Maxwell’s Equations; Magnetism of Matter

Figure 32-19a shows a capacitor, with circular plates, that is being charged. Point a (near one of the connecting wires) and point b (inside the capacitor gap) are equidistant from the central axis, as are point c (not so near the wire) and point d (between the plates but outside the gap). In Fig. 32-19b, one curve gives the variation with distance r of the magnitude of the magnetic field inside and outside the wire. The other curve gives the variation with distance r of the magnitude of the magnetic field inside and outside the gap. The two curves partially overlap. Which of the three points on the curves correspond to which of the four points of Fig. 32-19a?

Figure 32-20 shows a parallel-plate capacitor and the current in the connecting wires that are discharging the capacitor. Are the directions of (a) electric field $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$ and (b) displacement current i_d leftward or rightward between the plates? (c) Is the magnetic field at point P into or out of the page?

Figure 32-21 shows, in two situations, an electric field vector $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$ and an induced magnetic field line. In each, is the magnitude $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$ increasing or decreasing?

Figure 32-22a shows a pair of opposite spins orientations for an electron in an external magnetic field ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$. Figure 32-22b gives three choices for the graph of the energies associated with those orientations as a function of the magnitude ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$. Choices b and c consist of intersecting lines and choice of parallel lines. Which is the correct choice?

An electron in an external magnetic field ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$. has its spin angular momentum S_z antiparallel to ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$. If the electron undergoes a spin-flip so that S_z is then parallel th ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$, must energy be supplied to or lost by the electron?

Does the magnitude of the net force on the current loop of Figs. 32-12a and b increase, decrease, or remain the same if we increase (a) the magnitude of ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{e}\mathbf{x}\mathbf{t}}$ and (b) the divergence of ${\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}}_{\mathbf{ext}}$?

Figure 32-23 shows a face-on view of one of the two square plates of a parallel-plate capacitor, as well as four loops that are located between the plates. The capacitor is being discharged. (a) Neglecting fringing of the magnetic field, rank the loops according to the magnitude of $\mathbf{\oint }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{·}\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{s}}$ along them, greatest first. (b) Along which loop, if any, is the angle between the directions of $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{s}}$ constant (so that their dot product can easily be evaluated)? (c) Along which loop, if any, is B constant (so that B can be brought in front of the integral sign in Eq. 32-3)?

Figure 32-24 shows three loop models of an electron orbiting counterclockwise within a magnetic field. The fields are non-uniform for models 1 and 2 and uniform for model 3. For each model, are (a) the magnetic dipole moment of the loop and (b) the magnetic force on the loop directed up, directed down, or zero?

Replace the current loops of Question 8 and Fig. 32-24 with diamagnetic spheres. For each field, are (a) the magnetic dipole moment of the sphere and (b) the magnetic force on the sphere directed up, directed down, or zero?

Replace the current loops of Question 8 and Fig. 32-24 with paramagnetic spheres. For each field, are (a) the magnetic dipole moment of the sphere and (b) the magnetic force on the sphere directed up, directed down, or zero?

Figure 32-25 represents three rectangular samples of a ferromagnetic material in which the magnetic dipoles of the domains have been directed out of the page (encircled dot) by a very strong applied field B₀ . In each sample, an island domain still has its magnetic field directed into the page (encircled X ). Sample 1 is one (pure) crystal. The other samples contain impurities collected along lines; domains cannot easily spread across such lines. 

The applied field is now to be reversed and its magnitude kept moderate. The change causes the island domain to grow. (a) Rank the three samples according to the success of that growth, greatest growth first. Ferromagnetic materials in which the magnetic dipoles are easily changed are said to be magnetically soft; when the changes are difficult, requiring strong applied fields, the materials are said to be magnetically hard. (b) Of the three samples, which is the most magnetically hard?

Figure 32-26 shows four steel bars; three are permanent magnets. One of the poles is indicated. Through experiment ends a and d attract each other, ends c and f repel, ends e and h attract, and ends a and h attract. (a) Which ends are the north poles? (b) Which bar is not a magnet?

The magnetic flux through each of five faces of a die (singular of “dice”) is given by ${\mathit{\phi }}_{\mathbf{B}}\mathbf{=}\mathbf{\pm }\mathit{N}\mathit{W}\mathit{b}$, where N(= 1 to 5) is the number of spots on the face. The flux is positive (outward) for N even and negative (inward) for N odd. What is the flux through the sixth face of the die?

