Consider a cylindrical optical fiber with a core that has a graded index (GRIN) of refraction given by where is the distance to from the optical axis of the fiber. Show that this profile results in continual refocusing ofparaxialrays as shown in the figure below. If , determine the distance in millimetersbetween two consecutive focal points.
A thin ring with resistance and self inductance rotates with constant angular speed in an external uniform magnetic field perpendicular to the rotation axis. As a result, the magnetic flux created by the external field varies with time as Because of the resistance of the ring, energy is being continuously dissipated in the system. What average power in Wattsmust develop the external forces to keep the ring rotating at constant angular speed (on average).
Consider two balls of equal radii and masses but opposite charges, distributed uniformly over their volumes. Initially, the balls are at rest and far away from one another. Due to the Coulomb attraction, the balls start moving towards each other. The balls can be treated as charged clouds, that is to say, they can interpenetrate without friction. During the interaction, the maximum speed achieved by the balls is . What is the magnitude of their maximum accelerationin meters per second squared? The radius of the balls is .
An electron is accelerated by a potential difference of . It then enters a region with an inhomogeneous magnetic field generated by a system of coils carrying currents .
然后,我们执行一个类似的实验一个质子. We first reverse the current in all the coils generating the magnetic field. We then accelerate the proton with a potential difference of and it enters the region with the magnetic field. What must bein Voltsso that the trajectory of the proton is the same as that of the electron? Don't forget any sign changes!
Hint: Compute where is the arc length along the trajectory:
.
Details and assumptions
The proton to electron mass ratio is approximately .
In 1-d Newtonian mechanics the energy as a function of momentum for a particle is given by or, in terms of momentum, . In Einstein's theory of special relativity there is still energy and momentum, and they are still conserved, but the relationship between the two is different. In empty space, the energy of a particle as a function of its momentum and mass is given by where is the speed of light. The photon, the "particle" of light, has no mass - its energy satisfies in empty space. In a medium like water this can change - photons travel slower in a medium and their energy-momentum relation becomes , where is the speed of light in the medium. Our question is the following: A very high energy proton with enters a tank of water. What is the minimum magnitude of the proton's momentum, i.e. , inkg m/s, such that the proton can emit a photon with non-zero momentum and still conserve energy and momentum?
Details and assumptions