APhO 2010, Теория — Задача 2. Strong Resistive Electromagnets

Theoretical Competition — 25 April 2010 — Question Number 2 — Theoretical Question 2: Strong Resistive Electromagnets · 25 април 2010 г.

Условие

Resistive electromagnets are magnets with coils made of a normal metal such as copper or aluminum. Modern strong resistive electromagnets can provide steady magnetic fields higher than 30 tesla. Their coils are typically built by stacking hundreds of thin circular plates made of copper sheet metal with lots of cooling holes stamped in them; there are also insulators with the same pattern. When voltage is applied across the coil, current flows through the plates along a helical path to generate high magnetic fields in the center of the magnet.

In this question we aim to assess if a cylindrical coil (or solenoid) of many turns can serve as a magnet for generating high magnetic fields. As shown in Fig. 1, the center of the magnet is at O. Its cylindrical coil consists of $N$ turns of copper wire carrying a current $I$ uniformly distributed over the cross section of the wire. The coil’s mean diameter is $D$ and its length along the axial direction $x$ is $\ell$. The wire’s cross section is rectangular with width $a$ and height $b$. The turns of the coil are so tightly wound that the plane of each turn may be taken as perpendicular to the $x$ axis and $\ell = Na$. In Table 1, data specifying physical dimensions of the coil are listed.

In assessing if such a magnet can serve to provide high magnetic fields, two limiting factors must not be overlooked. One is the mechanical rigidity of the coil to withstand large Lorentz force on the field-producing current. The other is that the enormous amount of Joule heat generated in the wire must not cause excessive temperature rise. We shall examine these two factors using simplified models.

The Appendix at the end of the question lists some mathematical formulae and physical data which may be used if necessary.

(a) Part A. Magnetic Fields on the Axis of the Coil

Assume $b \ll D$ so that one may regard the wire as a thin strip of width $a$. Let O be the origin of $x$ coordinates. The direction of the current flow is as shown in Fig. 1.

(a) Find the $x$-component $B(x)$ of the magnetic field on the axis of the coil as a function of $x$ when the steady current passing through the coil is $I$. [1.0 point]

(b) Find the steady current $I_0$ passing through the coil if $B(0)$ is $10.0\ \mathrm{T}$. Use data given in Table 1 when computing numerical values. [0.4 point]

(c) Part B. The Upper Limit of Current

In Part B, we assume the length $\ell$ of the coil is infinite and $b \ll D$. Consider the turn of the coil located at $x = 0$. The magnetic field exerts Lorentz force on the current passing through the turn. Thus, as Fig. 2 shows, a wire segment of length $\Delta s$ is subject to a normal force $\Delta F_\mathrm{n}$ which tends to make the turn expand.

(c) Suppose that, when the current is $I$, the mean diameter of the expanded coil remains at a constant value $D'$ larger than $D$, as shown in Fig. 2.

Find the outward normal force per unit length $\Delta F_\mathrm{n}/\Delta s$. [1.2 point]

Find the tension $F_\mathrm{t}$ acting along the wire. [0.6 point] [1,8 т.]

(d) Neglect the coil’s acceleration during the expansion. Assume the turn will break when the wire’s unit elongation (i.e. tensile strain or fractional change of the length) is 60 % and tensile stress (i.e. tension per unit cross sectional area of the unstrained wire) is $\sigma_\mathrm{b} = 455\ \mathrm{MPa}$. Let $I_\mathrm{b}$ be the current at which the turn will break and $B_\mathrm{b}$ the corresponding magnetic field at the center O.

Find an expression for $I_\mathrm{b}$ and then calculate its value. [0.8 point]

Find an expression for $B_\mathrm{b}$ and then calculate its value. [0.4 point] [1,2 т.]

(e) Part C. The Rate of Temperature Rise

When the current $I$ is $10.0\ \mathrm{kA}$ and the temperature $T$ of the coil is $293\ \mathrm{K}$, assume that the resistivity, the specific heat capacity at constant pressure, and the mass density of the wire of the coil are, respectively, given by $\rho_\mathrm{e} = 1.72\times10^{-8}\ \Omega\cdot\mathrm{m}$, $c_p = 3.85\times10^{2}\ \mathrm{J/(kg\cdot K)}$ and $\rho_m = 8.98\times10^{3}\ \mathrm{kg\cdot m^{-3}}$.

