Съдържание

WoPhO 2012, Теория — Задача 1.6

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от WoPhO 2012

Theoretical 1: Magnetic Monopole · 1,3 т.

Увод

Our everyday experience shows that the magnetic poles always exist in pairs North and South. Breaking a magnet results in the appearance of a new pair of opposite magnetic poles on the two broken ends. The fundamental laws of physics, however, do not contradict the existence of magnetically charged particles called magnetic monopoles. The magnetic monopole is an object possessing just one magnetic pole, either North, or South, which are analogues of the positive and negative electric charges respectively. Thus, the magnetic field of the monopole is similar to the electric field created by a static electric charge, i.e. its force lines begin or end at the point where the monopole is located. This property is in contrast to the closed force lines of the magnetic field created by permanent magnets (magnetic dipoles) and electric currents. The concept of magnetic monopole was introduced in 1932 by the famous physicist Paul Dirac. On the basis of quantum mechanics he proved that the existence of magnetic monopoles can explain the existence of the elementary electric charge. That is why the physicists do not cease their efforts to discover magnetic monopoles experimentally.

In the following questions you are going to establish some properties of the magnetic monopoles by analyzing simple model situations (though experiments). You may assume that all laws of physics known to you apply to the magnetic monopoles, except the statement for closed force lines of the magnetic field. The velocities considered in this problem are much smaller than the speed of light and, therefore, you may neglect the relativistic effects on time, length and mass.

Use the following physical constants in your solution:

magnetic permeability of vacuum: $\mu_0 = 4\pi\times10^{-7}\ \mathrm{H/m}$ ;
electric permittivity of vacuum: $\epsilon_0 = 8.85\times10^{-12}\ \mathrm{F/m}$ ;
speed of light: $c = 2.998\times10^{8}\ \mathrm{m/s}$ ;
elementary electric charge: $e = 1.602\times10^{-19}\ \mathrm{C}$ ;
Planks constant: $h = 6.626\times10^{-34}\mathrm{J.s}$ .

Условие

A point particle of electric charge $q_e$ is confined to move along a circle without any resistance or friction, as shown in Figure 4. A monopole of magnetic charge $q_m$ passes through the plane of the circle by moving along its axis $Z$ from $z \to -\infty$ to $z \to \infty$ .

A monopole q_m moving to the right along the Z axis towards a circular ring perpendicular to the axis; a point charge q_e sits on the ring. — Figure 4: Question 6

(a) Obtain an expression for the change in $Z$ component, $\Delta L_z$ , of the angular momentum of the electrically charged particle during the whole motion of the magnetic monopole. Express your answer in terms of $q_e$ , $q_m$ and fundamental constants only. [1,3 т.]

Отговори

Решение

The rate of change of the angular momentum of the electric charge is equal to the torque of the electric force acting on it:

\frac{\mathrm{d}L_z}{\mathrm{d}t} = Fa = q_e aE

Here $E$ is the tangential component of the electric field created by the moving monopole. By using the result derived in Question 4, we have:

\frac{\mathrm{d}L_z}{\mathrm{d}t} = -\frac{\mu_0 q_m q_e a^2}{4\pi(z^2+a^2)^{3/2}}v.

Since the displacement of the monopole for a time interval d $t$ is:

\mathrm{d}z = v\mathrm{d}t

we can obtain the change of angular momentum by integration over $z$ :

\boxed{\Delta L_z = -\int_{-\infty}^{+\infty}\frac{\mu_0 q_m q_e a^2}{4\pi(z^2+a^2)^{3/2}}\mathrm{d}z = -\frac{\mu_0 q_m q_e}{2\pi}}.

The integration could be avoided by noticing that the torque of the electric force is proportional to the EMF induced in the circular loop:

\frac{\mathrm{d}L_z}{\mathrm{d}t} = q_e aE = q_e\frac{\mathcal{E}}{2\pi}.

From the Faraday’s law: $\mathcal{E} = -\dfrac{\mathrm{d}\Phi}{\mathrm{d}t}$ , it follows that:

\frac{\mathrm{d}L_z}{\mathrm{d}t} = -\frac{q_e}{2\pi}\frac{\mathrm{d}\Phi}{\mathrm{d}t}

where $\Phi$ is the flux of the magnetic field through the plane of the loop. Therefore, the change of the angular momentum of the electric charge for arbitrary time interval is proportional to the change of the magnetic flux through the loop for the same interval:

\Delta L_z = -\frac{q_e}{2\pi}\Delta\Phi

According to the formula: $\Phi = \dfrac{\mu_0 q_m}{2}\left(1+z/(z^2+a^2)^{1/2}\right)$ , derived in Question 4, the magnetic flux through the circle changes from 0 when $z \to -\infty$ to $\mu_0 q_m$ as $z \to +\infty$ . Thus, we obtain the same result:

\boxed{\Delta L_z = -\frac{\mu_0 q_m q_e}{2\pi}}

Marking scheme

Question	Element of the solution	Points	Comments
6 a	Relates d $L_z$ /d $t$ to the torque created by the electric force applied on the electric charge	0.3
	Provides expression for the torque on the electric charge through the instantaneous position z and velocity v of the monopole	0.3	Alternatively: Relates the torque to the induced EMF
	Uses the expression d $z$ = $v$ d $t$ for the displacement of the monopole along z	0.2	Uses the Faraday’s law and to derive relation between $\Delta Lz$ and $\Delta\Phi$
	Integrates d $Lz$ over monopole position z in proper limits	0.3	Calculates $\Delta\Phi$ for the whole motion
	Final expression for $\Delta L_z$	0.2
	Subtotal on Question 6 a	1.3

Оригинал в Архива: WoPhO_2012_Q11.pdf · официални решения: WoPhO_2012_S11.pdf