The potential energy due to gravitational field of earth will be maximum at

The correct answer is Infinite distance.

To understand why the gravitational potential energy is maximum at an infinite distance, let us look at the formula for gravitational potential energy.

The gravitational potential energy ($U$) of a system consisting of the Earth of mass $M$ and an object of mass $m$ separated by a distance $r$ from the center of the Earth is given by:

$U=-\frac{GMm}{r}$

where $G$ is the universal gravitational constant.

Let's analyze this formula:
1. The negative sign indicates that the gravitational force is attractive, and the zero reference point for potential energy is conventionally set at an infinite distance ($r\to \infty$).
2. As the distance $r$ increases, the fraction $\frac{GMm}{r}$ decreases in magnitude (gets closer to $0$).
3. Since there is a negative sign in front of the fraction, a decrease in the magnitude of the fraction means the overall value of $U$ becomes less negative (i.e., it increases).

At an infinite distance ($r=\infty$):

$U=-\frac{GMm}{\infty }=0$

For any finite distance $r$, the potential energy $U$ is negative ($U<0$). Since $0$ is greater than any negative number, the maximum possible value of the gravitational potential energy is $0$, which occurs at an infinite distance.