(e) Find an expression for the power density (i.e. power per unit volume) of heat generation in the coil and then calculate its value. Use data in Table 1. [0.5 point]

(f) Let $\dot{T}$ be the time rate of change of temperature in the coil. Find an expression for $\dot{T}$ and then calculate its value. [0.5 point]

(g) Part D. A Pulsed-Field Magnet

If the large current needed for a strong magnet lasts only for a short time, the temperature rise caused by excessive Joule heating may be greatly reduced. This idea is employed in a pulsed-field magnet.

Thus, as shown in Fig. 3, a capacitor bank of capacitance $C$ charged initially to a potential $V_0$ is used to drive the current $I$ through the coil. The circuit is equipped with a switch $K$. The inductance $L$ and resistance $R$ of the circuit are assumed to be entirely due to the coil. The construct and dimensions of the coil are the same as given in Fig. 1 and Table 1. Assume $R$, $L$, and $C$ to be independent of temperature and the magnetic field is the same as that of an infinite solenoid with $\ell \to \infty$.

(g) Find expressions for the inductance $L$ and resistance $R$. [0.6 point]

Calculate the values of $L$ and $R$. Use data given in Table 1. [0.4 point] [1 т.]

(h) At time $t = 0$, the switch $K$ is thrown to position 1 and the current starts flowing. For $t \ge 0$, the charge $Q(t)$ on the positive plate of the capacitor and the current $I(t)$ entering the positive plate are given by

in which $\alpha$ and $\omega$ are positive constants and $\theta_0$ is given by

Note that, if $Q(t)$ is expressed as a function of a new variable $t' \equiv (t + \theta_0/\omega)$, then $Q(t')$ and its time derivative $I(t)$ are identical in form except for an overall constant factor. The time derivative of $I(t)$ may therefore be obtained similarly without further differentiations.

Find $\alpha$ and $\omega$ in terms of $R$, $L$, and $C$. [0.8 point]

Calculate the values of $\alpha$ and $\omega$ when $C$ is $10.0\ \mathrm{mF}$. [0.4 point] [1,2 т.]

(i) Let $I_\mathrm{m}$ be the maximum value of $|I(t)|$ for $t > 0$. Find an expression for $I_\mathrm{m}$. [0.6 point]

If $C = 10.0\ \mathrm{mF}$, what is the maximum value $V_{0\mathrm{b}}$ of the initial voltage $V_0$ of the capacitor bank for which $I_\mathrm{m}$ will not exceed $I_\mathrm{b}$ found in Problem (d)? [0.4 point] [1 т.]

(j) Suppose the switch $K$ is moved instantly from position 1 to 2 when the absolute value of the current $|I(t)|$ reaches $I_\mathrm{m}$. Let $\Delta E$ be the total amount of heat dissipated in the coil from $t = 0$ to $\infty$ and $\Delta T$ the corresponding temperature increase of the coil. Assume the initial voltage $V_0$ takes on the maximum value $V_{0\mathrm{b}}$ obtained in Problem (i) and the electromagnetic energy loss is only in the form of heat dissipated in the coil.

Find an expression for $\Delta E$ and then calculate its value. [1.0 point]

Find an expression for $\Delta T$ and then calculate its value. Note that the value for $\Delta T$ must be compatible with the assumption of constant $R$ and $L$. [0.4 point] [1,4 т.]

Appendix

1. $\displaystyle\int_0^L \frac{dx}{(D^2+x^2)^{3/2}} = \frac{1}{D^2}\left\{\frac{L}{(D^2+L^2)^{1/2}}\right\}$

2. $\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

3. permeability of free space $\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\cdot m/A}$

Отговори

Решение

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Оригинал в Архива: 2.pdf · официални решения: 2_sol.pdf