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An electron released on the axis of a po...

An electron released on the axis of a positively charged ring at a large distance from the centre will

A

Not move

B

Do oscillatory motion

C

Do SHM

D

Do non-periodic motion

Text Solution

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The correct Answer is:
To solve the problem of an electron released on the axis of a positively charged ring at a large distance from the center, we can follow these steps: ### Step 1: Understand the Electric Field Due to the Ring At a large distance \( x \) from the center of the ring, the electric field \( E \) due to a uniformly charged ring can be expressed as: \[ E = \frac{k \cdot Q \cdot x}{(R^2 + x^2)^{3/2}} \] where \( k \) is Coulomb's constant, \( Q \) is the total charge of the ring, \( R \) is the radius of the ring, and \( x \) is the distance from the center of the ring along its axis. ### Step 2: Analyze the Force on the Electron The force \( F \) acting on the electron (which has charge \( -e \)) due to the electric field is given by: \[ F = -e \cdot E \] Substituting the expression for \( E \): \[ F = -e \cdot \frac{k \cdot Q \cdot x}{(R^2 + x^2)^{3/2}} \] The negative sign indicates that the force is directed towards the center of the ring (attraction). ### Step 3: Determine the Nature of Motion As the electron is released, it will experience a force directed towards the center of the ring. This force will cause the electron to accelerate towards the ring. As it approaches the ring, the electric field strength will change, but the force will always act towards the center. ### Step 4: Identify the Type of Motion Since the force acting on the electron is always directed towards the center, the electron will oscillate about the center of the ring. However, the relationship between the force and displacement is not linear (as it is proportional to \( \frac{1}{x^2} \)), which means the motion will not be simple harmonic motion (SHM). ### Step 5: Conclusion The electron will undergo oscillatory motion but not SHM. The oscillation will be periodic but not simple harmonic due to the nature of the force being inversely proportional to the square of the distance. ### Final Answer The electron will undergo oscillatory motion. ---
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