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Chapter 23: Problem 25

A proton, initially at rest, is accelerated through a potential difference of \(500 .\) V. What is its final velocity?

Short Answer

Expert verified
Answer: The final velocity of the proton is approximately \(3.09 \times 10^{5} m/s\).

Step by step solution

01

Identify the mass and charge of a proton

A proton has a mass of \(m_p = 1.67 \times 10^{-27} kg\) and a charge of \(q_p = 1.6 \times 10^{-19} C\).
02

Calculate the kinetic energy gained by the proton

Due to the potential difference, the proton will gain kinetic energy. The kinetic energy (KE) gained can be calculated using the formula: \(KE = q_p \times V\), where V is the potential difference. Plug in the values, we get: \(KE = (1.6 \times 10^{-19} C)(500V)\) \(KE = 8 \times 10^{-17} J\)
03

Use the kinetic energy formula to find the final velocity

The formula for kinetic energy is: \(KE = \frac{1}{2} m_p v^2\), where \(m_p\) is the mass of the proton and v is the final velocity. Let's solve for v using the obtained KE value: \(8 \times 10^{-17} J = \frac{1}{2} \times 1.67 \times 10^{-27} kg \times v^2\) Now, we can solve for v: \(v^2 = \frac{2 \times 8 \times 10^{-17}}{1.67 \times 10^{-27}}\) \(v^2 \approx 9.58 \times 10^{10}\) \(v \approx 3.09 \times 10^{5} m/s\)
04

Conclusion

The final velocity of the proton after being accelerated through a potential difference of 500 V is approximately \(3.09 \times 10^{5} m/s\).

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Most popular questions from this chapter

A classroom Van de Graaff generator accumulates a charge of \(1.00 \cdot 10^{-6} \mathrm{C}\) on its spherical conductor, which has a radius of \(10.0 \mathrm{~cm}\) and stands on an insulating column. Neglecting the effects of the generator base or any other objects or fields, find the potential at the surface of the sphere. Assume that the potential is zero at infinity.

An electron is accelerated from rest through a potential difference of \(370 \mathrm{~V}\). What is its final speed?

Consider an electron in the ground state of the hydrogen atom, separated from the proton by a distance of \(0.0529 \mathrm{nm}\) a) Viewing the electron as a satellite orbiting the proton in the electrostatic potential, calculate the speed of the electron in its orbit. b) Calculate an effective escape speed for the electron. c) Calculate the energy of an electron having this speed, and from it determine the energy that must be given to the electron to ionize the hydrogen atom.

The electric potential in a volume of space is given by \(V(x, y, z)=x^{2}+x y^{2}+y z\). Determine the electric field in this region at the coordinate (3,4,5) .

A \(10.0-\mathrm{V}\) battery is connected to two parallel metal plates placed in a vacuum. An electron is accelerated from rest from the negative plate toward the positive plate. a) What kinetic energy does the electron have just as it reaches the positive plate? b) What is the speed of the electron just as it reaches the positive plate?
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