# Eletromagnetismo - Hayt - 7ª Ed - Soluções - chapter03 7th solution

(Parte 1 de 2)

CHAPTER 3

3.1. An empty metal paint can is placed on a marble table, the lid is removed, and both parts are discharged (honorably) by touching them to ground. An insulating nylon thread is glued to the center of the lid, and a penny, a nickel, and a dime are glued to the thread so that they are not touching each other. The penny is given a charge of +5 nC, and the nickel and dime are discharged. The assembly is lowered into the can so that the coins hang clear of all walls, and the lid is secured. The outside of the can is again touched momentarily to ground. The device is carefully disassembled with insulating gloves and tools.

a) What charges are found on each of the ﬁve metallic pieces? All coins were insulated during the entire procedure, so they will retain their original charges: Penny: +5nC; nickel: 0; dime: 0. The penny’s charge will have induced an equal and opposite negative charge (-5 nC) on the inside wall of the can and lid. This left a charge layer of +5 nC on the outside surface which was neutralized by the ground connection. Therefore, the can retained a net charge of −5nC after disassembly.

b) If the penny had been given a charge of +5nC, the dime a charge of −2nC, and the nickel ac harge of −1nC, what would the ﬁnal charge arrangement have been? Again, since the coins are insulated, they retain their original charges. The charge induced on the inside wall of the can and lid is equal to negative the sum of the coin charges, or −2nC. This is the charge that the can/lid contraption retains after grounding and disassembly.

3.2. A point charge of 20 nC is located at (4,-1,3), and a uniform line charge of -25 nC/m is lies along the intersection of the planes x = −4 and z =6 .

a) Calculate D at (3,-1,0): The total ﬂux density at the desired point is

point charge line charge b) How much electric ﬂux leaves the surface of a sphere of radius 5, centered at the origin? This will be equivalent to how much charge lies within the sphere. First the point charge is at distance from the origin given by Rp = √ 16+1+9 = 5.1, and so it is outside. Second,

so the entire line charge is also outside the sphere. Answer: zero.

c) Repeat part b if the radius of the sphere is 10.

First, from part b, the point charge will now lie inside. Second, the length of line charge that lies inside the sphere will be given by 2y0, where y0 satisﬁes the equation,√

3.3. The cylindrical surface ρ =8 cm contains the surface charge density, ρs =5 e−20|z| nC/m2. a) What is the total amount of charge present? We integrate over the surface to ﬁnd:

b) How much ﬂux leaves the surface ρ =8 cm, 1cm <z < 5cm, 30◦ <φ< 90◦?W e just integrate the charge density on that surface to ﬁnd the ﬂux that leaves it.

leaving:

a) the inﬁnitely-long cylindrical surface ρ =7 : We use ρ0 aρ + z az

The total ﬂux through the cylindrical surface and the two end caps are, in this order:

ρ0 aρ · aρ z0 az · az ρdρdφ+ ρdρ where again, the actual values of ρ0 and z0 (7 and 10) did not matter.

surfaces to consider, only 2 will contribute to the net outward ﬂux. Why? First consider the planes at y =0 and 3. The y component of D will penetrate those surfaces, but will be inward at y =0 and outward at y =3 , while having the same magnitude in both cases. These ﬂuxes will thus cancel. At the x =0 plane, Dx =0 and at the z =0 plane, Dz =0 ,s o there will be no ﬂux contributions from these surfaces. This leaves the 2 remaining surfaces at x =2 and z =5 . The net outward ﬂux becomes:

z=5 · az dxdy

From the symmetry of the conﬁguration, we surmise that the ﬁeld will be everywhere z-directed, and will be uniform with x and y at ﬁxed z.F or ﬁnding the ﬁeld inside the charge, an appropriate Gaussian surface will be that which encloses a rectangular region deﬁned by −1 <x< 1, −1 <y < 1, and |z| <d /2. The outward ﬂux from this surface will be limited to that through the two parallel surfaces at ±z:

where the factor of 2 in the second integral account for the equal ﬂuxes through the

