Problem 835 - JEE Physics

Question

let \(\mathrm{V}\) and \(\mathrm{E}\) denote the gravitational potential and gravitational field at a point. Then the match the following \(\begin{array}{ll}\text { Table }-1 & \text { Table }-2\end{array}\) (A) \(\mathrm{E}=0, \mathrm{~V}=0\) (P) At center of spherical shell (B) \(\mathrm{E} \neq 0, \mathrm{~V}=0\) (Q) At center of solid sphere (C) \(\mathrm{V} \neq 0, \mathrm{E}=0\) (R) at centre of circular ring (D) \(\mathrm{V} \neq 0, \mathrm{E} \neq 0\) (S) At centre of two point masses of equal magnitude (T) None

Step-by-Step Solution

Answer

The correct matches are: (P) At center of spherical shell - (C) V ≠ 0, E = 0 (Q) At center of solid sphere - (C) V ≠ 0, E = 0 (R) at center of circular ring - (C) V ≠ 0, E = 0 (S) At center of two point masses of equal magnitude - (D) V ≠ 0, E ≠ 0

Step 1: Determine the scenario of V and E at the center of a spherical shell

At the center of a spherical shell, the gravitational field (E) is equal to 0, while the gravitational potential (V) is non-zero. Therefore, the correct match for the center of a spherical shell would be (C).

Step 2: Determine the scenario of V and E at the center of a solid sphere

At the center of a solid sphere, the gravitational field (E) is equal to 0, but the gravitational potential (V) is non-zero. Consequently, the correct match for the center of a solid sphere is also (C).

Step 3: Determine the scenario of V and E at the center of a circular ring

At the center of a circular ring, the gravitational field (E) is equal to 0, whereas the gravitational potential (V) is non-zero. Thus, the correct match for the center of a circular ring is once again (C).

Step 4: Determine the scenario of V and E at the center of two point masses of equal magnitude

At the center of two point masses of equal magnitude, the gravitational field (E) is not equal to 0, since the field from each mass is different. The gravitational potential (V) is also non-zero. Therefore, the correct match for this case would be (D).

Step 5: Identify matches that were not used and conclude the matching

From the above steps, we have not used (A) or (B). Additionally, we have not used (P), (Q), (R), (S), or (T). However, since (A), (B), and (C) have been thoroughly discussed in steps 1-4, we can safely conclude that the correct matches are as follows: (P) At center of spherical shell - (C) V ≠ 0, E = 0 (Q) At center of solid sphere - (C) V ≠ 0, E = 0 (R) at center of circular ring - (C) V ≠ 0, E = 0 (S) At center of two point masses of equal magnitude - (D) V ≠ 0, E ≠ 0 Thus, we've matched the appropriate scenarios of V and E with the points in Table-2.

Key Concepts

Spherical Shell

A spherical shell is a three-dimensional object that resembles a hollow ball, like an empty glass ornament. When it comes to gravitational potential ( V ) and gravitational field ( E ), the spherical shell offers some unique properties. If you are at the center of a spherical shell, the gravitational field ( E ) is zero. This occurs because the gravitational forces exerted by each part of the shell cancel each other out.

However, the gravitational potential ( V ) at the center is not zero. This is because gravitational potential is a measure of the work needed to move a unit mass from a point to infinity, and it accumulates through contributions from all the shell's mass, leading to a non-zero value. This unique characteristic makes the spherical shell a fascinating example in gravitational studies.

Solid Sphere

A solid sphere is a perfectly spherical object, with mass distributed throughout its volume. Imagine a solid rubber ball for reference. At the center of a solid sphere, similar to a spherical shell, the gravitational field ( E ) is zero.

This happens because while all the mass elements attract any particle toward the center, their attractions effectively cancel out. However, unlike the field, the gravitational potential ( V ) remains non-zero at the center. This occurs as all parts of the mass contribute to the potential, summing it up to a significant value despite no net force acting at the very center. Understanding these dynamics is crucial as it reveals how mass distribution affects gravitational characteristics.

Circular Ring

A circular ring is a simple structure made by arranging mass in a loop, akin to a hula hoop. At the precise center of this ring, the gravitational field ( E ), surprisingly, turns out to be zero. This is a result of symmetrical mass distribution, where gravitational forces from each side neutralize one another.

Nonetheless, the gravitational potential ( V ) at the center is not zero. Potential is cumulative; thus, the potential energy field reaches a finite value by summing contributions across the whole ring. This perfectly illustrates how geometric and symmetric mass layouts can result in differing gravitational field and potential characteristics.

Point Masses

Point masses are hypothetical objects occupying a very small space, yet possessing mass. When analyzing two point masses of equal magnitude, the gravitational dynamics differ from previous cases. If positioned centrally between them, the gravitational field ( E ) is definitely not zero.

This is because each mass exerts a gravitational force in the opposite direction, leading to a non-cancelling, combined field effect at the midpoint. The gravitational potential ( V ), however, does not cancel out and is non-zero as well. This is because potentials from each mass add algebraically. Point masses offer a unique perspective on gravitational interactions by highlighting how the vector nature of the field influences results.