Considering only the gravitational field of the Earth itself and neglecting the influence of other celestial bodies, at which location is the gravitational field strength due to the Earth effectively zero?

Introduction / Context:
This conceptual question deals with the gravitational field inside and around the Earth. In school physics, we often talk about the acceleration due to gravity g at the Earth surface, but g varies with distance from the centre of the Earth. Understanding where the gravitational field becomes zero helps build a correct mental picture of gravity inside a spherically symmetric body like the Earth.

Given Data / Assumptions:

• The Earth is assumed to be a spherically symmetric body of uniform or smoothly varying density.
• Only the gravitational field due to the Earth is considered; other bodies like the Moon and Sun are neglected.
• We are asked where the gravitational field strength due to Earth becomes zero.
• Options describe the centre, the surface, a finite height and infinite distance.

Concept / Approach:
According to the shell theorem in Newtonian gravitation, for a spherically symmetric mass distribution, the gravitational field inside a spherical shell of matter is zero. For a full sphere like the Earth, the gravitational field at a point inside the Earth depends only on the mass enclosed within a sphere of radius r, where r is the distance from the centre. As you move towards the centre, the effective mass that contributes to gravity decreases with r^3, while the distance r also decreases. The result is that the gravitational field inside a uniform sphere is proportional to r and goes to zero at r equal to zero, that is, at the centre. At the surface, g has its usual value of about 9.8 m/s^2. At greater distances, g decreases but only approaches zero as distance tends to infinity, never becoming exactly zero at a finite radius. Therefore, the gravitational field is exactly zero only at the centre of the Earth under these ideal conditions.

Step-by-Step Solution:
Step 1: Recall the shell theorem which states that a uniform spherical shell of matter exerts no net gravitational force on a particle located inside it. Step 2: For a point inside the Earth, think of the mass of the Earth as made up of a sphere of radius r and outer shells. Step 3: The outer shells contribute zero net gravitational field at that interior point due to the shell theorem. Step 4: Only the mass enclosed within radius r contributes to the gravitational field at that point. Step 5: For a uniform density Earth, enclosed mass is proportional to r^3, and gravitational field inside is proportional to r, which becomes zero when r equals zero. Step 6: At the very centre of the Earth, contributions from all directions cancel perfectly, resulting in zero net gravitational field. Step 7: Thus, the gravitational field strength due to the Earth is exactly zero only at its centre.

Verification / Alternative check:
Textbook graphs of gravitational field strength versus distance from the centre of the Earth show that g increases from zero at the centre to a maximum near the surface and then decreases with distance outside the Earth. While g decreases towards zero as distance tends to infinity, it never reaches exactly zero at any finite distance. Inside the Earth, however, the linear relationship between g and r leads to g equal to zero at the centre. This behaviour is a standard result in Newtonian gravitation and confirms that the centre is the only location of zero gravitational field due to the Earth alone.

Why Other Options Are Wrong:
At the Earth surface, the gravitational field is close to 9.8 m/s^2, not zero, and it is what we normally experience as weight. At a height equal to one Earth radius above the surface (twice the Earth radius from the centre), the gravitational field is reduced to one quarter of its surface value but is still not zero. At an infinite distance from the Earth, the gravitational field approaches zero but does not reach exactly zero at any finite distance; the question refers to an exact zero value.

Common Pitfalls:
Students sometimes think that gravity becomes zero at very high altitudes or at some specific distance like one Earth radius above the surface. Another confusion arises from mixing the ideas of near zero and exactly zero. In pure Newtonian theory, the only point where the Earth gravitational field is exactly zero is at the centre of the Earth. Remembering the shell theorem and the idea that contributions from all directions cancel at the centre helps avoid this confusion.

Final Answer:
Under the stated assumptions, the gravitational field due to the Earth is zero at the centre of the Earth.