Electromagnetic Radiation.

Figure 7.2:

What is the electric field produced at a point P by a charge $q$ located at a distance $r$ as shown in Figure 7.2? Anybody with a little knowledge of physics will tell us that this is given by Coulomb's law

$$\vec{E}=\frac{-q}{4 \pi \epsilon_0}\frac{\hat{e}_r}{r^2}$$ (7.1)

where $\hat{e}_r$ is an unit vector from P to the position of the charge. In what follows we shall follow the notation used in Feynman Lectures ( Vol. I, Chapter 28, Electromagnetic radiation).

In the 1880s J.C. Maxwell proposed a modification in the laws of electricity and magnetism which were known at that time. The change proposed by Maxwell unified our ideas of electricity and magnetism and showed both of them to be manifestations of a single underlying quantity. Further it implied that Coulomb's law did not tells us the complete picture. The correct formula for the electric field is

$$\vec{E}=\frac{-q}{4 \pi \epsilon_0}\left[ \frac{\hat{e}_{r'... ...ght) + \frac{1}{c^2} \frac{d^2}{dt^2} \hat{e}_{r'}\right]$$ (7.2)

This formula incorporates several new effects. The first is the fact that no information can propagate instantaneously. This is a drawback of Coulomb's law where the electric field at a distant point P changes the moment the position of the charge is changed. This should actually happen after some time. The new formula incorporates the fact that the influence of the charge propagates at a speed $c$. The electric field at the time $t$ is determined by the position of the charge at an earlier time. This is referred to as the retarded position of the charge $r'$, and $\hat{e}_{r'}$ also refers to the retarded position.

The first term in eq. (7.2) is Coulomb's law with the retarded position. In addition there are two new terms which arise due to the modification proposed by Maxwell. These two terms contribute only when the charge moves. The magnetic field produced by the charge is

$$\vec{B}=-\hat{e}_{r'}\times \vec{E}/c$$ (7.3)

A close look at eq. (7.2) shows that the contribution from the first two terms falls off as $1/{r'}^2$ and these two terms are of not of interest at large distances from the charge. It is only the third term which has a $1/r$ behaviour that makes a significant contribution at large distances. This term permits the a charged particle to influence another charged particle at a great distance through the $1/r$ electric field. This is referred to as electromagnetic radiation and light is a familiar example of this phenomena. It is obvious from the formula that only accelerating charges produce radiation.

The interpretation of the formula is substantially simplified if we assume that the motion of the charge is relatively slow, and is restricted to a region which is small in comparison to the distance $r$ to the point where we wish to calculate the electric field. We then have

$$\frac{d^2}{dt^2} \hat{e}_{r'}= \frac{d^2}{dt^2} \left( \fra... ...'} \right) \approx \frac{\ddot{\vec{r}^{'}_{\perp}}}{r}$$ (7.4)

where $\ddot{\vec{r}'_{\perp}}$ is the acceleration of the charge in the direction perpendicular to $\hat{e}_{r'}$. The parallel component of the acceleration does not effect the unit vector $\hat{e}_{r'}$ and hence it does not make a contribution here. Further, the motion of the charge makes a very small contribution to $r'$ in the denominator, neglecting this we replace $r'$ with the constant distance $r$.

The electric field at a time $t$ is related to $a(t-r/c)$ which is the retarded acceleration as

$$E(t)=\frac{-q}{4 \pi \epsilon_0 c^2 r} \, a(t-r/c) \, \sin \theta$$ (7.5)

where $\theta$ is the angle between the line of sight $\hat{e}_r$ to the charge and the direction of the retarded acceleration vector. The electric field vector is in the direction obtained by projecting the retarded acceleration vector on the plane perpendicular to $\hat{e}_r$ as shown in Figure 7.3.

Figure 7.3:

Problem 1: Show that the second term inside the bracket of eq.(7.2) indeed falls off as $1/{r'}^2$. Also show that the expression for electric field for an accelerated charge i.e. eq. (7.5) follows from it.

Solution 1: See fig. 7.4 ($r$ and $\theta$ can be treated as constants with respect to time).

Figure 7.4: