Questions and Answers of Introduction To Electrodynamics

A cop pulls you over and asks what speed you were going. “Well, officer, I cannot tell a lie: the speedometer read 4 × 108 m/s.” He gives you a ticket, because the speed limit on this highway is
Two positive point charges, qA and qB (masses mA and mB) 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
Two positive point charges, qA and qB (masses mA and mB) 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
(a) A point charge q is inside a cavity in an uncharged conductor (Fig. 2.45). Is the force on q necessarily zero?(b) Is the force between a point charge and a nearby uncharged conductor always
If the electric field in some region is given (in spherical coordinates) by the expression for some constant k, what is the charge density? k E(r) = [3f+ 2 sin 0 cos 0 sin þê + sin0 cos 4 ]
A stationary electric dipole p = p ˆz is situated at the origin. A positive point charge q (mass m) executes circular motion (radius s) at constant speed in the field of the dipole. Characterize the
Find the charge density σ(θ) on the surface of a sphere (radius R) that produces the same electric field, for points exterior to the sphere, as a charge q at the point a < R on the z axis.
Analyze the motion of a particle (charge q, mass m) in the magnetic field of a long straight wire carrying a steady current I.(a) Is its kinetic energy conserved?(b) Find the force on the particle,
An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, Φ=∫ B. da, I never specified the particular surface to be
Where is ∂B/∂t nonzero, in Figure 7.21(b) ?Exploit the analogy between Faraday’s law and Ampère’s law to sketch (qualitatively) the electric field. B (in) (a) B (in) (b) changing magnetic
For a point charge moving at constant velocity, calculate the flux integral ∫ E · da (using Eq. 10.75), over the surface of a sphere centered at the present location of the charge.Reference
Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed:(a) Derive the modified wave equation describing the
Consider a particle of charge q and mass m, free to move in the xy plane in response to an electromagnetic wave propagating in the z direction (Eq. 9.48—might as well set δ = 0).(a) Ignoring the
A time-dependent point charge q(t) at the origin, ρ(r, t) = q(t)δ3(r), is fed by a current J(r, t) = −(1/4π)(q/r 2) ˆr, where q ≡ dq/dt.(a) Check that charge is conserved, by confirming that
An expanding sphere, radius R(t) = vt (t > 0, constant v) carries a charge Q, uniformly distributed over its volume. Evaluate the integral Qeff =ρ(r, tr) dτ with respect to the center. Show that
A uniformly charged rod (length L, charge density λ) slides out the x axis at constant speed v. At time t = 0 the back end passes the origin (so its position as a function of time is x = vt, while
We are now in a position to treat the example in Sect. 8.2.1 quantitatively. Suppose q1 is at x1 = −vt and q2 is at y = −vt (Fig. 8.3, with t < 0). Find the electric
Develop the potential formulation for electrodynamics with magnetic charge (Eq. 7.44). You’ll need two scalar potentials and two vector potentials. Use the Lorenz gauge. Find the retarded
A positive charge q is fired head-on at a distant positive charge Q (which is held stationary), with an initial velocity v0. It comes in, decelerates to v = 0, and returns out to infinity. What
A point charge q, of mass m, is attached to a spring of constant k. At time t = 0 it is given a kick, so its initial energy is Now it oscillates, gradually radiating away this energy.(a) Confirm
With the inclusion of the radiation reaction force (Eq. 11.80), Newton’s second law for a charged particle becomes where F is the external force acting on the
An ideal electric dipole is situated at the origin; its dipole moment points in the Ẑ direction, and is quadratic in time: (a) Use the method of Section 11.1.2 to determine the (exact)
An electric dipole rotates at constant angular velocity ω in the x y plane. (The charges, ±q, are at the magnitude of the dipole moment is p = 2qR.)(a) Find the interaction
Use the result of Prob. 10.34 to determine the power radiated by an ideal electric dipole, p(t), at the origin. Check that your answer is consistent with Eq. 11.22, in the case of sinusoidal time
Prove or disprove (with a counterexample) the following Theorem: Suppose a conductor carrying a net charge Q, when placed in an external electric field Ee, experiences a force F; if the external
(a) Consider an equilateral triangle, inscribed in a circle of radius a, with a point charge q at each vertex. The electric field is zero (obviously) at the center, but (surprisingly) there are three
Find the potential on the rim of a uniformly charged disk (radius R, charge density σ).
(a) Show that(b) Let θ(x) be the step functionShow that dθ/dx = δ(x). d - x (8(x)) = -8(x). dx
Evaluate the following integrals:(a)(b)(c)(d) ²₂(2x + 3) 8 (3x) dx.
Calculate the curls of the vector functions in Prob. 1.15.Data from problem 1.15 (a) va = x² + 3xz² ŷ - 2xz 2. (b) V, =xyx+2yz ý+3zxê. (c) Vc = y²x + (2xy + z²) ŷ + 2yz î.
