Breaking Newton's third law: electromagnetic instances

Considering a point-like charge q₁ at rest, located at the origin of the coordinate system, the electric field at a point P located at a point $\vec{r}$ is given by

$$\begin{eqnarray}&&{\vec{E}}_{1}=\displaystyle \frac{{q}_{1}}{4\pi {\epsilon }_{o}}\displaystyle \frac{1}{{r}^{2}}\hat{r},\ \end{eqnarray} \tag{ 2.1 }$$

where $\hat{r}=\vec{r}/r$ and $4\pi {\epsilon }_{o}$ is a constant. An important feature of this Coulomb field is its dependence on the inverse-distance squared and it is traditionally represented by means of field-lines. In three-dimensional space, these lines are distributed uniformly in all directions and, for a positive charge, they can be represented as in figure 1(a).

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If a second charge q₂ is placed close to it, there will be an interaction between the two charges and a force arises. Coulomb's law states that, in empty space, the modulus of the electric force acting between two point-like charges at rest q₁ and q₂, separated by a distance r, is given by

$$\begin{eqnarray}&&| \vec{F}| =\displaystyle \frac{{q}_{1}{q}_{2}}{4\pi {\epsilon }_{0}}\displaystyle \frac{1}{{r}^{2}}.\end{eqnarray} \tag{ 2.2 }$$

This result represents, in fact, the moduli of two different forces, as indicated in figure 1(b), where F₁ is exerted by charge q₂ and felt by q₁ and F₂ is exerted by q₁ and felt by q₂. Their moduli are equal, $| \vec{{F}_{1}}| =| \vec{{F}_{2}}|$, and given by equation (2.2). As this interaction is mutual and ascribed to electric fields, one writes

$$\begin{eqnarray}&&\vec{{F}_{1}}={q}_{1}\vec{{E}_{2}}\quad\mathrm{and}\quad\vec{{F}_{2}}={q}_{2}\vec{{E}_{1}}.\end{eqnarray} \tag{ 2.3 }$$

As the forces ${\vec{F}}_{1}$ and ${\vec{F}}_{2}$ have the same moduli and opposite directions, one has ${\vec{F}}_{1}=-{\vec{F}}_{2}$, indicating that the Coulomb forces corroborates the Newtonian action–reaction principle, whereby forces always exist in opposite pairs.

It is well known that if a charge moves with constant velocity, the field-lines deviate from spherical symmetry. The new configuration can be derived rigorously from Maxwell's equations and, for a given velocity $\vec{v}$, the intensity of field produced by a point-like charge q₁ at a point P is given by

$$\begin{eqnarray}&&\vec{{E}_{1}}(r,\theta )=\displaystyle \frac{{q}_{1}}{4\pi {\epsilon }_{o}{r}^{2}}\displaystyle \frac{\left[1-\tfrac{{v}^{2}}{{c}^{2}}\right]}{{\left[1-\tfrac{{v}^{2}}{{c}^{2}}{\sin }^{2}\theta \right]}^{3/2}}\hat{r},\end{eqnarray} \tag{ 2.4 }$$

where $\vec{r}$ describes the positions of P relative to the charge and θ is the angle between $\vec{r}$ and $\vec{v}$ [1]. This result reduces to equation (2.1) for $v\to 0$. This field is no longer spherically symmetric, but still radial and, around the plane perpendicular to the movement, it becomes more intense than the corresponding Coulomb field. This means that the field lines become more concentrated in that region. For example, if the charge moves with constant velocity v along the y-axis, the lines of figure 1(a) tilt, approximating the xz-plane and become more concentrated there [2]. This feature of the field lines can be understood with the help of a pictorial image, in which space would be a kind of uniform net, similar to a graph paper. Considering just a few lines, figure 2(a) shows both this background net and the lines in the yz and xz planes for a charge at rest. For a moving charge, the net contracts along the y-axis and the same lines acquire a higher inclination, as discussed in [3] and shown in figure 2(b).

