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Chapter 11: Problem 36
Find the inductance of two identical inductors in parallel.
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The wires themselves in a circuit can have resistance, inductance, and capacitance. Would "stray" inductance and capacitance be most important for low-frequency or for high-frequency circuits? For simplicity, assume that the wires act like they're in series with an inductor or capacitor.
In our by-now-familiar neuron, the voltage difference between the inner and outer surfaces of the cell membrane is about \(V_{\text {out }}-V_{\text {in }}=-70 \mathrm{mV}\) in the resting state, and the thickness of the membrane is about \(6.0 \mathrm{~nm}\) (i.e., only about a hundred atoms thick). What is the electric field inside the membrane?(answer check available at lightandmatter.com)
Use Gauss' law to find the field inside an infinite cylinder with radius \(b\) and uniform charge density \(\rho\). (The external field has the same form as the one in problem 46.)(answer check available at lightandmatter.com)
(a) Use Gauss' law to find the fields inside and outside an infinite cylindrical surface with radius \(b\) and uniform surface charge density \(\sigma\).(answer check available at lightandmatter.com) (b) Show that there is a discontinuity in the electric field equal to \(4 \pi k \sigma\) between one side of the surface and the other, as there should be (see page 628). (c) Reexpress your result in terms of the charge per unit length, and compare with the field of a line of charge. (d) A coaxial cable has two conductors: a central conductor of radius \(a\), and an outer conductor of radius \(b\). These two conductors are separated by an insulator. Although such a cable is normally used for time-varying signals, assume throughout this problem that there is simply a DC voltage between the two conductors. The outer conductor is thin, as in part c. The inner 46. (a) Use Gauss' law to find the fields inside and outside an infinite cylindrical surface with radius \(b\) and uniform surface charge density \(\sigma\).(answer check available at lightandmatter.com) (b) Show that there is a discontinuity in the electric field equal to \(4 \pi k \sigma\) between one side of the surface and the other, as there should be (see page 628). (c) Reexpress your result in terms of the charge per unit length, and compare with the field of a line of charge. (d) A coaxial cable has two conductors: a central conductor of radius \(a\), and an outer conductor of radius \(b\). These two conductors are separated by an insulator. Although such a cable is normally used for time-varying signals, assume throughout this problem that there is simply a DC voltage between the two conductors. The outer conductor is thin, as in part c. The inner
Light is a wave made of electric and magnetic fields, and the fields are perpendicular to the direction of the wave's motion, i.e., they're transverse. An example would be the electric field given by \(\mathbf{E}=b \hat{\mathbf{x}} \sin c z\), where \(b\) and \(c\) are constants. (There would also be an associated magnetic field.) We observe that light can travel through a vacuum, so we expect that this wave pattern is consistent with the nonexistence of any charge in the space it's currently occupying. Use Gauss's law to prove that this is true.
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