d please Let us calculate the forces acting on a particle in a beam of positivel
ID: 1521698 • Letter: D
Question
d please
Let us calculate the forces acting on a particle in a beam of positively charged particles having a velocity v and carrying charges Q. We assume that there are no externally applied electric or magnetic fields. We shall consider positive particles, but it will be a simple matter to apply our results to negative ones. First, there is an outward E, as in the figure. This is a simple case of electrostatic repulsion. Also, the beam current produces an azimuthal B, and v cross B points inward as in the figure. Thus, there is an outward electric force QE and an inward magnetic force Qv cross B. Experimentally, electron and ion beams always diverge, when left to themselves, but the divergence is slight when the particle velocity v approaches the velocity of light c. We consider a particle situated at the edge of a beam of radius R. The current is I and the particle velocity is v. Calculate the charge per meter, lambda, of the beam. Calculate the outward electric force QE. Calculate the inward magnetic force QvB. Calculate the net force. You should find that the net force points outward and is proportional to 1 - sigma_0mu_0v^2 = 1 - v^2/c^2. Then the net force tends to zero as v rightarrow c. If the particles are negative, E is inward instead of outward, but QE is again outward. Also, B points in the opposite direction and Qv cross B again points inward. In practice, a vacuum is never perfect. Let us say the ions are positive. If their energy is of the order of tens of electron-volts or more, they ionize the residual gas, forming low-energy positive ions and low-energy electrons. These positive ions drift away from the positive beam. The low-energy electrons, however, remain trapped in the beam and neutralize part for the space charge, thereby reducing E. The magnetic force then tends to pinch the beam. This phenomenon is called the pinch effect. It is also called gas focusing. With gas focusing, some of the particles in the beam are scattered away, in colliding with the gas molecules.Explanation / Answer
d)
net force (lorentz force)=electric force +magnetic force
=qE-qvB
=q(E-vB)
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