Figure 32-27 shows a closed surface. Along the flat top face, which has a radius of 2.0 cm, a perpendicular magnetic field $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}$ of magnitude 0.30 T is directed outward. Along the flat bottom face, a magnetic flux 0.70 mWb is directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

A Gaussian surface in the shape of a right circular cylinder with end caps has a radius of 12.0 cm and a length of 80.0 cm. Through one end there is an inward magnetic flux $\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{\mu }\mathit{W}\mathit{b}$. At the other end, there is a uniform magnetic field 1.60 mT, normal to the surface and directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the net magnetic flux through the curved surface?

Two wires, parallel to a z-axis and a distance 4r apart, carry equal currents i in opposite directions, as shown in Figure. A circular cylinder of radius r and length L has its axis on the z-axis, midway between the wires. Use Gauss’ law for magnetism to derive an expression for the net outward magnetic flux through the half of the cylindrical surface above the x axis. (Hint: Find the flux through the portion of the xz plane that lies within the cylinder.)

The induced magnetic field at a radial distance 6.0 mm from the central axis of a circular parallel-plate capacitor is $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{ }\mathit{T}$. The plates have a radius 3.0 mm. At what rate $\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}\mathbf{/}\mathit{d}\mathit{t}$ is the electric field between the plates changing?

A capacitor with square plates of edge length L is being discharged by a current of 0.75 A. Figure 32-29 is a head-on view of one of the plates from inside the capacitor. A dashed rectangular path is shown. If L = 12cm, W = 4.0 cm , and H = 2.0 cm , what is the value $\mathbf{\oint }\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{\times }\mathit{d}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{s}}$ of around the dashed path?

Uniform electric flux. Figure 32-30 shows a circular region of radius R = 3.00 cm in which a uniform electric flux is directed out of the plane of the page. The total electric flux through the region is given by ${\mathbf{\varnothing }}_{\mathbf{E}}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{m}\mathbf{V}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}\mathit{t}$, where is t in seconds. (a)What is the magnitude of the magnetic field that is induced at a radial distance 2.00 cm? (b)What is the magnitude of the magnetic field that is induced at a radial distance 5.00 cm?

Figure 32-30

The figure shows a circular region of radius $\mathit{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{cm}}$in which an electric flux is directed out of the plane of the page. The flux encircled by a concentric circle of radius r is given by ${\mathit{\Phi }}_{\mathbf{E}\mathbf{,}\mathbf{enc}}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{600}\mathbf{ }\mathbf{\text{Vm/s}}\mathbf{)}\mathbf{(}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{)}\mathit{t}$, where $\mathit{r}\mathbf{\le }\mathit{R}$and t  is in seconds. (a)What is the magnitude of the induced magnetic field at a radial distance $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{cm}}$? (b)What is the magnitude of the induced magnetic field at a radial distance $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{cm}}$?

Uniform electric flux. Figure 32-30 shows a uniform electric field is directed out of the page within a circular region of radius $\mathit{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{cm}}$. The field magnitude is given by, $\mathit{E}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{50}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathbf{ }\mathbf{\text{V/ms}}\mathbf{)}\mathit{t}$ where t is in seconds. (a)What is the magnitude of the induced magnetic field at a radial distance 2.00 cm? (b) What is the magnitude of the induced magnetic field at a radial distance 5.00 cm?

The figure shows an electric field is directed out of the page within a circular region of radius $\mathit{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{cm}}$. The field magnitude is $\mathit{E}\mathbf{=}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{ }\mathbf{\text{V/ms}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{)}\mathit{t}$, where t is in seconds and r is the radial distance $\left(\mathbf{r}\le \mathbf{R}\right)$. What is the magnitude of the induced magnetic field at a radial distance $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{cm}}$? What is the magnitude of the induced magnetic field at a radial distance $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{cm}}$?

