# Can electromagnetic wave cause a charged particle to move with constant velocity?

I'm reading through Kleppner and Kolenkow's Introduction to Mechanics (2nd ed), and in it, they go through the example of an electron subject to an oscillating electric field (example 1.11). The example and the math is straightforward, but at the end of the explanation, they say:

Here is the problem statement:

The ionosphere is a region of electrically neutral gas, composed of positively charged ions and negatively charged electrons, that surrounds the Earth at a height of approximately 200 km (120 mi). If a radio wave passes through the ionosphere, its electric ﬁeld accelerates the charged particles. Because the electric ﬁeld oscillates in time, the charged particles tend to jiggle back and forth. The problem is to ﬁnd the motion of an electron of charge−e and mass m which is initially at rest, and which is suddenly subjected to an electric ﬁeld E = $E_0$ sinωt (ω is the frequency of oscillation in radians/second).

And the evaluation portion (the italicized one) writes:

RESULT: The electron is initially at rest, $x_0$ = $v_0$ = 0, so we have x(t) = ($a_0$/ω) t − ($a_0$/$ω^2$) (sinωt).

The result is interesting: the second term oscillates and corresponds to the jiggling motion of the electron that we predicted. The ﬁrst term, however, corresponds to motion with uniform velocity, so in addition to the jiggling motion the electron starts to drift away. Can you see why?

I'm not sure I'm able to see why, as they say. I understand that the drift comes as a result of the integration step. If you double integrate a sine, you have a constant term that then becomes the drift term, but I'm not seeing an "aha" explanation as they seem to suggest. I feel like I'm missing an insight that they are referring to. Am I?

• It might have to do with the assumption that $v_0=0$. If for example $v_0=-\frac{a_0}{\omega}$ then you don't get a drift velocity anymore. – BioPhysicist Mar 9 '18 at 4:22
• But the particle was at rest and started drifting after the radio wave passed through it (according to the question). – suiz Mar 9 '18 at 5:11
• There's an answer and the question on Reddit here: reddit.com/r/Physics/comments/5aaiy4/… – Rob Mar 9 '18 at 9:55
• I know this is the problem. I was just trying to consider a different scenario to help build insight. Based on the answer by @StephenG, if you have this specific initial velocity, then you have equal times of positive and negative velocities rather than all positive with one point in time where v is 0. Hence the drift is no longer present. – BioPhysicist Mar 9 '18 at 13:18
• @Rob the answer got on reddit wasn't satisfactory. Actually there was no definite answer given. StephenG's answer is very much intuitive and satisfactory. – suiz Mar 10 '18 at 13:55

The ﬁrst term, however, corresponds to motion with uniform velocity, so in addition to the jiggling motion the electron starts to drift away. Can you see why?

It's all to do with velocity.

The particle initially moves away with a positive force on it until the force has completed half it's cycle and then the force becomes negative.

But by the time the force becomes negative the electron has already acquired a forward momentum which will not be reversed during the negative cycle !

All the negative part of the force cycle can do is slow it down, but it will always have a positive velocity until right at the end of the complete cycle, when velocity becomes zero again (at which point we start another cycle and repeat this process).

So averaged over every complete cycle it always has a non-negative velocity and on average this moves it forward and is the reason for that linear term in the displacement equation.

If there is a force acting on it which is equal and opposite to the force by field it can maintain constant velocity