I have learnt that change in potential energy in any conservative force field is equal to the negative of work done by that field through a distance. While applying the same for gravitational potential energy, I move a mass $m$ in the field of a large immovable mass $M$ from infinity to a point P which is a distance $x$ from the source mass, I obtain the work done as
$$W = -\frac{GMm}{x}$$
But now, the change in potential energy will be negative of this work done and hence the it would be
$$\Delta\text{PE (from infinity to P)} = -W = \frac{GMm}{x}$$
The problem is that this energy term is taken negative in the expression of total energy of a satellite. I don't know what I have done wrong?
Follow up question: while calculating total energy of the satellite in a orbit of radius r at a point p(say), we take potential energy term as the work done from infinity to that point but we have launched the
satellite from earth surface to a height h (r=R+h, where R is the radius of earth) then why not take change in potential energy from surface to that point ?
Best Answer
What you did wrong was to have a minus sign in your first equation, the work done should have been positive.
Imagine the satellite started at infinity and was stationary.
It has zero potential energy and no kinetic energy, so the total energy is zero.
If it then fell to the point $x$ near the mass $M$, the gravitational field has done positive work of value $$W = \frac{GMm}{x}$$ (the direction of the force and movement are the same).
This is potential energy lost by the gravitational field, so the field now has $W = -\frac{GMm}{x}$ stored in it. This energy is often regarded as part of the energy of the satellite
The energy has been converted to kinetic energy, so the total is still zero. .