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Chapter 20: Problem 35

How strong an electric field is needed to accelerate electrons in an X-ray tube from rest to one-tenth the speed of light in a distance of \(5.0 \mathrm{cm} ?\)

Short Answer

Expert verified
The strength of the electric field necessary to accelerate the electrons is 5.1 x \(10^4\, N/C\).

Step by step solution

01

Set the Velocity of Electron

Since it is stated that the electrons are accelerating to one-tenth the speed of light, the final velocity \(v\) of the electrons is \(v = c/10\), where \(c = 3 \times 10^8\, m/s\) is the speed of light. So, \(v = 3 \times 10^7\, m/s\).
02

Calculate the Acceleration of Electron

Since the electrons start from rest, their initial velocity \(u = 0\). The distance \(s\) over which they are accelerated is 5 cm = 0.05 m. Using the second equation of motion \(v^2 = u^2 + 2as\), where \(a\) is acceleration, \(s\) is distance, \(v\) is the final velocity and \(u\) is the initial velocity, we can solve for acceleration. Substituting \(u = 0\), \(v = 3 \times 10^7\, m/s\) and \(s = 0.05\, m\) into the equation, we have \(a = v^2 /2s = (3 \times 10^7)^2 /(2 \times 0.05) = 9 \times 10^{15}\, m/s^2\).
03

Calculate the Electric Field Strength

The force \(F\) experienced by the electron is given by \(F = ma\), where \(m = 9.11 \times 10^{-31} kg\) is the mass of the electron and \(a = 9 \times 10^{15}\, m/s^2\) is acceleration. So, \(F = 9.11 \times 10^{-31} \times 9 \times 10^{15} = 8.2 \times 10^{-15} N\). The strength of the electric field \(E\) is then given by \(E = F/e\), where \(e = 1.6 \times 10^{-19}\, C\) is the charge on an electron. So, \(E = 8.2 \times 10^{-15} / 1.6 \times 10^{-19} = 5.1 \times 10^4\, N/C\).

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Most popular questions from this chapter

An electron at Earth's surface experiences a gravitational force \(m_{e} g .\) How far away can a proton be and still produce the same force on the electron? (Your answer should show why gravity is unimportant on the molecular scale!)

Earth carries a net charge of about \(-5 \times 10^{5} \mathrm{C}\). How many more electrons are there than protons on Earth?

A -1.0 - \(\mu\) C charge experiences a \(10 \hat{\imath}\) -N electric force in a certain electric field. What force would a proton experience in the same field?

The electric field on the axis of a uniformly charged ring has magnitude \(380 \mathrm{kN} / \mathrm{C}\) at a point \(5.0 \mathrm{cm}\) from the ring center. The magnitude \(15 \mathrm{cm}\) from the center is \(160 \mathrm{kN} / \mathrm{C} ;\) in both cases the field points away from the ring. Find (a) the ring's radius and (b) its charge.

Under what circumstances is the path of a charged particle a parabola? A circle?
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