Съдържание

WoPhO 2012, Теория — Задача 1.5

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от WoPhO 2012

Theoretical 1: Magnetic Monopole · 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

The analogy between electric and magnetic charges is found also in the way they interact with external magnetic and electric fields respectively. Similarly to the Lorentz force acting on an electric charge moving in magnetic field, the magnetic charge experiences a force when it moves in electric field.

(a) Propose and analyze a thought experiment in order to derive an expression for the Lorentz force acting on a monopole of magnetic charge $q_m$ moving with a velocity $\mathbf{v}$ in electric field of intensity $\mathbf{E}$ . Use vector notations in your final answer in order to specify both, the magnitude and the direction of the force. When describing your thought experiment, use proper drawings and short comments to them instead of a lengthy text. [1 т.]

Отговори

Решение

a. A simple model system is shown in Figure S5. A static electric charge $q_e$ is located at the origin of the coordinate system. A magnetic monopole $q_m$ with instantaneous position-vector r moves with velocity v. The electric field, felt by the monopole, is:

\mathbf{E} = \frac{q_e\mathbf{r}}{4\pi\varepsilon_0 r^3}

Static charge q_e at the origin and monopole q_m above it at position r, moving with velocity v to the right; at q_m the field E points up along r and the force F is drawn; at q_e the field E′ and force F′ are drawn as a double arrow pointing down-right. — Figure S5

On its turn, the moving magnetic monopole creates electric field at the origin of the coordinates system:

\mathbf{E}' = -\frac{\mu_0 q_m\mathbf{v}\times(-\mathbf{r})}{4\pi r^3} = \frac{\mu_0 q_m\mathbf{v}\times\mathbf{r}}{4\pi r^3}

Therefore, the electric charge experiences an electric force:

\mathbf{F}' = q_e\mathbf{E}' = \frac{\mu_0 q_m q_e\mathbf{v}\times\mathbf{r}}{4\pi r^3}

According to the Newton’s third law the “Lorenz” force, applied on the monopole, is:

\mathbf{F} = -\mathbf{F}' = -\frac{\mu_0 q_m q_e\mathbf{v}\times\mathbf{r}}{4\pi r^3} = -\mu_0\varepsilon_0 q_m\mathbf{v}\times\mathbf{E}

or, taking into account that $\mu_0\varepsilon_0 = 1/c^2$ :

\boxed{\mathbf{F} = -\frac{1}{c^2}q_m\mathbf{v}\times\mathbf{E}}

Marking scheme

Question	Element of the solution	Points	Comments
5 a	Clear drawing for a system composed of a moving magnetic monopole and a stationary electric charge	0.2	Follow the same marking scheme for alternative systems.
	Provides expression for the force F ' acting on the electric charge	0.2
	Provides expression for the force F acting on the moving monopole	0.2
	Applies the Newton's third law for F and F'	0.2
	Provides vector expression for F	0.2
	Subtotal on question 5 a	1.0

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