Съдържание

WoPhO 2012, Теория — Задача 1.3

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от WoPhO 2012

Theoretical 1: Magnetic Monopole · 2,1 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

When at rest, the magnetic monopole creates static magnetic field, similar to the electric field produced by a static electric charge. The magnetic induction $\mathbf{B}$ at a point of position- vector $\mathbf{r}$ , relative to the monopole (see Figure 2), is given by the equation:

\mathbf{B} = \frac{k_m q_m\mathbf{r}}{r^3}, \qquad (2)

A monopole q_m with the position vector r drawn from it to a point where the field vector B is shown as a bold arrow pointing along r. — Figure 2: Question 3

where $k_m$ is a coefficient of proportionality and $r = |\mathbf{r}|$ .

(a) By analyzing the system described in Question 2, express the unknown coefficient $k_m$ through the provided fundamental constants. [1,6 т.]

(b) Formulate by means of equation the Gauss law for the flux $\Phi$ of the magnetic induction created by the magnetic monopole. [0,5 т.]

Отговори

Решение

a. The monopole, on its turn, exercises a force $\mathbf{F}'$ on the current loop. That force can be expressed by considering first the Ampere’s force applied on a small element d $s$ of the loop:

\mathrm{d}F_z' = -(IB\mathrm{d}s)\sin\alpha = -\frac{k_m q_m I\mathrm{d}s}{r^2}\sin\alpha = -\frac{k_m q_m Ia\mathrm{d}s}{\left(a^2+z^2\right)^{3/2}}.

Current loop with current I and radius a; the monopole q_m on the Z axis at distance z; the line r from the loop element ds to q_m makes angle α with the axis; at ds the field B of the monopole and the force element dF′ on the loop are drawn as bold arrows. — Figure S2.

After integration along the circle we obtain:

F_z' = -\frac{2\pi k_m q_m Ia^2}{\left(a^2+z^2\right)^{3/2}}.

According to the Newton’s third law:

F_z' = -F_z

This condition is satisfied provided that:

\boxed{k_m = \frac{\mu_0}{4\pi}}

b. Now, consider a spherical surface of radius $r$ surrounding the magnetic monopole. The flux of magnetic induction through that surface is:

\boxed{\Phi = B(r)\times(4\pi r^2) = \mu_0 q_m}

That last relation is a formulation of the Gauss law in the case of magnetic monopoles.

Marking scheme

Question	Element of the solution	Points	Comments
3 a	Provides expression for the Ampere's force on an infinitesimal element of the loop	0.5	Assign 0.5 for the whole part if $k_m$ is related to $\mu_0$ on the basis of a dimensional analysis.
	Projects the force onto axis Z	0.2
	Integration (summation) along the loop	0.1
	Final expression for F'z	0.1
	Applies the Newton's third law for Fz and F'z	0.5
	Correct expression for $k_m$	0.2
	Subtotal on Question 3 a	1.6
3 b	Applies the definition of a magnetic flux for a spherical surface around the monopole	0.3	Assign the full mark if the magnetic flux is calculated by integration over arbitrary closed surface
	Correct formulation of the Gauss law	0.2
	Subtotal on Question 3 b	0.5

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