Does the sun do any work on earth, when earth revolves around the sun in a perfectly circular orbit?

Hint: Recall what is work and how we define work in the world of physics. Does the shape of the orbit have to do something or the force experienced by earth will provide you with the answer. Getting answers to these easy yet fundamental questions will help us solve this problem.

Complete answer:
Let’s first go to the definition of work. Work is said to be done when a force acting on a certain object has some resulting displacement on the object relative to its initial position. Or in simple terms, when a force acting on an object displaces it by a certain distance, work is said to be done.
Work is a scalar quantity, it does not possess any specific direction.
Mathematically, work is the dot product or scalar product of force vector and displacement vector.
Or $W = \overrightarrow F \cdot \overrightarrow s $
Where, $W$ is the work done by the force $\overrightarrow F $ after displacement of $\overrightarrow s $ of the object.
$ \Rightarrow W = Fs\cos \theta $
Where $\theta $ is the angle between two vectors.
If the angle is $0^\circ$, work done will be maximum, that is, if the force is acting in the same direction as displacement.
If the angle is $180^\circ $ work done will be the negative maximum because force is acting opposite to the direction of displacement.

But at an angle $90^\circ $ between the vectors, net work done will be zero.
Now let’s look at the problem. If earth is rotating around a completely circular path, then the centripetal force or the gravitational force experienced by earth due to the sun, is at an angle $90^\circ $ from the direction of motion.
Because the displacement vector at any point along earth’s orbit is tangent drawn at that point. In circular motion, centripetal force acts along the radius of the circle.

Thus work done by the sun on earth when revolving in a perfectly circular orbit will be zero.

Note: The required centripetal force to keep earth rotating around the sun is provided by their gravitational attraction. However earth does not revolve in a perfectly circular path around the sun, the trajectory is somewhat elliptical. Thus some amount of work is done by the sun on earth, because the angle between force and displacement/ direction of motion will be slightly more or less than $90^\circ $.