What is meant by continuous charge distribution?

The electric field II: Continuous charge distributions

Comprehension tasks

19.1 • 

Right or wrong? a) The electric field caused by a homogeneously charged thin spherical shell is zero at all points within the shell. b) In electrostatic equilibrium, the electric field must be zero everywhere inside a conductor. c) When the total charge on a conductor is zero, the charge density at any point on the surface of the conductor must be zero.

19.2 •• 

A single point charge q is at the center of both an imaginary cube and an imaginary sphere. What is the relationship between the electrical flow through the surface of the cube and the flow through the surface of the sphere? Explain your answer.

19.3 •• 

Explain why the electric field between the center point and the surface of a homogeneously charged full sphere with r increases instead of decreasing with \ (1 / r ^ {2} \).

19.4 •• 

The total charge on the conductive spherical shell in Figure 19.37 is zero. The negative point charge in the center carries the charge q. Which direction does the electric field have in the following areas? a) \ (r r> r_ {1} \), c) \ (r> r_ {2} \). Explain your answer.

19.5 •• 

The conductive spherical shell in Figure 19.37 is grounded on the outside. The negative point charge in the center carries the charge q. Which of the following statements is true? a) The charge on the inner surface of the spherical shell is +q, and the charge on the outer surface is -q. b) The charge is on the inner surface of the spherical shell q, and the charge on the outer surface is zero. c) The charge on both surfaces of the spherical shell is +q. d) The charge on both surfaces of the spherical shell is zero.

19.6 •• 

The conductive spherical shell in Figure 19.37 is grounded on the outside. The negative point charge in the center carries the charge q. Which direction does the electric field have in the following areas? a) \ (r r> r_ {1} \), c) \ (r> r_ {2} \). Explain your answer.

Estimation and approximation tasks

19.7 •• 

In this chapter we derived the expression for the electric field on the axis of a homogeneously charged disk; is at one point on the axis
$$ | E | = \ frac {\ sigma} {2 \, \ varepsilon_ {0}} \, \ left [1- \ left (1+ \ frac {r ^ {2}} {z ^ {2}} \ right) ^ {- 1/2} \ right] \,. $$
At a great distance (\ (| z | \ gg r \)) the field approaches the expression \ (E \ approx (1/4 \ uppi \ varepsilon_ {0}) (q / z ^ {2}) \), very close to the disk (\ (| z | \ ll r \)) the field strength runs approximately as in an infinitely extended charged plane with \ (| E | \ approx \ sigma / (2 \, \ varepsilon_ {0}) \ ). You have a disk with a radius of 2.5 cm with a homogeneous surface charge density of 3.6 \ (\ upmu \) C / m \ ({} ^ {2} \). Using both the exact expression and the approximate expressions, find the electric field strength on the axis at a) 0.010 cm, b) 0.040 cm and c) 5.0 m from the disc. Compare the two values ​​in each case and assess how good the approximation is in their respective range of validity.

Calculation of E. from Coulomb’s law

19.8 •• 

A homogeneous line charge with the linear charge density \ (\ lambda = \ text {3 {,} 5 \, nC / m} \) extends over the x-Axis of x = 0 to \ (x = \ text {5 {,} 0 \, m} \). a) What is the total charge? Calculate the electric field on the x-Axis at b) \ (x = \ text {6 {,} 0 \, m} \), c) \ (x = \ text {9 {,} 0 \, m} \) and d) \ (x = \ text {250 \, m} \). e) Determine the field at \ (x = \ text {250 \, m} \); use the approximation that the charge is a point charge at \ (x = \ text {2 {,} 5 \, m} \) and compare your result with the exact calculation from part d. (To do this, you must assume that the numerical values ​​given apply to more than two significant digits.) Is your approximate result greater or smaller than the exact result? Explain your answer.

19.9 •• 

a) Show that the electric field strength E. on the axis of a ring charge of radius a Has maximum and minimum values ​​at \ (z = + a / \ sqrt {2} \) and \ (z = -a / \ sqrt {2} \). b) Sketch the course of E. above z for positive and negative z-Values. c) Find the maximum value of E..

19.10 •• 

A line charge with a homogeneous linear charge density λ extends along the x-Axis from \ (x = x_ {1} \) to \ (x = x_ {2} \) (with \ (x_ {1} x-Component of the electric field on a point of y-Axis is given by
$$ E_ {x} = \ frac {1} {4 \ pi \ varepsilon_ {0}} \, \ frac {\ lambda} {y} \, (\ mathrm {cos \>} \ theta_ {2} - \ mathrm {cos \>} \ theta_ {1}) \,. $$
Here \ (\ theta_ {1} = \ mathrm {atan \>} (x_ {1} / y) \) and \ (\ theta_ {2} = \ mathrm {atan \>} (x_ {2} / y ) \) and \ (y \ neq 0 \).

19.11 •• 

A ring of radius \ (r _ {\ text {R}} \) has a charge distribution \ (\ lambda (\ theta) = \ lambda_ {0} \, \ mathrm {sin \>} \ theta \) (Figure 19.38) . a) In which direction does the electric field point in the center of the ring? b) What is the amount of the field in the center of the ring?

19.12 ••• 

A thin hemispherical shell of radius \ (r _ {\ mathrm {K}} \) has a homogeneous surface charge density σ. Find the electric field in the center of the base of the hemispherical shell.

Gaussian law

19.13 • 

We consider a homogeneous electric field \ (\ varvec {E} = (2 {,} 00 \, \ text {kN / C}) \, \ varvec {\ widehat {x}} \). a) How large is the electric flux of this field through a square area with a side length of 10 cm, which is on the x-Axis is centered and its normal in the positive x-Direction points? b) What is the electric flux through the same square if its normal with the y-Axis an angle of 30 \ ({} ^ {\ circ} \) and with the z-Axis encloses an angle of 90 \ ({} ^ {\ circ} \)?

19.14 •• 

Since Newton's law of gravitation and Coulomb's law have the same distance dependency in the form of a \ ((1 / r ^ {2}) \) law, one can also use an analogy to Gauss's law for the electrical flow Find an expression for the vector flow of the gravitational field. The gravitational field G at one point has been defined in Chapter 4 as the force per unit mass on a test mass m0 in this point. For a field-generating mass m at the origin of the coordinate system the gravitational field at a location \ (\ boldsymbol {\ widehat {r}} \) is given by
$$ \ boldsymbol {G} = - \ varGamma \, \ frac {m} {r ^ {2}} \, \ boldsymbol {\ widehat {r}} \,. $$
Calculate the flow of the gravitational field through a spherical surface with the radius \ (r _ {\ text {K}} \) and the center at the origin of the coordinates. Show that in analogy to Gauss law for the electric flow for gravity \ (\ varPhi _ {\ rm grav} = - \, 4 \ uppi \, \ varGamma \, m _ {\ text {inside}} \) applies.

19.15 •• 

An imaginary perpendicular circular cone (Figure 19.39) with the base angle θ and the base radius \ (r _ {\ mathrm {K}} \) is located in a charge-free area with a homogeneous electric field E. (the field lines run vertically, parallel to the cone axis). What is the relationship between the number of field lines per unit area that penetrate the base of the cone and the number of field lines per unit area that penetrate the surface of the cone? Apply Gauss’s Law. (The field lines shown in the figure are only representative examples.)

19.16 •• 

In a certain area of ​​the earth's atmosphere, the electric field above the earth's surface was measured with the following results: 150 N / C at a height of 250 m and 170 N / C at a height of 400 m. In both cases the electric field is directed downwards towards the earth. Calculate the space charge density of the atmosphere between 250 and 400 m assuming that it is homogeneous in this area. (The curvature of the earth can be neglected. Why?)

