Concept explainers
A charged particle moves along a straight line in a uniform electric field E with a speed v. If the motion and the electric field are both in the x direction, (a) show that the magnitude of the acceleration of the charge q is given by
(b) Discuss the significance of the dependence of the acceleration on the speed. (c) If the particle starts from rest at x = 0 at t = 0, find the speed of the particle and its position after a time t has elapsed. Comment on the limiting values of v and x as t →∞.
(a)
The magnitude of acceleration of the charge.
Answer to Problem 4P
It is proved that the acceleration of the charged particle is
Explanation of Solution
Write the equation for the relativistic momentum.
Here,
Write the equation for relativistic force.
Here,
Substitute equation (I) in (II).
Write the equation for the force in terms of electric field.
Here,
Conclusion:
Substitute equation (IV) in (III) and rearrange.
Hence, the given equation for the acceleration of the charged particle is proved.
(b)
The significance of dependence of acceleration on speed.
Answer to Problem 4P
It signifies that no particle can move with a speed greater than the speed of light.
Explanation of Solution
Equation (V) gives the expression for the acceleration of the charged particle.
Conclusion:
From equation (V), as
(c)
The speed and position of the particle.
Answer to Problem 4P
The speed of the particle is
Explanation of Solution
Rearrange equation (V) to separate the variables.
Conclusion:
Integrate the above equation by giving proper limits.
Simplify further.
The limiting behavior of v as
Here, as
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Chapter 2 Solutions
Modern Physics
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