.15, find the potential at the center, using infinity as your reference point.

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.

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 Prob. 2.2a, Ex. 2.1, and Prob. 2.6, respectively. Suppose that we changed the right-hand charge in Fig. 2.34a to –q; what then is the potential at P? What field does that suggest? Compare your answer to Prob. 2.2b, and explain carefully any discrepancy.

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).

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 p. Use your result to calculate the electric field at this point. (Assume that z > L/2.)

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.

Check that Eq. 2.29 satisfies Poisson's eqqation, by applying the Laplacian and using Eq. 1.102.

(a) Check that the results of Exs. 2.4 and 2.5, 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 consistent with Eq. 2.33.

(c) Check that the result of Ex. 2.7 is consistent with boundary conditions 2.34 and 2.36.

(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 configuration of four charges?

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 spherical volume of radius a. Notice what happens as a → ∞.

Here is a fourth way of computing the energy of a uniformly charged sphere: Assemble the sphere layer by layer, each time bringing in an infinitesimal charge dq from far away and smearing it uniformly over the surface, thereby increasing the radius. How much work dW does it take to build up the radius by an amount dr? Integrate this to find the work necessary to create the entire sphere of radius R and total charge q.

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

(b) Using Eq. 2.47 and the results of Ex. 2.8.

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 potential at the center, using infinity as the reference point.

(c) Now the outer surface is touched to a grounding wire, which lowers its potential to zero (same as at infinity). How do your answers to (a) and (b) change?

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 surface charges σa, σb, and σR.

(b) What is the field outside the conductor?

(c) What is the field within each cavity?

(d) What is the force on qa and qb?

(e) Which of these answers would change if a third charge, qc, were brought near the conductor?

Two large metal plates (each of area A) are held a distance d apart. Suppose we put a charge Q on each plate; what is the electrostatic pressure on the plates?

A metal sphere of radius R carries a total,charge Q. What is the force of repulsion between the "northern" hemisphere and the "southern" hemisphere?

Find the capacitance per unit length of two coaxial metal cylindrical tubes, of radii a and b (Fig.2.53).

Suppose the plates of a parallel-plate capacitor move closer together by an infinitesimal distance 6, as a result of their mutual attraction.

(a) Use Eq. 2.52 to express the amount of work done by electrostatic forces, in terms of the field E, and the area of the plates, A.

(b) Use Eq. 2.46 to express the energy lost by the field in this process. (This problem is supposed to be easy, but it contains the embryo of an alternative derivation of Eq. 2.52, using conservation of energy.)

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