Applications of Gauss’s law for spherical symmetry

19.17 • 

A thin, non-conductive spherical shell with a radius \ (r _ {\ text {{K}}, 1} \) carries a total charge q1that is evenly distributed on its surface. A second, larger spherical shell with the radius \ (r _ {\ rm {K}, 2} \), which is concentric to the first, carries a charge q2which is also evenly distributed on its surface. a) Apply Gauss law and determine the electric field in the areas \ (r r _ {\ text {{K}}, 2} \). b) How do you have to choose the ratio of the charges \ (q_ {1} / q_ {2} \) and their relative signs so that the electric field for \ (r> r _ {\ text {{K}}, 2} \ ) is zero? c) Sketch the electric field lines for the situation in subtask b, if q1 is positive.

19.18 •• 

Consider the solid conductive sphere and the concentrically arranged conductive spherical shell from Figure 19.40. The spherical shell carries the charge \ (- 7q \), the full sphere the charge \ (+ 2q \). a) Which charge is on the outer surface of the spherical shell, which on the inner one? b) Now a metal wire is drawn in between the ball socket and the ball. What charge is there on the sphere and on the surfaces of the spherical shell after the electrostatic equilibrium has been reached? Does the electric field on the surface of the ball change when the wire is pulled in? If so, in what way? c) We now return to the charge distribution of subtask a, connect the spherical shell to the earth with a metal wire and interrupt the connection again. Which charges are now on the sphere and on the surfaces of the spherical shell?

19.19 •• 

A non-conductive sphere with the radius \ (r _ {\ rm {K}} \) has a space charge density that is proportional to the distance from the center: \ (\ rho = B \, r \) for \ (r \ leq r _ {\ text {{K}}} \). In it is B. a constant. For \ (r> r _ {\ rm {K}} \) \ (\ rho = 0 \). a) Determine the total charge on the sphere by plotting the charges on spherical shells of thickness \ (\, \ mathrm {d} r \) and volume \ (4 \ uppi \, r ^ {2} \, \, \ mathrm {d} r \) add up. b) Determine the electric field E. inside and outside of the charge distribution. c) Sketch the electric field E. depending on the distance r to the center of the sphere.

19.20 •• 

Repeat Exercise 19.19 for the space charge density \ (\ rho = C / r \) for \ (r r_ { \ text {{K}}} \) (C. is a constant).

Applications of Gauss’s law for cylindrical symmetry

19.21 •• 

For an internship, you build a Geiger-Müller counter tube to detect ionizing radiation. The counter tube consists of a long cylindrical tube with a thin metal wire stretched along its axis. The wire has a thickness of 0.500 mm, the inner diameter of the counter tube is 4.00 cm. The counter tube is filled with a diluted gas in which a gas discharge (a voltage breakdown in the gas) takes place when the electric field strength reaches a value of \ (\ text {5 {,} 50} \ cdot 10 ^ {6} \, \ text {N / C} \) reached. Determine the maximum value of the linear charge density on the wire at which voltage breakdown does not yet occur. Assume that the counter tube and wire are infinitely long.

19.22 •• 

An infinitely long, non-conductive, massive cylinder with the radius \ (r _ {\ text {{{Z}}} \) has a homogeneous space charge density of \ (\ rho (r) = \ rho_ {0} \). Show that the electric field is through
$$ \ begin {aligned} E & = \ frac {\ rho} {2 \ varepsilon_ {0}} \, r _ {{} _ {\ bot}} \ qquad (0 \ leq r _ {{} _ {\ bot} } r _ {\ text {{Z}}}) \ end {aligned} $$
given is. \ (r _ {{} _ {\ bot}} \) is the distance from the longitudinal axis of the cylinder.

19.23 •• 

Figure 19.41 shows the cross-section of an infinitely long coaxial cable. The inner conductor has a linear charge density of 6.00 nC / m, the outer conductor is uncharged. a) Find the electric field for all values r the distance from the axis of the concentric cylinder system. b) What are the surface charge densities on the inner and outer surfaces of the outer conductor?

