Massachusetts Institute Of Technology

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department

Physics 8.022

September 18, 2009

Problem Set 2 Due: Friday, September 25, at 4:45 pm (in the problem set boxes located at junction of buildings 8 and 16, on the third floor.) Reading: Chapter 2 in Purcell’s text Electricity and Magnetism, 2nd Edition; Chapter 2 in Shey’s book Div, Grad, Curl, and All That. Vector Calculus Fact Sheet: You may make use of the vector calculus fact sheet which, among other things, contains the gradient, divergence, and curl operators in Cartesian, cylindrical, and spherical coordinates. Problem 1 “Superposition Principle” Purcell, problem #2.29 Problem 2 “Practice With Gradients” Calculate the gradient of each of these scalar fields: (a) xyz. (b) x2 + y 2 + z 2 . (c) 1/r (in spherical coordinates). (d) (cos θ)/r2 (in spherical coordinates). Problem 3 “Practice With Divergences” Calculate the divergence of each of these vector fields: (a) xˆx + yˆy + zˆp z. (b) (ˆ xy − yˆx)/ x2 + y 2 . (c) rˆ/r2 (in spherical coordinates). ˆ (d) rˆ(2 cos θ)/r3 + θ(sin θ)/r3 (in spherical coordinates). Problem 4 “Divergence of a Curl” Purcell, Problem #2.16 1

Problem 5 “Field of a Long Cylinder of Uniform Charge Density” Consider a very long cylinder of radius R that is filled with a uniform charge density ~ both inside and ρ. Use the following two different approaches to find the electric field, E, outside the cylinder. (a) Apply Gauss’ law ~ ·E ~ = 4πρ. (b) Integrate Poisson’s equation: ∇ Be sure that the E field inside and the E field outside match at the boundary (i.e., at R). Problem 6 “Practice With Curls” Calculate the curl of each of these vector fields: (a) xˆyz + yˆxz + zˆxy. (b) xˆxy + yˆy 2 + zˆyz. (c) (1/r2 )ˆ r (in spherical coordinates). (d) (1/R)φˆ (in cylindrical coordinates). Problem 7 “Stokes’s Theorem in Action” Consider the vector field F~ = xˆz 2 + yˆx2 − zˆy 2 . H (a) Calculate F~ · d~r around a square path with corners (x0 ± s/2, y0 ± s/2, 0). The square has center (x0 , y0 , 0), side length s, and its sides are parallel to the x- and y-axes. The sense of rotation of the path is counter-clockwise as viewed from the +z direction. (b) Divide your answer to (a) by the area of the square, and take the limit as s → 0. ~ × F~ at the center of the square. (c) Calculate ∇ ~ × F~ evaluated at (d) Verify that your answer to (b) is equal to the normal component of ∇ the center of the square. Problem 8 “Gauss’s Theorem in Action” Consider a vector field F~ = rˆ r (in spherical coordinates), and a closed surface S that is a cube with one corner at the origin and the opposite corner at (b, b, b). Verify Gauss’s theorem, I Z ~ ~ · F~ )dV , F ·n ˆ dS = (∇ for this particular case by performing both the surface integral on the left side, and the volume integral on the right side, and showing that they are equal.

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Problem 9 “Potential Averaged Over the Surface of a Sphere” On page 64 of Purcell there is a lengthy discussion of some of the properties of Laplace’s equation (∇2 ϕ = 0). In particular, Purcell makes the following claim: “If ϕ satisfies Laplace’s equation, then the average value of ϕ over the surface of any sphere (not necessarily a small sphere) is equal to the value of ϕ at the center of the sphere.”

