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WoPhO 2012, Теория — Задача 1.2

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Theoretical 1: Magnetic Monopole · 1,1 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

Electric current $I$ circulates along a circular loop of radius $a$ . A monopole of magnetic charge $q_m$ is situated on the axis of the loop at a point of coordinate $z$ relative to its center, as shown in Figure 1 . The positive direction of the axis $Z$ and the direction of current circulation are related through the right-hand rule.

(a) Find out an expression for the $z$ -component $F_z$ of the force acting on the monopole. [1,1 т.]

A circular current loop drawn in perspective with the current I marked by an arrow along the loop and the radius a; the horizontal axis Z passes through the loop centre, and a monopole q_m sits on the axis to the right of the loop. — Figure 1: Question 2

Отговори

Решение

a. The magnetic field of the current is found by means of the Biot–Savart law. The magnetic field d $B$ created by a small current element of length d $s$ is:

\mathrm{d}B = \frac{\mu_0 I\mathrm{d}s}{4\pi r^2}

Current loop with current I, radius a and length element ds; the monopole q_m on the Z axis at distance z from the centre; the line r from ds to q_m makes angle α with the axis; the field element dB at q_m is drawn as a bold arrow. — Figure S1.

(See Figure S1 for the notations).

The projection of the magnetic induction on the axis $Z$ is:

\mathrm{d}B_z = \mathrm{d}B\sin\alpha = \frac{\mu_0 Ia\mathrm{d}s}{4\pi\left(a^2+z^2\right)^{3/2}}

since $\sin\alpha = a/r$ and $r = \left(a^2+z^2\right)^{1/2}$ . We obtain the total magnetic field at a point $z$ by integrating the above expression along the circle:

B_z = \frac{\mu_0 Ia^2}{2\left(a^2+z^2\right)^{3/2}}

and find respectively the force acting on the monopole:

\boxed{F_z = \frac{\mu_0 q_m Ia^2}{2\left(a^2+z^2\right)^{3/2}}}

Marking scheme

Question	Element of the solution	Points
2 a	Applies the Biot-Savart law for an infinitesimal element of the loop	0.5
	Projects correctly dB onto axis Z	0.2
	Correct integration (summation) along the loop	0.2
	Final expression for Fz	0.2
	Subtotal on Question 2 a	1.1

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