Prove that:[A × (B × C)] + [B × (C × A)] + [C × (A × B)] = 0. Under what conditions does A × (B × C) = (A × B) × C?
Prove or disprove (with a counterexample) the following Theorem: Suppose a conductor carrying a net charge Q, when placed in an external electric field Ee, experiences a force F; if the external
What is the minimum-energy configuration for a system of N equal point charges placed on or inside a circle of radius R? Because the charge on a conductor goes to the surface, you might think the N
A circular ring in the xy plane (radius R, centered at the origin) carries a uniform line charge λ. Find the first three terms (n = 0, 1, 2) in the multipole expansion for V(r, θ).
A solid sphere, radius R, is centered at the origin. The “northern” hemisphere carries a uniform charge density ρ0, and the “southern” hemisphere a uniform charge density −ρ0. Find the
Buckminsterfullerine is a molecule of 60 carbon atoms arranged like the stitching on a soccer-ball. It may be approximated as a conducting spherical shell of radius R = 3.5Å. A nearby electron would
A point charge Q is “nailed down” on a table. Around it, at radius R, is a frictionless circular track on which a dipole p rides, constrained always to point tangent to the circle. Use Eq. 4.5 to
The space between the plates of a parallel-plate capacitor is filled with dielectric material whose dielectric constant varies linearly from 1 at the bottom plate (x = 0) to 2 at the top plate (x =
Prove the following uniqueness theorem: A volume V contains a specified free charge distribution, and various pieces of linear dielectric material, with the susceptibility of each one given. If the
Calculate the magnetic field at the center of a uniformly charged spherical shell, of radius R and total charge Q, spinning at constant angular velocity ω.
Consider a plane loop of wire that carries a steady current I; we want to calculate the magnetic field at a point in the plane. We might as well take that point to be the origin (it could be inside
First write B(r) as a Taylor expansion about the center of the loop: where r0 is the position of the dipole and ∇0 denotes differentiation with respect to r0. Put this into the Lorentz force law
What current density would produce the vector potential, A = k ˆφ (where k is a constant), in cylindrical coordinates?
Prove Alfven’s theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field
Imagine two parallel infinite sheets, carrying uniform surface charge +σ (on the sheet at z = d) and −σ (at z = 0). They are moving in the y direction at constant speed v (a)
An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit length λ, uniformly distributed over its surface. Surrounding it, at
(a) Show that(b) Show that [ƒ(V x A) · da = [[A × (Vƒ)] · da + f ƒA. dl. $ . P
Evaluate the following integrals:(a)(b)(c)d. (3x² - 2x - 1) 8(x − 3) dx.
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
Use Gauss’s law to find the electric field inside a uniformly charged solid sphere (charge density ρ).
Find the electric field inside a sphere that carries a charge density proportional o the distance from the origin, ρ = kr, for some constant k. (Fig. 2.25). Find the electric field in the three
Suppose the conductivity of the material separating the cylinders in Ex. 7.2 is not uniform; specifically, σ(s) = k/s, for some constant k. Find the resistance between the cylinders.
Imagine a uniform magnetic field, pointing in the z direction and filling all space (B = B0 ẑ). A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby
An infinite wire carrying a constant current I in the ẑ direction is moving in the y direction at a constant speed v. Find the electric field, in the quasistatic approximation, at the instant the
A circular wire loop (radius r, resistance R) encloses a region of uniform magnetic field, B, perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with
An infinite wire runs along the z axis; it carries a current I (z) that is a function of z (but not of t), and a charge density λ(t) that is a function of t (but not of z).(a) By examining the
A certain transmission line is constructed from two thin metal “ribbons,” of width w, a very small distance h << w apart. The current travels down one strip and back along the other. In
Two concentric spherical shells carry uniformly distributed charges +Q (at radius a) and −Q (at radius b > a). They are immersed in a uniform magnetic field B = B0 ẑ(a) Find the angular
A point charge q is located at the center of a toroidal coil of rectangular cross section, inner radius a, outer radius a + w, and height h, which carries a total of N tightly-wound turns and current
Consider an ideal stationary magnetic dipole m in a static electric field E. Show that the fields carry momentumSo far, this is valid for any localized static configuration. For a current confined to
A circular disk of radius R and mass M carries n point charges (q), attached at regular intervals around its rim. At time t = 0 the disk lies in the xy plane, with its center at the origin, and is
Figure 2.35 summarizes the laws of electrostatics in a “triangle diagram” relating the source (ρ), the field (E), and the potential (V). Figure 5.48 does the same for magnetostatics, where the
Find the (Lorenz gauge) potentials and fields of a time-dependent ideal electric dipole p(t) at the origin. (It is stationary, but its magnitude and/or direction are changing with time.) Don’t
A parallel-plate capacitor C, with plate separation d, is given an initial charge (±)Q0. It is then connected to a resistor R, and discharges, Q(t) = Q0e−t/RC.(a) What fraction of its initial