# Radiation by an accelerated charge

**Alternating magnetic field, electromagnetic radiation, electromagnetic induction**

An electric field moves with the velocity that the particle had while emitting that field.

**Electric field of a charge that stops**

Suppose a charge q has been moving at constant speed v_{0} in the x direction for a long time. Suddenly it stops after a short period of constant deceleration. Electric field of a charge travels radially outward and moves with the velocity that the charge had while emitting that field.

The velocity versus time graph

The electric field

Assuming that the v_{0} ≪ c, we can neglect the relativistic compression of the field lines. Time t=0 it was the moment when deceleration began, and position x=0 it was the position of the particle at that moment. The particle has been moved a little farther on before coming to a stop, Δx=1/2*v_{0}Δt_{1}. That distance is very small compared with the other distances in the picture.

We now examine the electric field at a time t=Δt_{2}≫Δt_{1}. The electric field reaches a distance R=cΔt_{2}. Thus, the field at a distance greater than R=cΔt_{2}, in region I must be a field of a charge which has been moving and is still moving at the constant speed v_{0}. That field appears to come from the point x=v_{0}Δt_{2} on the x axis. That is where the particle would be now if it hadn't stopped. On the other hand, the field at a distance less than c(Δt_{2}-Δt_{1}), in region II must be a field of a charge at rest close to the position x=0 (exactly at x=1/2*v_{0}Δt_{1}).

What must the field be like in the transition region, the spherical shell of thickness cΔt_{1} between region I and region II? A field line segment AB lies on a cone around the x axis which includes a certain amount of flux from the charge q. Due to the Gauss' law, if CD makes the same angle θ with the axis, the cone on which it lies includes that same amount of flux. (Because the v_{0} ≪ c, we can neglect the relativistic compression of the field lines.) That's why AB and CD must be parts of the same field line, connected by a segment BC. The line segment BC shows us the direction of the filed E within the shell. This field E within the shell has both a radial component E_{r} and a transverse component E_{θ}. From the geometry of the figure their ratio is easily found.

Due to the Gauss' law, E

_{r}must have the same value within the shell thickness that it does in region II near B. That's why E

_{r}=q/4πε

_{0}R

^{2}=q/4πε

_{0}c

^{2}Δt

_{2}

^{2}, and substituting this in Eq. 1 we obtain

_{0}/Δt

_{1}= a, the magnitude of the (negative) acceleration, and cΔt

_{2}= R, so our result can be written

_{θ}is proportional to 1/R, not to 1/R

^{2}! As time goes on and R increases, the transverse field E

_{θ}will eventually become very much stronger than E

_{r}. Accompanying this transverse (that is, perpendicular to R) electric field will be a magnetic field of equal strength perpendicular to both R and E. This is a general property of an electromagnetic wave.

We now calculate the energy stored in the transverse electric field above, in the whole spherical shell. The energy density is

^{2}cΔt

_{1}, and the average value of sin

^{2}θ over a sphere is 2/3. The total energy of the transverse electric field is consequently

Total energy in transverse electromagnetic field is

_{1}is the duration of the deceleration, and is also the duration of the electromagnetic pulse a distant observer measures, we can say that the power radiated during the acceleration process was

_{rad}itself turns out to be Lorentz-invariant, which is sometimes very handy. That is because P

_{rad}is energy/time, and energy transforms like time, each being the fourth component of a four-vector. We have here a more general result than we might have expected. Equation 7 correctly gives the instantaneous rate of radiation of energy by a charged particle moving with variable acceleration-for instance, a particle vibrating in simple harmonic motion. It applies to a wide variety of radiating systems from radio antennas to atoms and nuclei.

radiation, accelerated, charge, electromagnetic, induction, wave, waves, physics, science,

**Radiating Charge simulation - PhET INTERACTIVE SIMULATIONS**

Purcell appendix B in the CGS system