Suppose that a parallel-plate capacitor has circular plates with a radius $\mathit{R}\mathbf{=}\mathbf{30}\mathbf{ }\mathbf{\text{mm}}$ and, a plate separation of  $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{mm}}$. Suppose also that a sinusoidal potential difference with a maximum value of  $\mathbf{150}\mathbf{ }\mathbf{\text{V}}$ and, a frequency of  $\mathbf{60}\mathbf{ }\mathbf{\text{Hz}}$ is applied across the plates; that is,

$\mathit{V}\mathbf{=}\mathbf{(}\mathbf{150}\mathbf{ }\mathbf{\text{V}}\mathbf{)}\mathbf{\text{sin}}\mathbf{\left[}\mathbf{2}\mathbf{\pi }\mathbf{(}\mathbf{60}\mathbf{ }\mathbf{\text{Hz}}\mathbf{)}\mathbf{t}\mathbf{\right]}$

(a) Find ${\mathbf{\text{B}}}_{\mathit{m}\mathit{a}\mathit{x}}\left(\mathit{R}\right)$, the maximum value of the induced magnetic field that occurs at $\mathit{r}\mathbf{=}\mathit{R}$.   

(b) Plot ${\mathbf{B}}_{\mathbf{max}}\left(\mathbf{r}\right)$  for  $\mathbf{0}\mathbf{<}\mathit{r}\mathbf{<}\mathbf{10}\mathbf{ }\mathbf{\text{cm}}$.

Question: A parallel-plate capacitor with circular plates of radius 40 mm is being discharged by a current of 6.0 A . At what radius (a) inside and (b) outside the capacitor, the gap is the magnitude of the induced magnetic field equal to 75% of its maximum value? (c) What is that maximum value?

At what rate must the potential difference between the plates of a parallel-plate capacitor with a$\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{\mu }\mathbf{\text{F}}$ capacitance be changed to produce a displacement current of $\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{ }\mathit{A}$?

A parallel-plate capacitor with circular plates of the radius $\mathit{R}$is being charged. Show that, the magnitude of the current density of the displacement current is

${\mathit{J}}_{\mathbf{d}}\mathbf{=}{\mathit{\epsilon }}_{\mathbf{0}}\left(\frac{dE}{dt}\right)\mathit{f}\mathit{o}\mathit{r}\mathbf{ }\mathit{r}\mathbf{⩽}\mathit{R}$.

Prove that the displacement current in a parallel-plate capacitor of capacitance  $\mathit{C}$ can be written as ${\mathit{i}}_{\mathbf{d}}\mathbf{=}\mathit{C}\mathbf{(}\mathbf{dV}\mathbf{/}\mathbf{dt}\mathbf{)}$, where $\mathit{V}$ is the potential difference between the plates.

A parallel-plate capacitor with circular plates of radius $\mathbf{0}\mathbf{.}\mathbf{10}\mathbf{\text{m}}$ is being discharged. A circular loop of radius $\mathbf{0}\mathbf{.}\mathbf{20}\mathbf{ }\mathit{m}$ is concentric with the capacitor and halfway between the plates. The displacement current through the loop is $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{A}$. At what rate is the electric field between the plates changing?

A silver wire has resistivity $\mathbf{\rho }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathbf{ }\mathit{\Omega }\mathit{m}$ and a cross-sectional area of $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{m}{\mathit{m}}^{\mathbf{2}}$. The current in the wire is uniform and changing at the rate of  $\mathbf{2000}\mathbf{ }\mathit{A}\mathbf{/}\mathit{s}$when the current is $\mathbf{100}\mathbf{ }\mathit{A}$. (a)What is the magnitude of the (uniform) electric field in the wire when the current in the wire is $\mathbf{100}\mathbf{ }\mathit{A}$? (b)What is the displacement current in the wire at that time? (c) What is the ratio of the magnitude of the magnetic field due to the displacement current to that due to the current at a distance $\mathit{r}$ from the wire?

The circuit in Fig. consists of switch S, a $\mathbf{12}\mathbf{.}\mathbf{0}\mathit{V}$  ideal battery, a $\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{M}\mathit{\Omega }$ resistor, and an air-filled capacitor. The capacitor has parallel circular plates of radius $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{c}\mathit{m}$ , separated by $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{m}\mathit{m}$ . At time $\mathit{t}\mathbf{=}\mathbf{0}$ , switch S is closed to begin charging the capacitor. The electric field between the plates is uniform. At $\mathit{t}\mathbf{=}\mathbf{250}\mathit{\mu }\mathit{s}$, what is the magnitude of the magnetic field within the capacitor, at a radial distance $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{c}\mathit{m}$?

The figure 32-30 shows a circular region of radius $\mathbf{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{cm}$  in which a displacement current is directed out of the page. The displacement current has a uniform density of magnitude  (a) What is the magnitude of the magnetic field due to displacement current at a radial distance  2.00 cm ?(b) What is the magnitude of the magnetic field due to displacement current at a radial distance  5.00 cm?

The figure shows  a circular region of radius $\mathbf{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$ in which a uniform displacement current  ${\mathit{i}}_{\mathbf{d}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{ }\mathit{A}$ is out of the page.(a)What is the magnitude of the magnetic field due to displacement current at a radial distance $\mathit{r}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{c}\mathit{m}$?(b)What is the magnitude of the magnetic field due to displacement current at a radial distance  $\mathbf{r}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$?

The figure 32-20 shows a circular region of radius$\mathit{R}\mathbf{=}\mathbf{3}\mathbf{cm}$  in which a displacement current is directed out of the page. The magnitude of the density of this displacement current is ${\mathit{J}}_{\mathbf{d}}\mathbf{=}\mathbf{(}\raisebox{1ex}{$\mathbf{4}\mathbf{ }\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}\mathbf{)}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{)}$}\right.$, where  is the radial distance  $\left(r\le R\right)$.(a) What is the magnitude of the magnetic field due to displacement current at  $\mathbf{2}\mathbf{cm}$?(b) What is the magnitude of the magnetic field due to displacement current at $\mathbf{5}\mathbf{cm}$ ?

The figure 32-20 shows a circular region of radius $\mathit{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$  in which a displacement current is directed out of the page. The magnitude of the density of this displacement current is ${\mathit{J}}_{\mathbf{d}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{)}$, where $\mathit{r}$ is the radial distance $\mathit{r}\mathbf{\le }\mathit{R}$. (a) What is the magnitude of the magnetic field due to displacement current at $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$? (b) What is the magnitude of the magnetic field due to displacement current at $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$?

Fig 32-20 

Figure 32-30 shows a circular region of radius $\mathit{R}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$ in which a displacement current ${\mathit{i}}_{\mathbf{d}}$ is directed out of the page. The magnitude of the displacement current is given by  ${\mathbf{i}}_{\mathbf{d}}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{A}\mathbf{)}\mathbf{(}\mathbf{r}\mathbf{/}\mathbf{R}\mathbf{)}$, where $\mathit{r}$ is the radial distance $(r\le R)$. (a) What is the magnitude of the magnetic field due to ${\mathit{i}}_{\mathbf{d}}$  at radial distance $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$? (b) What is the magnitude of the magnetic field due to  ${\mathit{i}}_{\mathbf{d}}$ at radial distance $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{cm}$?

In Fig. 32-32, a parallel-plate capacitor has square plates of edge length $\mathbf{L}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{m}$. A current of $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{A}$ charges the capacitor, producing a uniform electric field $\overrightarrow{\mathbf{E}}$ between the plates, with perpendicular to the plates. (a) What is the displacement current  ${\mathbf{i}}_{\mathbf{d}}$ through the region between the plates? (b)What is $\mathbf{dE}\mathbf{/}\mathbf{dt}$ in this region? (c) What is the displacement current encircled by the square dashed path of edge length $\mathbf{d}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{50}\mathit{m}$? (d)What is $\mathbf{\oint }\overrightarrow{\mathbf{B}}\mathbf{.}\mathbf{d}\overrightarrow{\mathbf{s}}$ around this square dashed path?

The magnitude of the electric field between the two circular parallel plates in Fig. is $\mathit{E}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{)}\mathbf{-}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{t}\mathbf{)}$, with $\mathit{E}$ in volts per meter and $\mathit{t}$ in seconds. At  $\mathbf{t}\mathbf{=}\mathbf{0}$, $\overrightarrow{\mathbf{E}}$ is upward. The plate area is $\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{2}}\mathbf{\hspace{0.17em}}{\mathbf{m}}^{\mathbf{2}}$. For $\mathit{t}\mathbf{\ge }\mathbf{0}$, (a) What is the magnitude and (b) What is the direction (up or down) of the displacement current between the plates, and (c) What is the direction of the induced magnetic field clockwise or counter-clockwise in the figure?

