物理学引论 作业四

物理学引论

陆一洲 5140309557

Chapter 28 Question 15

How might you measure the magnetic dipole moment of the Earth?

先测量出北极点或者南极点的磁场强度，再使用公式 $\mathrm{\mu}=2\pi R^3B/\mu_0$ 求出地球的磁矩。

Chapter 28 Question 23

Suppose you have three iron rods, two of which are magnetized but the third is not. How would you determine which two are the magnets without using any additional objects?

两两放一起，再将其中一只倒过来再放。如果有哪一对之间出现了相斥，那这一对，就是磁化了的铁棍。

Chapter 28 Problems 14

A long pair of insulated wires serves to conduct $28.0\ \mathrm{A}$ of dc current to and from an instrument. If the wires are of negligible diameter but are $2.8\ \mathrm{mm}$ apart, what is the magnetic field $10.0\ \mathrm{cm}$ from their midpoint, in their plane (Fig. 28-36)? Compare to the magnetic field of the Earth.

$$B=\frac{\mu_0I}{2\pi\left(r-\frac{1}{2}d\right)}-\frac{\mu_0I}{2\pi\left(r+\frac{1}{2}d\right)}=1.6\times 10^{-6}\ \mathrm{T} \\ \frac{B}{B_E}=0.031$$

Chapter 28 Problems 24

A triangular loop of side length $a$ carries a current $I$ (Fig. 28-41). If this loop is placed a distance $d$ away from a very long straight wire carrying a current $I'$, determine the force on the loop.

$$F=F_1-2F_2=\frac{\mu_0 II'}{\pi}\left(\frac{a}{2d}-\frac{\sqrt{3}}{3}\ln\left(1+\frac{\sqrt{3}a}{2d}\right)\right)$$

Chapter 28 Problems 31

A coaxial cable consists of a solid inner conductor of radius $R_1$, surrounded by a concentric cylindrical tube of inner radius $R_2$ and outer radius $R_3$ (Fig. 28-42). The conductors carry equal and opposite currents $I_0$ distributed uniformly across their cross sections. Determine the magnetic field at a distance $R$ from the axis for:
(a) $R<R_1$ ;
(b) $R_1<R<R_2$ ;
(c) $R_2<R<R_3$ ;
(d) $R>R_3$ .
(e) Let $I_0 = 1.50\ \mathrm{A}$, $R_1 = 1.00\ \mathrm{cm}$, $R_2 = 2.00\ \mathrm{cm}$, and $R_3 = 2.50\ \mathrm{cm}$. Graph $B$ ftom $R = 0$ to $R = 3.00\ \mathrm{cm}$.

(a)

$$B=\frac{\mu_0I_0R}{2\pi R_1^2}$$
(b) $$B=\frac{\mu_0I_0}{2\pi R}$$
(c) $$B=\frac{\mu_0I_0}{2\pi R}\frac{R_3^2-R^2}{R_3^2-R_2^2}$$
(d) $$B=0$$
(e)

Chapter 29 Question 11

What is the advantage of placing the two insulated electric wires carrying ac close together or even twisted about each other?

由于两根导线中的电流是反向的，所以如此做事为了尽可能地抵消产生的磁场。

Chapter 29 Problem 8

(a) If the resistance of the resistor in Fig. 29-38 is slowly increased, what is the direction of the current induced in the small circular loop inside the larger loop?
(b) What would it be if the small loop were placed outside the larger one, to the left?

(a) 逆时针；
(b) 顺时针。

Chapter 29 Problem 13

While demonstrating Faraday's law to her class, a physics professor inadvertently moves the gold ring on her finger from a location where a $0.80$-$\mathrm{T}$ magnetic field points along her finger to a zero-field location in $45\ \mathrm{ms}$. The $1.5$-$\mathrm{cm}$-diameter ring has a resistance and mass of $55\ \mathrm{\mu\Omega}$ and $15\ \mathrm{g}$, respectively.
(a) Estimate the thermal energy produced in the ring due to the flow of induced current.
(b) Find the temperature rise of the ring, assuming all of the thermal energy produced goes into increasing the ring's temperature. The specific heat of gold is $129\ \mathrm{J/kg \cdot C^{\circ}}$.

(a)

$$Q=\frac{(\Delta B)^2\pi^2d^4}{16R\Delta t}\doteq8.1\ \mathrm{mJ}$$
(b)
$$\Delta T=\frac{Q}{mc}\doteq 4.2\times10^{-3}\ \mathrm{C}^{\circ}$$

Chapter 29 Problem 16

A power line carrying a sinusoidally varying current with frequency $f = 60\ \mathrm{Hz}$ and peak value $10 = 55\ \mathrm{kA}$ runs at a height of $7.0\ \mathrm{m}$ across a farmer's land (Fig. 29-41). The farmer constructs a vertically oriented $2.0$-$\mathrm{m}$-high $10$-turn rectangular wire coil below the power line. The farmer hopes to use the induced voltage in this coil to power $120$-$\mathrm{Volt}$ electrical equipment, which requires a sinusoidally varying voltage with frequency $f = 60\ \mathrm{Hz}$ and peak value $V_0 = 170\ \mathrm{V}$. What should the length $l$ of the coil be? Would this be unethical?

$$l=\frac{\mathscr{E}_0}{N\mu_0I_0f\ln(1.4)}\doteq12\ \mathrm{m}$$

Chapter 29 Problem 33

A conducting rod rests on two long frictionless parallel rails in a magnetic field $\vec{B}$ ($\perp$ to the rails and rod) as in Fig. 29-44.
(a) If the rails are horizontal and the rod is given an initial push, will the rod travel at constant speed even though a magnetic field is present?
(b) Suppose at $t = 0$, when the rod has speed $v = v_0$, the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance $R$ and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.

(a) 是；
(b)

$$m\frac{\mathrm{d}v}{\mathrm{d}t}=-\frac{B^2l^2}{R}v\ \ \Rightarrow\ \ v(t)=v_0e^{-\frac{B^2l^2t}{mR}}$$

Chapter 29 Problem 34

Suppose a conducting rod (mass $m$, resistance $R$) rests on two frictionless and resistanceless parallel rails a distance $l$ apart in a uniform magnetic field $\vec{B}$ ($\perp$ to the rails and to the rod) as in Fig. 29-44. At $t = 0$, the rod is at rest and a source of emf is connected to the points a and b. Determine the speed of the rod as a function of time if
(a) the source puts out a constant current $I$,
(b) the source puts out a constant emf $\mathscr{E}_0$.
(c) Does the rod reach a terminal speed in either case? If so, what is it?

(a)

$$v(t)=\frac{BIlt}{m}$$
(b) $$v(t)=\frac{\mathscr{E}_0}{Bl}(1-e^{-\frac{B^2l^2t}{mR}})$$
(c) $$v_t=\frac{\mathscr{E}_0}{Bl}$$