Outside the charge, the Gaussian surface is the same, except that the parallel boundaries at ±z occur at |z| >d /2. As a result, the calculation is nearly the same as before, with the only change being the limits on the total charge integral:

Solve for Dz to ﬁnd the constant values:

3.7. Volume charge density is located in free space as ρv =2 e−1000r nC/m3 for 0 <r < 1 m, and ρv =0 elsewhere. a) Find the total charge enclosed by the spherical surface r =1 m: To ﬁnd the charge we integrate:

Integration over the angles gives a factor of 4π. The radial integration we evaluate using tables; we obtain

b) By using Gauss’s law, calculate the value of Dr on the surface r =1 m: The gaussian surface is a spherical shell of radius 1 m. The enclosed charge is the result of part a.

We thus write 4πr2Dr = Q,o r

Dr = Q

3.8. Use Gauss’s law in integral form to show that an inverse distance ﬁeld in spherical coordinates,

D = Aar/r, where A is a constant, requires every spherical shell of1m thickness to contain 4πA coulombs of charge. Does this indicate a continuous charge distribution? If so, ﬁnd the charge density variation with r.

The net outward ﬂux of this ﬁeld through a spherical surface of radius r is

ar · ar r2 sinθdθdφ =4 πAr = Qencl

We see from this that with every increase in r by one m, the enclosed charge increases by 4πA (done). It is evident that the charge density is continuous, and we can ﬁnd the density indirectly by constructing the integral for the enclosed charge, in which we already found the latter from Gauss’s law:

To obtain the correct enclosed charge, the integrand must be ρ(r)= A/r2.

3.9. A uniform volume charge density of 80µC/m3 is present throughout the region 8mm <r <

10mm. Let ρv =0 for 0 <r < 8mm. a) Find the total charge inside the spherical surface r =1 0 m: This will be

c) If there is no charge for r> 10 m, ﬁnd Dr at r =2 0 m: This will be the same computation as in part b, except the gaussian surface now lies at 20 m. Thus

3.10. Volume charge density varies in spherical coordinates as ρv =( ρ0 sinπr)/r2, where ρ0 is a constant. Find the surfaces on which D =0 .

3.1. In cylindrical coordinates, let ρv =0 for ρ< 1 m, ρv =2 sin(2000πρ)n C/m3 for 1m < ρ< 1.5mm, and ρv =0 for ρ> 1.5mm. Find D everywhere: Since the charge varies only with radius, and is in the form of a cylinder, symmetry tells us that the ﬂux density will be radially-directed and will be constant over a cylindrical surface of a ﬁxed radius. Gauss’ law applied to such a surface of unit length in z gives:

a) for ρ< 1 m, Dρ =0 , since no charge is enclosed by a cylindrical surface whose radius lies within this range.

b) for 1mm <ρ< 1.5mm, we have

3.1. (continued) c) for ρ> 1.5mm, the gaussian cylinder now lies at radius ρ outside the charge distribution, so the integral that evaluates the enclosed charge now includes the entire charge distribution. To accomplish this, we change the upper limit of the integral of part b from ρ to 1.5 m, ﬁnally obtaining:

3.12. The sun radiates a total power of about 2 × 1026 watts (W). If we imagine the sun’s surface to be marked oﬀ in latitude and longitude and assume uniform radiation, (a) what power is radiated by the region lying between latitude 50◦ N and 60◦ N and longitude 12◦ W and 27◦ W? (b) What is the power density on a spherical surface 93,0,0 miles from the sun in W/m2?

ascertain that D will be radially-directed and will vary only with radius. Thus, we apply Gauss’ law to spherical shells in the following regions: r< 2: Here, no charge is enclosed,

b) Determine ρs0 such that D =0 at r =7 m. Since ﬁelds will decrease as 1/r2, the question could be re-phrased to ask for ρs0 such that D =0 at all points where r> 6m .I n this region, the total ﬁeld will be

3.14. The sun radiates a total power of about 2 × 1026 watts (W). If we imagine the sun’s surface to be marked oﬀ in latitude and longitude and assume uniform radiation, (a) what power is radiated by the region lying between latitude 50◦ N and 60◦ N and longitude 12◦ W and 27◦ W? (b) What is the power density on a spherical surface 93,0,0 miles from the sun in W/m2?

