I get asked that question a lot, and I also see similar question on Physics Forums. Kids who are either still in high school, or starting their undergraduate years are asking which area of study should they pursue. In fact, I've seen cases where students ask whether they should do "theoretical physics" or "engineering", as if there is nothing in between those two extremes!
My response has always been consistent. I why them why can't they have their cake and eat it too?
This question often arises out of ignorance of what physics really encompasses. Many people, especially high school students, still think of physics as being this esoteric subject matter, dealing with elementary particles, cosmology, wave-particle duality, etc.. etc., things that they don't see involving everyday stuff. On the other hand, engineering involves things that they use and deal with everyday, where the product are often found around them. So obviously, with such an impression, those two areas of study are very different and very separate.
I try to tackle such a question by correcting their misleading understanding of what physics is and what a lot of physicists do. I tell them that physics isn't just the LHC or the Big Bang. It is also your iPhone, your medical x-ray, your MRI, your hard drive, your silicon chips, etc. In fact, the largest percentage of practicing physicists are in the field of condensed matter physics/material science, an area of physics that study the basic properties of materials, the same ones that are used in modern electronics. I point to them many of the Nobel Prize in physics that were awarded to condensed matter physicists or for invention of practical items (graphene, lasers, etc.). So already, the idea of having to choose between doing physics, and doing something "practical and useful" may not be mutually exclusive.
Secondly, I point to different areas of physics in which physics and engineering smoothly intermingle. I've mentioned earlier about the field of accelerator physics, in which you see both physics and engineering come into play. In fact, in this field, you have both physicists and electrical engineers, and they often do the same thing. The same can be said about those in instrumentation/device physics. In fact, I have also seen many high energy physics graduate students who work on detectors for particle colliders who looked more like electronics engineers than physicists! So for those working in this field, the line between doing physics and doing engineering is sufficiently blurred. You can do exactly what you want, leaning as heavily towards the physics side or engineering side as much as you want, or straddle exactly in the middle. And you can approach these fields either from a physics major or an electrical engineering major. The point here is that there are areas of study in which you can do BOTH physics and engineering!
Finally, the reason why you don't have to choose to major in either physics or engineering is because there are many schools that offer a major in BOTH! My alma mater, the University of Wisconsin-Madison (Go Badgers!) has a major called AMEP - Applied Mathematics, Engineering, and Physics - where with your advisor, you can tailor a major that straddles two of more of the areas in math, physics, and engineering. There are other schools that offer majors in Engineering Physics or something similar. In other words, you don't have to choose between physics or engineering. You can just do BOTH!