Question

A charge \( Q \) is fixed in position. Another charge \( q \) is brought near charge \( Q \) and released from rest. Which of the following graphs is the correct representation of the acceleration of the charge \( q \) as a function of its distance \( r \) from charge \( Q \)?

• A.
• B.
• C.
• D.

Hint:

For Coulomb’s law, remember the force between two charges is inversely proportional to the square of the distance between them.

Answer 1

The Correct Option is A

The acceleration \( a \) of a charge \( q \) in an electric field due to another fixed charge \( Q \) can be expressed using Coulomb's Law. The force \( F \) between two charges is given by: \[ F = \frac{k \cdot |Q \cdot q|}{r^2} \] where \( k \) is the Coulomb's constant. The acceleration \( a \) of the charge \( q \) is then: \[ a = \frac{F}{m} = \frac{k \cdot |Q \cdot q|}{m \cdot r^2} \] where \( m \) is the mass of the charge \( q \). From the equation, it is clear that the acceleration \( a \) is inversely proportional to the square of the distance \( r \). As \( r \) increases, \( a \) decreases rapidly, and as \( r \) decreases, \( a \) increases sharply. This relationship can be depicted graphically as a hyperbolic function of acceleration versus distance. Therefore, the correct graph representation is the one where acceleration decreases with the square of the distance, depicted as \(\frac{1}{r^2}\). The correct option is the first graph, where acceleration is inversely proportional to \( r^2 \).