19.24 ••• 

Take another look at your Geiger-Müller counter tube from Exercise 19.21. Ionizing radiation has generated an ion and an electron in the counter tube at a distance of 2.00 cm from the longitudinal axis of the wire. The wire should be positively charged and have a linear charge density of 76.5 pC / m. a) What is the speed of the electron in this case when it hits the wire? b) Compare the electron speed with the final speed of the ion when it hits the inner surface of the counter tube. Explain your answer.

Electric charges and fields on conductor surfaces

19.25 • 

An uncharged copper coin is in a homogeneous electric field with a strength of 1.60 kN / C, which is perpendicular to its surface. a) Determine the charge density on each face of the copper coin assuming that the faces are flat. b) Find the total charge on one of the faces if the radius of the coin is 1.00 cm.

19.26 •• 

150 N / C was measured for the downward electric field immediately above the earth's surface. a) What is the sign of the total charge on the earth's surface under these conditions? b) Which total charge on the earth's surface does this measurement correspond to?

19.27 •• 

If the strength of an electric field in air is about \ (3 {,} 0 \ cdot 10 ^ {6} \, \ text {N / C} \), then the air becomes ionized and electrically conductive. This phenomenon is called dielectric breakdown (or dielectric discharge). A charge of 18 \ (\ upmu \) C is placed on a conductive sphere. Up to what minimum radius can the ball just hold this charge without breaking through?

general tasks

19.28 •• 

Consider the three concentric metal balls or spherical shells shown in Figure 19.42. Sphere 1 is a solid sphere with the radius \ (r _ {\ text {K, 1}} \), sphere 2 is a hollow sphere with the inner radius \ (r _ {\ text {K, 2}} \) and the outer radius \ ( r _ {\ text {K, 3}} \), and sphere 3 is a hollow sphere with the inner radius \ (r _ {\ text {K, 4}} \) and the outer radius \ (r _ {\ text {K, 5} } \). At the beginning all three balls are uncharged. Then a negative charge \ (- q_ {0} \) is placed on ball 1 and a positive charge \ (+ q_ {0} \) on ball 3. a) In which direction does the electric field in the space between balls 1 and 2 point when electrostatic equilibrium has been established? b) How big is the charge on the inner surface of sphere 2? Enter the sign of this charge. c) How big is the charge on the outer surface of sphere 2? d) How big is the charge on the inner surface of sphere 3? e) How big is the charge on the outer surface of sphere 3? f) Set E. in dependence of r represent.

19.29 •• 

A thin, non-conductive, homogeneously charged spherical shell with a radius r (Figure 19.43a) carries a total charge q. A small circular plug is removed from the surface. a) State the magnitude and direction of the electric field in the center of the hole b) The plug is reinserted into the hole (Figure 19.43 b). Use the result of part a to calculate the electrostatic force on the plug. c) Using the magnitude of the force, calculate the electrostatic pressure (force / unit area) that tries to expand the sphere.

19.30 •• 

An infinitely extended plane in the x-z-Plane carries a homogeneous surface charge density \ (\ sigma_ {1} = + \ text {65 \, nC / m $ {} ^ {2} $} \). A second infinitely extended plane with a homogeneous charge density of \ (\ sigma_ {2} = + \ text {45 \, nC / m $ {} ^ {2} $} \) intersects the x-zLevel in the zAxis and closes with the x-z-Plane an angle of 30 \ ({} ^ {\ circ} \) (Figure 19.44). Determine the electric field at a) \ (x = \ text {6 {,} 0 \, m} \), \ (y = \ text {2 {,} 0 \, m} \) and b) at \ (x = \ text {6 {,} 0 \, m} \), \ (y = \ text {5 {,} 0 \, m} \).

19.31 •• 

A quantum mechanical consideration of the hydrogen atom shows that the electron in this atom can be seen as a smeared negative charge distribution with the form \ (\ rho (r) = - \ rho_ {0} \, {\ rm e} ^ {- 2 \, r / a} \).In it is r the distance from the core and a the first Bohr radius (\ (a = \ text {0 {,} 0529 \, nm} \)). Remember that the nucleus of the hydrogen atom is made up of a proton, which you can think of as a positive point charge. a) Calculate \ (\ rho_ {0} \) taking into account the fact that the atom is uncharged. b) Give the electric field as a function of the distance r from the core.

