Question

A negatively charged particle revolves in a clockwise direction around a positively charged sphere. The work done on the negatively charged particle by the electric field of the sphere is a) positive. b) negative. c) zero.

Short answer

a) positive b) negative c) zero Answer: c) zero

Step-by-step solution

Identify the forces acting on the particle

The negatively charged particle will be attracted towards the positively charged sphere due to the electrostatic force between them. This force will act along the radial direction, towards the center of the sphere.

Determine the direction of motion

The particle is given to be revolving in a clockwise direction around the sphere. As it moves along the circular path, its direction of motion will always be tangent to the circle at any point on the path.

Relationship between force and direction of motion

We need to determine the angle between the electrostatic force (towards the center of the sphere) and the direction of motion (tangent to the circle). The angle between the radial direction (force) and the tangential direction (motion) is always 90 degrees.

Calculate the work done

Now that we know the angle between the force and the direction of motion (90 degrees), we can apply the formula for work done, which is given by: Work done = Force × Displacement × cos(angle) Since the angle between force and displacement is 90 degrees, cos(90) = 0. Therefore, Work done = Force × Displacement × 0

Conclusion

Since the work done is given by the product of force, displacement, and cos(angle), and the angle between the force and displacement is 90 degrees, the work done on the negatively charged particle by the electric field of the sphere is zero. So, the correct answer is: c) zero.

Key concepts

Understanding Electrostatic Force

Electrostatic force is a fundamental interaction between objects with electric charge. In our scenario, a negatively charged particle experiences attraction towards a positively charged sphere due to this invisible but powerful force. Imagine two magnets with opposite poles facing each other; they irresistibly pull towards one another. Similarly, charged particles exhibit electrostatic attraction or repulsion depending on whether their charges are opposite or identical respectively. This force is described quantitatively by Coulomb's law, which states that the force between two point charges varies directly with the product of their charges and inversely with the square of the distance between them. It's important to note that this force acts along the line connecting the charges, which in this case, is the radial line extending from the center of the sphere to the particle.

In the case of circular motion, as in our textbook problem, the electrostatic force provides the centripetal force necessary to keep the particle in its circular path. This is crucial for understanding why the particle doesn't simply fly off in a straight line; it's constantly being 'pulled' back towards the center by the electrostatic force.

Circular Motion Dynamics

Circular motion involves an object moving along a curved path, like the negatively charged particle orbiting our positively charged sphere. The key point here is that even though the particle is accelerating due to its change in direction, it doesn't necessarily mean there's work being done in the context of physics. Work, in the physical sense, requires displacement in the direction of the force. Since our particle is moving perpendicular to the electrostatic force—meaning there's a 90-degree angle between the displacement (path of the particle) and the force (radial)—the work calculated using the formula involves the cosine of 90 degrees. But cos(90º) equals zero, which indicates that the force, despite being present and central to the particle's motion, does not perform work on the particle as it travels in its circular path.

This distinction can be confusing but is vital for physics students. The particle is indeed using energy to maintain its speed and stay in motion, but that's not attributed to work done by electrostatic force. Instead, this continuous motion is a consequence of the inertia of the particle and the constant redirection provided by the electrostatic force acting as a centripetal force.

The Work-Energy Principle in Electrostatic Contexts

The work-energy principle is a cornerstone of classical mechanics, stating that work done on an object results in a change in kinetic energy. If work is positive, the object gains kinetic energy; if negative, it loses kinetic energy; if zero, the kinetic energy remains unchanged. Applying this concept to our problem, it becomes apparent that since the work done on the negatively charged particle by the electric field is zero, its kinetic energy does not increase or decrease while it moves in circular motion around the charged sphere.

It's essential to clarify that the particle's constant speed in circular motion is not due to a lack of forces acting on it; on the contrary, the electrostatic force is indispensable in maintaining its circular trajectory. However, since this force always acts perpendicular to the motion, it only changes the direction of velocity, not its magnitude, thus doesn't contribute to the work done. This perfectly illustrates the work-energy principle where, despite continuous motion and force application, the kinetic energy of an object can remain constant if the work done is zero.