Concept explainers
a.
The mass of the particle.
The mass of the particle is
Given:
The charge of the ring
The charge of the particle
The radius of the ring
The frequency of oscillation
Also, that the displacement is much small than the radius of the ring
Formula Used:
Electric field by a ring as a function of x.
E is the electric field.
k is a constant.
q is the charge of the ring.
a is the radius of the ring.
x is the distance from center of the ring.
Calculations:
As
Now force is
As the negatively charged particle experiences a restoring force, the motion will be a simple harmonic motion.
Equating
Relating with the acceleration of a particle executing a simple harmonic motion.
This is the differential equation of a simple harmonic motion.
Hence
Solving for m
Substituting the values in the equation.
Conclusion:
The mass of the particle is
b.
The frequency of the motion if the radius of the ring is doubled.
The frequency is
Given:
The frequency is doubled.
Formula Used:
Angular frequency is
k is a constant.
q is the charge of the ring.
a is the radius of the ring.
x is the distance from center of the ring.
Calculations:
Now comparing the angular frequency when the radius is doubled.
Conclusion:
The frequency is
b.
The frequency of the motion if the radius of the ring is doubled.
The frequency is
Given:
The frequency is doubled.
Formula Used:
Angular frequency is
k is a constant.
q is the charge of the ring.
a is the radius of the ring.
x is the distance from center of the ring.
Calculations:
Now comparing the angular frequency when the radius is doubled.
Conclusion:
The frequency is
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Physics for Scientists and Engineers
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