In Fig. 28-58, an electron of mass m, charge -e, and low (negligible) speed enters the region between two plates of potential difference V and plate separation d, initially headed directly toward the top plate. A uniform magnetic field of magnitude B is normal to the plane of the figure. Find the minimum value of B, such that the electron will not strike the top plate.
The minimum value of B such that the electron will not strike the top plate is
Consider the equations of electric force and magnetic force. The magnetic force opposes the motion of electron, so to prevent the electron from striking, the magnetic force must be greater than the electric force.
Consider the equation for potential energy of electron just before it strikes the top plate as,
Now, according to the conservation of energy principle is as follows:
Rearranging for velocity derive the equation as:
Now, consider the equation for magnetic force as
Substituting for the velocity, we get
Now, to prevent the electron from striking the top plate, FB must be greater than FE.
So, to find the minimum magnetic field, let us assume those to be equal.
Rearranging it for magnetic field, solve as:
A cyclotron with dee radius 53.0 cm is operated at an oscillator frequency of 12.0 MHz to accelerate protons.
(a) What magnitude B of magnetic field is required to achieve resonance?
(b) At that field magnitude, what is the kinetic energy of a proton emerging from the cyclotron? Suppose, instead, that B = 1.57 T.
(c) What oscillator frequency is required to achieve resonance now?
(d) At that frequency, what is the kinetic energy of an emerging proton?
Figure shows a wire ring of radius that is perpendicular to the general direction of a radially symmetric, diverging magnetic field. The magnetic field at the ring is everywhere of the same magnitude , and its direction at the ring everywhere makes an angle with a normal to the plane of the ring. The twisted lead wires have no effect on the problem. Find the magnitude of the force the field exerts on the ring if the ring carries a current .
A certain particle is sent into a uniform magnetic field, with the particle’s velocity vector perpendicular to the direction of the field. Figure 28-37 gives the period T of the particle’s motion versus the inverseof the field magnitude B. The vertical axis scale is set by , and the horizontal axis scale is set by what is the ratio m/q of the particle’s mass to the magnitude of its charge?
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