As a parallel-plate capacitor with circular plates $\mathbf{20}\mathbf{ }\mathbf{\text{cm}}$ in diameter is being charged, the current density of the displacement current in the region between the plates is uniform and has a magnitude of $\mathbf{20}\mathbf{ }\mathbf{A}\mathbf{/}{\mathbf{m}}^{\mathbf{2}}$. (a) Calculate the magnitude $\mathit{B}$ of the magnetic field at a distance  $\mathbf{r}\mathbf{=}\mathbf{50}\mathbf{ }\mathbf{mm}$ from the axis of symmetry of this region. (b) Calculate $\mathbf{dE}\mathbf{/}\mathbf{dt}$  in this region.

A capacitor with parallel circular plates of the radius  $\mathbf{R}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{ }\mathbf{cm}$ is discharging via a current of 12.0 A . Consider a loop of radius  R/3  that is centered on the central axis between the plates. (a)How much displacement current is encircled by the loop? The maximum induced magnetic field has a magnitude of  12.0 mT. (b)At what radius inside and (c)outside the capacitor gap is the magnitude of the induced magnetic field  3.00 mT?

In Fig, a uniform electric field $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{E}}$ collapses. The vertical axis scale is set by $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{ }\mathbf{N}\mathbf{/}\mathbf{C}$, and the horizontal axis scale is set by  ${\mathit{t}}_{\mathbf{s}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{\mu }\mathbf{s}$. Calculate the magnitude of the displacement current through a $\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{ }{\mathbf{m}}^{\mathbf{2}}$ area perpendicular to the field during each of the time intervals a,b, and cisshown on the graph.

Question: Figure 32-35a shows the current i   that is produced in a wire of resistivity$\mathbf{1}\mathbf{.}\mathbf{62}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathit{\Omega }\mathit{m}$ . The magnitude of the current versus time i_s = 10.0A is shown in Figure b. The vertical axis scale is set by t_s = 50.0ms and the horizontal axis scale is set by . Point P is at radial distance 9.00 mm from the wire’s centre. (a) Determine the magnitude of the magnetic field ${\overrightarrow{\mathbf{B}}}_{\mathbf{i}}$ at point  P due to the actual current  i in the wire at  t= 20 ms, (b) At t = 40 ms , and (c) t = 60 ms . Next, assume that the electric field driving the current is confined to the wire. Then Determine the magnitude of the magnetic field at point P due to the displacement current  i_d in the wire at (d) t = 20 ms  , (e) At t = 40 ms , and (f) At   t= 60 ms . At point P at 20 s, what is the direction (into or out of the page) of (g)   ${\overrightarrow{\mathbf{B}}}_{\mathbf{i}}$ and (h) ${\overrightarrow{\mathbf{B}}}_{\mathbf{id}}$?

In Fig. 32-36, a capacitor with circular plates of radius  $\mathit{R}\mathbf{=}\mathbf{18}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$  is connected to a source of emf $\mathit{\xi }\mathbf{=}{\mathit{\xi }}_{\mathbf{m}}\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{\omega }\mathit{t}$ , where  ${\mathit{\xi }}_{\mathbf{m}}\mathbf{=}\mathbf{220}\mathbf{ }\mathbf{\text{V}}$ and $\mathit{\omega }\mathbf{=}\mathbf{130}\mathbf{ }\mathbf{rad}\mathbf{/}\mathbf{s}$. The maximum value of the displacement current is  ${\mathit{i}}_{\mathbf{d}}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{60}\mathbf{ }\mathit{\mu }\mathbf{\text{A}}$ . Neglect fringing of the electric field at the edges of the plates. (a) What is the maximum value of the current i  in the circuit? (b) What is the maximum value of $\mathbf{d}{\mathit{\varphi }}_{\mathbf{E}}\mathbf{/}\mathbf{dt}$ , where  ${\mathit{\varphi }}_{\mathbf{E}}$ is the electric flux through the region between the plates? (c) What is the separation d  between the plates? (d) Find the maximum value of the magnitude of  $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}$ between the plates at a distance  $\mathit{r}\mathbf{=}\mathbf{11}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$ from the center.