3.15. Volume charge density is located as follows: ρv =0 for ρ< 1m m and for ρ> 2 m, ρv =4 ρµ C/m3 for 1 <ρ< 2 m.

a) Calculate the total charge in the region 0 <ρ<ρ 1,0 <z <L , where 1 <ρ 1 < 2 m: Weﬁnd

4ρρd ρdφdz = 8πL

where ρ1 is in meters.

b) Use Gauss’ law to determine Dρ at ρ = ρ1: Gauss’ law states that 2πρ1LDρ = Q, where Q is the result of part a.T hus

where ρ1 is in meters.

c) Evaluate Dρ at ρ =0 .8mm, 1.6mm, and 2.4mm: At ρ =0 .8mm, no charge is enclosed bya cylindrical gaussian surface of that radius, so Dρ(0.8mm) = 0.A t ρ =1 .6mm, we evaluate the part b result at ρ1 =1 .6t o obtain:

At ρ =2 .4, we evaluate the charge integral of part a from .001 to .002, and Gauss’ law is written as

3.16. In spherical coordinates, a volume charge density ρv =1 0e−2r C/m3 is present. (a) Determine D. (b) Check your result of part a by evaluating ∇· D.

a) apply Gauss’ law to ﬁnd the total ﬂux leaving the closed surface of the cube. We call the surfaces at x =1 .2 and x =1 the front and back surfaces respectively, those at y =1 .2 and y =1 the right and left surfaces, and those at z =1 .2 and z =1 the top and bottom surfaces. To evaluate the total charge, we integrate D · n over all six surfaces and sum the results. We note that there is no z component of D,s o there will be no outward ﬂux contributions from the top and bottom surfaces. The ﬂuxes through the remaining four are

front right b) evaluate ∇· D at the center of the cube: This is

c) Estimate the total charge enclosed within the cube by using Eq. (8): This is

3.18. State whether the divergence of the following vector ﬁelds is positive, negative, or zero: (a) the thermal energy ﬂow in J/(m2 − s) at any point in a freezing ice cube; (b) the current density in A/m2 in a bus bar carrying direct current; (c) the mass ﬂow rate in kg/(m2 − s) below the surface of water in a basin, in which the water is circulating clockwise as viewed from above.

3.19. A spherical surface of radius 3 m is centered at P(4,1,5) in free space. Let D = xax C/m2. Use the results of Sec. 3.4 to estimate the net electric ﬂux leaving the spherical surface: We

3.20. Suppose that an electric ﬂux density in cylindrical coordinates is of the form D = Dρ aρ.

Describe the dependence of the charge density ρv on coordinates ρ, φ, and z if (a) Dρ = f(φ,z); (b) Dρ =( 1/ρ)f(φ,z); (c) Dρ = f(ρ).

3.21. Calculate the divergence of D at the point speciﬁed if

∂Dφ∂φ + ∂Dz∂z

c) D =2 r sinθ sinφar + r cosθ sinφaθ + r cosφaφ at P(3,45◦,−45◦): In spherical coordinates, we have r sinθ ∂ ∂θ

(sinθDθ)+ r sinθ ∂Dφ

cos2θ sinφ sinθ − sinφ

3.2. (a) A ﬂux density ﬁeld is given as F1 =5 az.E valuate the outward ﬂux of F1 through the hemispherical surface, r = a,0 <θ <π /2, 0 <φ< 2π. (b) What simple observation would have saved a lot of work in part a? (c) Now suppose the ﬁeld is given by F2 =5 zaz. Using the appropriate surface integrals, evaluate the net outward ﬂux of F2 through the closed surface consisting of the hemisphere of part a and its circular base in the xy plane. (d) Repeat part c by using the divergence theorem and an appropriate volume integral.

