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WoPhO 2012, Теория — Задача 1.4

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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

A moving electric charge creates magnetic field. Likewise, the moving magnetic monopole produces electric field with circular force-lines (i.e. a vortex field) concentric with the direction of motion of the monopole (see Figure 3). Consider a monopole of magnetic charge $q_m$ moving along a straight line with a constant velocity $\mathbf{v}$ .

A monopole q_m moving with velocity v along a dashed axis; the position vector r from q_m to a circular electric force line around the axis makes angle θ with v; the circle is labelled “electric force line”. — Figure 3: Question 4

(a) Derive an expression for the intensity $\mathbf{E}$ of the electric field created by the monopole in a point of position-vector $\mathbf{r}$ making an angle $\theta$ with the vector of velocity, as shown in Figure 3. Use vector notations in your final answer in order to specify both, the magnitude, and the direction of the electric field. [1,7 т.]

(b) Suppose that a positive electric charge $q_e$ and a positive magnetic charge $q_m$ are moving toward you, perpendicularly to the sheet of paper. Draw an arbitrary force line in your answer sheet for the magnetic field created by the electric charge. Draw separately another arbitrary force line for the electric field created by the magnetic charge. Indicate the directions of the two lines. [0,4 т.]

Отговори

Решение

a. The question could be addressed differently depending on the choice of the frame of reference.

I approach Laboratory reference system – the monopole moves with velocity v along $Z$ –axis.

Consider the circular force-line of the electric field passing through the point of interest (see Figure S3). The radius of the force line is related to the instantaneous values of $r$ and $\theta$ as $a = r\sin\theta$ . The origin of the coordinate system is chosen at the center of the circle. The flux of the magnetic field through the plane of the force line is:

\Phi = \mu_0 q_m\frac{\Omega}{4\pi}

Monopole q_m moving with velocity v along the Z axis; the position vector r makes angle θ with v and ends on a circular force line of radius a; the spherical cap subtending solid angle Ω is shaded; the field E is drawn tangent to the circle; the distance −z from q_m to the circle's plane is marked. — Figure S3

where

\Omega = 2\pi(1-\cos\theta)

is the solid angle enclosed by the conical surface of half-aperture $\theta$ . Taking into account that $\cos\theta = -z/(z^2+a^2)^{1/2}$ we obtain:

\Phi = \frac{\mu_0 q_m}{2}\left(1+z/(z^2+a^2)^{1/2}\right)

The EMF, induced along the force-line, is:

\mathcal{E} = -\frac{\mathrm{d}\Phi}{\mathrm{d}t} = -\frac{\mu_0 q_m a^2}{2(z^2+a^2)^{3/2}}v

where $v = \mathrm{d}z/\mathrm{d}t$ is the speed of the monopole. The minus sign indicates that the EMF acts against the circulation direction defined by the right-hand rule, as shown in Figure S3.

The EMF is related to the tangential component $E$ of the electric-filed by:

\mathcal{E} = E(2\pi a)

Therefore:

E = -\frac{\mu_0 q_m a}{4\pi(z^2+a^2)^{3/2}}v = -\frac{\mu_0 q_m v\sin\theta}{4\pi r^2}

The “–“ sign indicates that the direction of the vector E is against the positive direction of the force line, as shown in Figure S3. Therefore, in a vector form, the electric field created by the moving monopole, is:

\boxed{\mathbf{E} = -\frac{\mu_0 q_m\mathbf{v}\times\hat{\mathbf{r}}}{4\pi r^2} = -\frac{\mu_0 q_m\mathbf{v}\times\mathbf{r}}{4\pi r^3}}

where $\hat{\mathbf{r}} = \mathbf{r}/r$ is the unit-vector along the position-vector r.

II approach A “shortcut” solution – the rest-system of the monopole

Consider a stationary point electric charge $q_e$ situated at the position defined by the vector r relative to the monopole (see Figure S4). In the monopole rest system the electric charge moves with a velocity –v, against the positive direction of axis $Z$ . Since only static magnetic field is present at that system (the monopole is at rest) the electric charge experiences a Lorenz force:

\mathbf{F} = q_e(-\mathbf{v})\times\mathbf{B}(\mathbf{r}) = -\frac{\mu_0 q_e q_m\mathbf{v}\times\mathbf{r}}{4\pi r^3}

Two drawings. Left, “monopole rest frame”: charge q_e at position r from q_m moving with −v, with the field B along r and the force F perpendicular. Right, “laboratory frame”: q_m moving with v, the charge q_e on an electric force line with the field E and force F drawn at q_e. — Figure S4

In a non-relativistic approximation, the same force is sensed by the electric charge in the laboratory reference system. In that system, however, it is due to the presence of an electric field at the point where the electric charge is situated. By following the definition of the intensity of electric field, we obtain:

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

III approach An energetic balance approach

Suppose that the magnetic monopole moves along the axis of a circular current-carrying loop, similar to that shown in Figure S1. According to the result of Question 2, the monopole experiences a force:

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

Since the monopole + current system is closed, the total instantaneous power of all forces in the system is zero:

F_z v + \mathcal{E}I = 0

where $\mathcal{E}$ is the EMF induced in the loop. Thus, we obtain:

\mathcal{E} = -\frac{\mu_0 q_m a^2}{2(z^2+a^2)^{3/2}}v

After that we recover all the subsequent steps described in the first approach in order to find E.

b. The two drawings below indicate the directions of the force lines of the magnetic and electric fields created by moving positive electric and magnetic charges respectively.

Two circular force lines with direction arrows: left, the magnetic field B around a charge q_e moving out of the page (anti-clockwise); right, the electric field E around a monopole q_m moving out of the page (clockwise).

Marking scheme

Question	Element of the solution	Points	Alternatively:	Alternatively:
4 a	Calculates the magnetic flux through the area enclosed by the force line	0.5	Formulates the energy balance equation: 0.8 and provides expression for the induced EMF: 0.2	Expression for the Lorenz force experienced by an electric charge in the monopole rest-frame 1.0
	Applies the Faraday's law and finds an expression for the EMF induced along the force line	0.5
	Relates the induced EMF to the tangential component of the electric field	0.3	0.3	Uses the invariance of force and applies the definition of electric field intensity in the laboratory frame: 0.5
	Correct expression for the tangential component $E$	0.2	0.2
	Vector expression for the electric field intensity	0.2	0.2	Correct vector expression for the electric field intensity: 0.2
	Subtotal on Question 4 a	1.7
4 b	Draws a circular magnetic force line for the moving electric charge and indicates an anti-clockwise direction on it	0.2
	Draws a circular electric force line for the moving magnetic monopole and indicates a clockwise direction on it	0.2
	Subtotal on Question 4 b	0.4

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