Electrons in a television's CRT are accelerated from rest by an electric field through a potential difference of \(2.5 \mathrm{kV} .\) In contrast to an oscilloscope, where the electron beam is deflected by an electric ficld, the beam is deflected by a magnetic field. (a) What is the specd of the electrons? (b) The beam is deflected by a perpendicular magnetic field of magnitude $0.80 \mathrm{T}$. What is the magnitude of the acceleration of the electrons while in the field? (c) What is the speed of the electrons after they travel 4.0 mm through the magnetic field? (d) What strength electric field would give the electrons the same magnitude acceleration as in (b)? (c) Why do we have to use an clectric ficld in the first place to get the electrons up to speed? Why not use the large acceleration due to a magnetic field for that purpose?

Step by step solution

01

Find the speed of the electrons after acceleration

To find the speed of the electrons after acceleration through the electric field, first, find their kinetic energy using the formula \(KE = eV\), where \(e\) is the charge of the electron, and \(V\) is the potential difference. Then, use the kinetic energy and the mass of the electron to find their speed using the formula \(KE = \frac{1}{2}mv_a^2\).

02

Calculate the magnetic force and acceleration

To find the magnetic force acting on the electrons, use the formula \(F_m = ev_aB\), where \(F_m\) is the magnetic force, \(v_a\) is the speed of the electrons calculated in step 1, and \(B\) is the magnetic field strength. After finding the magnetic force, find the acceleration using the formula \(a_m = \frac{F_m}{m}\), where \(m\) is the mass of the electron.

03

Find the speed of the electrons after traveling through the magnetic field

Since the magnetic force acts perpendicular to the speed of the electrons, it does not change their magnitude. Thus, the speed of the electrons after traveling through the magnetic field remains the same as the speed calculated in step 1 \((v_a)\).

04

Calculate the strength of an equivalent electric field

To find the strength of an electric field that gives the same magnitude of acceleration as the magnetic field, use the formulas \(F_e = ea_e\) and \(F_e = eE\), where \(F_e\) is the electric force, \(E\) is the electric field strength, and \(a_e\) is the equivalent acceleration. With the calculated magnetic acceleration, \(a_m\), we can find the equivalent electric field strength \(E = \frac{a_m}{e}\).

05

Discuss the role of electric and magnetic fields

Electric fields are used to accelerate the electrons because they act parallel to the electric field, and thus, they can change the speed (either increase or decrease) of the electrons. In contrast, magnetic fields always act perpendicular to the electron's speed, so they can only change the direction of the electrons but not their magnitude. That is why electric fields are used to accelerate the electrons, and magnetic fields are used to deflect (change the direction) of the electrons.

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