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The field lines of a charge moving with velocity $\vec{v}$ along the y-axis are shown in figure 3(a) and its value at point P is given by equation (2.4). Placing a second charge q₂ at point P, keeping it at rest, one has the situation given in figure 3(b). Again, calling ${\vec{F}}_{1}$ the force it exerts on q₁ and ${\vec{F}}_{2}$, the force felt by q₂ owing to q₁, one has

$$\begin{eqnarray}| {\vec{F}}_{1}| & = & {q}_{1}\displaystyle \frac{{q}_{2}}{4\pi {\epsilon }_{o}{r}^{2}},\ \\ | {\vec{F}}_{2}| & = & {q}_{2}\displaystyle \frac{{q}_{1}}{4\pi {\epsilon }_{o}{r}^{2}}\displaystyle \frac{\left[1-\tfrac{{v}^{2}}{{c}^{2}}\right]}{{\left[1-\tfrac{{v}^{2}}{{c}^{2}}{\sin }^{2}\theta \right]}^{3/2}}.\end{eqnarray} \tag{ 2.5 }$$

These results show that ${F}_{1}\ne {F}_{2}$ and hence Newton's third law fails to hold in this instance. In spite of its importance, this result may not look too impressive to students, due to the fact that just the moduli of these forces are different. Cases involving different directions, discussed in the sequence, may be more striking.

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The features of acceleration relevant for this discussion are mostly qualitative and can be discussed employing field lines only. This allows one to avoid the complications associated with formalism.

The electric field lines of a charge which had the modulus of its velocity changed has been discussed by Purcell [3]. One takes a charge in uniform motion along the y-axis and assumes that it hits an obstacle, which makes it stop [4]. At the instant the charge begins to stop, this information begins to propagate all over space, with the speed of light. This information is carried by a spherical bubble, whose radius increases with the speed of light c. Some time later the charge reaches rest and a second bubble begins to propagate, informing all points around the charge that it stopped completely. At any instant later, space becomes divided into three regions, as shown in figure 4. Region R_I, internal to the smaller bubble, corresponds to all points informed that the charge is at rest, region R_II corresponds to points not reached by the first information bubble, whereas R_III is a transition region, associated with the fact that the deceleration process is not instantaneous. This last region lies between two spherical surfaces, with radii $c{t}_{F}$ and $c{t}_{I}$, where t_F − t_I is the time interval of the deceleration process.

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The existence of these three regions allows one to sketch the field lines. All points belonging to R_I are informed that charge is at rest and the field inside the first sphere is electrostatic. In R_II, the field is that of a charge with uniform motion, since points there are not informed that an acceleration has occurred. For this reason, field lines in R_II point to a spot ahead of charge, which would be the position of the charge in the absence of acceleration. In order to understand what happens with lines in R_III, one resorts to Gauss' law, which states that the field lines may only spring from or die in charges. As there are no charges in R_III, field-lines must be continuous there. The field lines of a charge which was moving with velocity $\vec{v}$ and was brought to rest are represented in figure 5(a).

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The borders of R_III are spherical surfaces, with radii increasing with the speed of light. Inside this region, the electric field has important components orthogonal to the radial direction, which are carried in the expansion process. The change of the electric field in time, caused by the movement of R_III, gives rise to a magnetic field, according to the Ampère–Maxwell law, which is perpendicular to the electric field. These are features of an electromagnetic wave existing inside R_III.

Acceleration associated with changes in the direction of the velocity is present in both the classical hydrogen atom and in the scattering of charged particles by heavy targets. A particularly important instance of the latter kind is Rutherford scattering. It is quite common in scattering of charged particles, which the direction of velocity is modified. In figure 5(b), the electric field lines are shown for a charged particle which was initially moving along the y-axis and deflected afterwards by α, moving along the dashed red line.

In general, accelerating charges radiate and the qualitative description presented here rely on deformations of field lines which propagate inside expanding bubbles. These deformations are generated by changes in the velocity of charged particles. In these examples, just single pulse waves were considered. If one wants to create a monochromatic electromagnetic wave, with a well defined frequency, the acceleration of the charge must be periodic, as in the simple harmonic motion.

From a didactical point of view, simulators in physics lessons can both help the visualization of the situations discussed above and allow the inclusion of many other instances. For example, the simulator Radiation 2D, which displays the time development of electric field lines in time, was constructed by means of a mathematical method in agreement with electromagnetic theory [5]. Similar versions can be found in specialized sites³ and can be used in order to clarify this subject. In figure 6, it is shown the field lines of a moving charge brought to rest, as discussed in section 3.1. It is interesting to follow the direction of the lines both inside and outside the circle. As in figure 5(a), field lines inside the bubble are radial and point to the center, where the charge is, whereas the external lines point to the right of the charge.

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This simulator is very useful for showing the deformation of electric field lines because it allows the initial velocity of charge to be chosen at will. In general, it is also possible to specify the trajectory of the charge, as line, dipole oscillation, figure 7, circle, figure 8, and others.

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