3.23. a) A point charge Q lies at the origin. Show that div D is zero everywhere except at the origin. For a point charge at the origin we know that D = Q/(4πr2)ar. Using the formula for divergence in spherical coordinates (see problem 3.21 solution), we ﬁnd in this case that

r2 d dr

The above is true provided r> 0. When r =0 ,w eh ave a singularity in D,s o its divergence is not deﬁned.

b) Replace the point charge with a uniform volume charge density ρv0 for 0 <r <a . Relate ρv0 to Q and a so that the total charge is the same. Find div D everywhere: To achieve the same net charge, we require that (4/3)πa3ρv0 = Q,s o ρv0 =3 Q/(4πa3)C /m3. Gauss’ law tells us that inside the charged sphere

Qr3

Thus

Dr = Qr r2 d dr as expected. Outside the charged sphere, D = Q/(4πr2)ar as before, and the divergence is zero.

3.24. (a) A uniform line charge density ρL lies along the z axis. Show that ∇· D =0 everywhere except on the line charge. (b) Replace the line charge with a uniform volume charge density ρ0 for 0 <ρ<a . Relate ρ0 to ρL so that the charge per unit length is the same. Then ﬁnd ∇· D everywhere.

3.25. Within the spherical shell, 3 <r < 4m , the electric ﬂux density is given as

D =5 (r −3)3ar C/m2 a) What is the volume charge density at r =4 ?I n this case we have

r2 d dr

b) What is the electric ﬂux density at r =4 ? Substitute r =4 into the given expression to ﬁnd D(4) = 5ar C/m2 c) How much electric ﬂux leaves the sphere r =4 ? Using the result of part b, this will be Φ=4 π(4)2(5) = 320π C d) How much charge is contained within the sphere, r =4 ?F rom Gauss’ law, this will be the same as the outward ﬂux, or again, Q = 320π C.

3.26. If we have a perfect gas of mass density ρm kg/m3, and assign a velocity U m/s to each diﬀerential element, then the mass ﬂow rate is ρmU kg/(m2 − s). Physical reasoning then leads to the continuity equation, ∇· (ρmU)= −∂ρm/∂t. (a) Explain in words the physical interpretation of this equation. (b) Show that ∮ s ρmU · dS = −dM/dt, where M is the total mass of the gas within the constant closed surface, S, and explain the physical signiﬁcance of the equation.

a) Find ρv for r =0 .06 m: This radius lies within the ﬁrst region, and so

r2 d dr r2 d dr b) Find ρv for r =0 .1m : This is in the region where the second ﬁeld expression is valid. The 1/r2 dependence of this ﬁeld yields a zero divergence (shown in Problem 3.23), and so the volume charge density is zero at 0.1 m.

c) What surface charge density could be located at r =0 .08 m to cause D =0 for r> 0.08 m? The total surface charge should be equal and opposite to the total volume charge. The latter is

So now

3.28. Repeat Problem 3.8, but use ∇· D = ρv and take an appropriate volume integral. 3.29. In the region of free space that includes the volume 2 <x ,y,z < 3,

a) Evaluate the volume integral side of the divergence theorem for the volume deﬁned above: In cartesian, we ﬁnd ∇· D =8 xy/z3. The volume integral side is now

vol

8xy b. Evaluate the surface integral side for the corresponding closed surface: We call the surfaces at x =3 and x =2 the front and back surfaces respectively, those at y =3 and y =2 the right and left surfaces, and those at z =3 and z =2 the top and bottom surfaces. To evaluate the surface integral side, we integrate D · n over all six surfaces and sum the results. Note that since the x component of D does not vary with x, the outward ﬂuxes from the front and back surfaces will cancel each other. The same is true for the left and right surfaces, since Dy does not vary with y. This leaves only the top and bottom surfaces, where the ﬂuxes are:

3.30. Let D =2 0ρ2 aρ C/m2. (a) What is the volume charge density at the point P(0.5,60◦,2)? (b) Use two diﬀerent methods to ﬁnd the amount of charge lying within the closed surface

(Parte 1 de 2)