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WoPhO 2012, Теория — Задача 1.1

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Всички материали от WoPhO 2012

Theoretical 1: Magnetic Monopole · 0,8 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 exposed to external magnetic field of induction $\mathbf{B}$ , a monopole of magnetic charge $q_m$ experiences a force:

\mathbf{F} = q_m\mathbf{B} \qquad (1)

(a) Derive the unit of magnetic charge in terms of the basic SI units: kilogram, meter, second, ampere. [0,8 т.]

Отговори

Решение

a. The simplest approach is to compare the expression

\mathbf{F} = q_m\mathbf{B}

for the force acting on the monopole with the Ampere’s formula for the magnetic force acting on an element d of a conductor carrying current $I$ :

\mathbf{F} = I\mathbf{d}\times\mathbf{B}

The equivalence of the dimensionality of the right-hand sides of the two equations shows that:

\boxed{[q_m] = \mathrm{A}\cdot\mathrm{m}\ (\text{ampere}\times\text{meter})}

Alternatively, one may use the expression for the Lorenz force applied on a moving electric charge;

\mathbf{F} = q\mathbf{v}\times\mathbf{B}

which gives:

[q_m] = \mathrm{C}\cdot\mathrm{m}\cdot\mathrm{s}^{-1} = \mathrm{A}\cdot\mathrm{m}

since A = C·s $^{-1}$ .

Marking scheme

Question	Element of the solution	Points	Comments
1 a	Compares the formula for the force applied on the monopole with a relevant formula for the magnetic force applied to a conductor or a moving electric charge	0.5
	Correct final expression for $[q_m]$	0.3	Deduce 0.1 if C.s $^{-1}$ is not converted to A
	Subtotal on Question 1 a	0.8

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