19.32 •• 

A stationary ring of radius \ (r _ {\ text {R}} \) lies in the y-zLevel and carries a positive charge qwhich is evenly distributed along its length. A point particle with the mass m and a negative charge -q is in the center of the ring. a) Show that for \ (x \ ll r _ {\ text {R}} \) the electric field along the ring axis is proportional to x is. b) Determine the force on the particle with the mass m as a function of x. c) Show that the particle has a small displacement in positive x-Direction carries out a harmonic oscillation. d) What is the frequency of this movement?

19.33 •• 

A homogeneously charged, non-conductive full sphere with the radius a and the center point in the origin of coordinates has a space charge density ρ. a) Show that at a point inside the sphere at a distance r from the center through the electric field
$$ \ boldsymbol {E} = \ frac {\ rho} {3 \, \ varepsilon_ {0}} \, r \, \ boldsymbol {\ widehat {r}} $$
given is. b) Now material is removed from the sphere, so that a spherical cavity with the radius \ (b = a / 2 \) and the center at \ (x = b \) on the xAxis is created (Figure 19.45). Calculate the electric field at points 1 and 2, which are shown in Figure 19.45. (Hint: Replace the sphere with the cavity with two homogeneous spheres with equally large positive and negative charge densities.)

19.34 •• 

Consider a simple but surprisingly accurate model of the hydrogen molecule: two positive point charges, each with a + chargee, are inside a sphere of radius r with a homogeneous charge density and the total charge \ (- 2e \). The two point charges are spatially symmetrical, that is, they are each arranged equidistant from the center of the sphere (Figure 19.46). Determine the distance a from the center of the sphere, at which the resulting force on each of the two point charges is zero.

19.35 ••• 

Show that the electric field in the cavity of Exercise 19.33b is homogeneous and through
$$ \ boldsymbol {E} = \ frac {\ rho} {3 \, \ varepsilon_ {0}} \, b \, \ boldsymbol {\ widehat {x}} $$
is described.

19.36 ••• 

A small Gaussian surface in the shape of a cube with faces parallel to the x-y-, the x-z- and the y-zPlane (Figure 19.47), is located in a space in which the electric field is parallel to x-Axis is. a) Show with the aid of the Taylor series (neglecting all terms from the second order) that the total flow of the electric field from the Gaussian surface through
$$ \ varPhi _ {\ text {el}} = \ frac {\ partial E_ {x}} {\ partial x} \, \ Updelta V $$
given is; \ (\ Updelta V \) is the volume enclosed by the Gaussian surface. b) Show with Gauss’s law and the results from subtask a that the following applies:
$$ \ frac {\ partial E_ {x}} {\ partial x} = \ frac {\ rho} {\ varepsilon_ {0}} \,. $$
Here ρ is the space charge density within the cube.
Comment: This equation is the one-dimensional version of Gauss’s theorem. The corresponding result for examples in which the direction of the electric field is not limited to one-dimensional problems is
$$ \ varPhi _ {\ mathrm {el}} = \ left (\ frac {\ partial E_ {x}} {\ partial x} + \ frac {\ partial E_ {y}} {\ partial y} + \ frac { \ partial E_ {z}} {\ partial z} \ right) \, \ Updelta V \ ,, $$
where the sum of the partial derivatives in the round brackets, as usual in vector analysis, is summarized as \ (\ nabla \ cdot \ boldsymbol {E} \) and as the divergence of the vector field E. referred to as. The symbol \ (\ nabla \) (called Nabla) is the vector operator
$$ \ nabla = \ frac {\ partial} {\ partial x} \, \ boldsymbol {\ widehat {x}} + \ frac {\ partial} {\ partial y} \, \ boldsymbol {\ widehat {y}} + \ frac {\ partial} {\ partial z} \, \ boldsymbol {\ widehat {z}} \,. $$