Purcell then goes on to give a clever “word proof” that this is so. This result is important in that it is directly related to Earnshaw’s theorem which states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges (Samuel Earnshaw, 1842). In this problem, we will prove mathematically that, in electrostatics, the following two statements are true: (a) For an arbitrary collection of charges located entirely outside of a sphere of radius R, the potential averaged over the surface of the sphere is given by X qk hϕiS = dk k where dk is the distance of each charge from the center of the sphere. Note: this is the same as ϕ evaluated at the center of the sphere. (b) For an arbitrary collection of charges located entirely inside of a sphere of radius R, the potential averaged over the surface of the sphere is given by 1X qk hϕiS = R k Hint: You need prove these results for only a single charge at some arbitrary distance from the center of the sphere. Everything else follows from the principle of superposition. In case you are not using Mathematica, Maple, or some table to look up the integral, you may find the following dimensionless integral helpful: Z 1  −1/2 2 if a > 1 (a − ξ)2 + (1 − ξ 2 ) dξ = 2 if a < 1 or a −1 where a is a dimensionless real number > 0. Problem 10 “Gauss’ Law Applied to a Non-Spherically Symmetric Field” There is an interesting footnote at the bottom of page 24 in Purcell. Referring to the derivation of Gauss’ law from Coulomb’s law, Purcell comments: “There is one difference, inconsequential here, but relevant to our latter study of the fields of moving charges. Gauss’ law is obeyed by a wider class of fields than those represented by the electrostatic field. In particular, a field that is inverse-square in r but not spherically symmetric can satisfy Gauss’ law. In other words, Gauss’ law alone does not imply the symmetry of the field of a point source which is implicit in Coulomb’s law.”

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We illustrate that point here with the field of a point charge, q, that is moving with uniform speed, v. We may or may not have time to derive this result later, but for now take it on faith that the E field of a uniformly moving charge is given by 1 − β2 ~ r) = q rˆ E(~ r2 (1 − β 2 sin2 θ)3/2 where β is a constant equal to v/c (where c is the speed of light), and θ is the angle between the direction of motion of the electron and the point where the field is being measured (all at a fixed instant of time). For purposes of this problem you may think of the field and the surface integral as being calculated for a stationary point charge with a field described as above. See Fig. 5.14 on page 184 of Purcell to get a better idea of what the field-line configuration looks like. For a point charge, q, with a field described by the above non-symmetric field, located anywhere within a sphere, compute the surface integral: Z ~ r ) · dA ~ E(~ S

and show that it equals 4πq, independent of the value of γ. With no loss of generality, it may be helpful to take θ = 0 to lie along the z axis and use spherical coordinates to carry out the integration. In case you are not using Mathematica, Maple, or some table to look up the integral, you may find the following dimensionless integral helpful: Z 1  −3/2 2 1 + ξ 2 (a2 − 1) dξ = a −1 where a2 > 1.

Supplemental Problems Not to be turned in – for discussion in recitation sections Problem S1 “Potential at the Center and Corner of a Uniformly Charged Cube” Purcell, problem #2.30 Problem S2 “Work Done to Assemble A Uniformly Charged Disk” Compute the potential energy of a uniformly charged (thin) disk of radius R. (a) First review Purcell’s derivation of the potential, ϕ, at an arbitrary point on the edge of a thin disk. (b) Use this result to compute the work done to bring up a new thin annulus of charge. Integrate to find the potential energy U . 4

Problem S3 “Potential of an Electric Dipole” Compute the potential ϕ(x, y, z) of a dipole charge configuration. The dipole consists of a charge +q located at z = a/2 and a charge −q located at z = −q. (a) Write down ϕ(x, y, z) (i.e., in Cartesian coordinates). (b) Expand ϕ(x, y, z) in a Maclaurin series (i.e., a Taylor series about a = 0) to first order in a. (c) Convert your results to spherical coordinates: ϕ(r, θ, φ). ~ ~ and compare your results with (d) Compute ∇ϕ(r, θ, φ), in spherical coordinates, to find E, what you found in Problem #6 of pset #1. Problem S4 “Visualizing vector fields” Sketch the field lines of the following vector fields. Be sure to indicate directions. (a) (ˆ xx + yˆy)/(x2 + y 2 ). (b) xˆy − yˆx. (c) zˆ × ~r, where ~r is the position vector xˆx + yˆy + zˆz. This could be a good problem to try with Mathematica. This can be done with the built-in programs called “VectorPlot” and/or “StreamDensityPlot”. Problem S5 “The Gradient Theorem in Action” Let T = xy 2 , and take point A to be theRorigin and point B to be (2, 1, 0). Choose any ~ · d~r = T (B) − T (A). path you wish from A to B, and verify that ∇T Problem S6 “Curl-Free and Divergence-Free Fields” ~ × G? ~ If so, give an For each of the following fields F~ , is it possible to write F~ = ∇ ~ that works. Is it possible to write F~ = ∇s ~ for some scalar field s? If so, give example of G an example of s that works. (a) F~ = C zˆ where C is a constant. (b) F~ = xˆy. (c) F~ = rˆr (in spherical coordinates). (d) F~ = xˆyz + yˆzx + zˆxy.

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