: Assume the average value of the vertical component of Earth’s magnetic field is $\mathbf{43}\mathbf{ }\mathit{\mu }\mathbf{T}$  (downward) for all of Arizona, which has an area of ${\mathbf{\text{2.95×10}}}^{\mathbf{5}}\mathbf{ }{\mathbf{km}}^{\mathbf{2}}$. (a)What is the magnitude and (b) What is the direction (inward or outward) of the net magnetic flux through the rest of Earth’s surface (the entire surface excluding Arizona)?

In New Hampshire the average horizontal component of Earth’s magnetic field in 1912 was $\mathbf{16}\mathbf{ }\mathit{\mu }\mathbf{T}$, and the average inclination or “dip” was $\mathbf{73}\mathbf{°}$. What was the corresponding magnitude of Earth’s magnetic field?

Figure  is a one-axis graph along which two of the allowed energy values (levels) of an atom are plotted. When the atom is placed in a magnetic field of $\mathbf{0}\mathbf{.}\mathbf{500}\mathbf{ }\mathit{T}{\mathit{E}}_{\mathbf{2}}$ , the graph changes to that of Figure b because of the energy associated with ${\overrightarrow{\mathbf{\mu }}}_{\mathbf{o}\mathbf{r}\mathbf{b}}\mathbf{.}\overrightarrow{\mathbf{B}}$. Level ${\mathit{E}}_{\mathbf{1}}$ is unchanged, but level splits into a (closely spaced) triplet of levels. What are the allowed values of  ${\mathit{m}}_{\mathbf{1}}$ associated with (a) Energy level ${\mathit{E}}_{\mathbf{1}}$ and (b) Energy level  ${\mathit{E}}_{\mathbf{2}}$? (c) In joules, what amount of energy is represented by the spacing between the triplet levels?

If an electron in an atom has an orbital angular momentum with $\mathit{m}\mathbf{=}\mathbf{0}$, what are the components (a) ${\mathit{L}}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$ and (b) ${\mathit{\mu }}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$? If the atom is in an external magnetic field $\overrightarrow{\mathbf{B}}$ that has magnitude $\mathbf{35}\mathbf{ }\mathbf{mT}$ and is directed along the z  axis, what are (c) the energy ${\mathit{U}}_{\mathbf{o}\mathbf{r}\mathbf{b}}$ associated with $\overrightarrow{{\mathbf{\mu }}_{\mathbf{o}\mathbf{r}\mathbf{b}}}$and (d) the energy ${\mathit{U}}_{\mathbf{s}\mathbf{p}\mathbf{i}\mathbf{n}}$ associated with $\overrightarrow{{\mathbf{\mu }}_{\mathbf{s}}}$? (e) If, instead, the electron has $\mathit{m}\mathbf{=}\mathbf{-}\mathbf{3}$, what are (e) ${\mathit{L}}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$? (f)  ${\mathit{\mu }}_{\mathbf{o}\mathbf{r}\mathbf{b}\mathbf{,}\mathbf{z}}$(g) ${\mathit{U}}_{\mathbf{o}\mathbf{r}\mathbf{b}}$ and (h)${\mathit{U}}_{\mathbf{s}\mathbf{p}\mathbf{i}\mathbf{n}}$ ?

What is the energy difference between parallel and antiparallel alignment of the z component of an electron’s spin magnetic dipole moment with an external magnetic field of magnitude 0.25 T, directed parallel to the z axis?

(a) What is the measured component of the orbital magnetic dipole moment of an electron with ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{1}\mathbf{?}$  (b) What is the measured component of the orbital magnetic dipole moment of an electron with ${\mathit{m}}_{\mathbf{l}}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{?}$

An electron is placed in a magnetic field $\overrightarrow{B}$ that is directed along a $z$ axis. The energy difference between parallel and antiparallel alignments of the $z$ component of the electron’s spin magnetic moment with $\overrightarrow{B}$  is $6.00\times {10}^{-25}$J. What is the magnitude of $\overrightarrow{B}$ ?

Assume that an electron of mass m and charge magnitude e moves in a circular orbit of radius r about a nucleus. A uniform magnetic field  is then established perpendicular to the plane ofthe orbit. Assuming also that the radius of the orbit does not change and that the change in the speed of the electron due to field is small, find an expression for the change in the orbital magnetic dipole moment of the electron due to the field.