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3: Path is a straight line. Particle is accelerated due to electric force in the direction opposite to the direction as in case 1. Speed and kinetic energy of the particle increase by same amount as in the case 1.
4: Path is a semicircular curve of greater radius of curvature due to increased speed. Particle is accelerated due to magnetic force. This acceleration is centripetal acceleration without any change in speed and kinetic energy of the particle.
5: Path is a straight line. Particle is accelerated due to electric force in the direction opposite to the direction as in case 1. Speed and kinetic energy of the particle increase by same amount as in the case 1 or 3.
We see that the particle follows consecutive larger semicircular path due to increase in the speed at the end of semicircular journey. The resulting path of charged particle, therefore, is a spiral path – not circular.
What should be the frequency at which the electrical oscillator changes sign? As per the account given in the previous section, the particle is required to be accelerated after completion of every semicircular journey of charged particle in the Dee. Does it mean that electrical polarity should be changed twice for one revolution in the magnetic field? Answer is no. Though particle is speeded up twice in a cycle, it requires change of direction of electric field only once. One of the directions is the existing direction and other is the reversed or changed direction. See the figure. Count the numbers of “change of directions” involved and numbers of revolutions. There are 7 occasions each when electric field has one of two possible directions. On the other hand, there are 7 revolutions counted from the beginning. Clearly, numbers of changes in directions are equal to numbers of revolutions. This means that frequency of electric oscillator should be equal to frequency of revolutions.
From the perspective of energy also, it is required that energy is added up to the moving charge at its natural frequency. This is the principle involved in resonance phenomena. We can pump energy to a periodic or oscillating system by supplying energy in small quantity at the natural frequency of the system. Hence, frequency of electrical oscillator is :
$$\nu =\frac{qB}{2\pi m}$$
Note that periodic properties of spiral motion are exactly same as that of circular motion of a charged particle in magnetic field. The frequency at which the charged particle completes spiral revolution is independent of the velocity. It is a very important feature of motion of charged particle in magnetic field. So even if the speed of the particle is increased with every passage through the gap, the time taken to reach the gap consecutively is same. It is the core consideration here allowing us to have a fixed frequency of electrical oscillator for a given magnetic field or conversely allowing us to have a constant magnetic field for a given frequency of electric oscillator. Of course, these constant values are determined keeping in mind the specific charge (charge and mass ratio) and size of the cyclotron.
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