Problem 723 - JEE Physics

Question

What is the intensity of gravitational field at the center of spherical shell (A) \(\left(\mathrm{Gm} / \mathrm{r}^{2}\right)\) (B) \(\mathrm{g}\) (C) zero (D) None of these

Step-by-Step Solution

Answer

The intensity of the gravitational field at the center of a spherical shell is zero, as the gravitational field created by each mass element of the shell cancels out due to symmetry. Therefore, the correct answer is (C) zero.

Step 1: Find Gravitational Field Intensity Formula

We need to recall the formula for the intensity of the gravitational field, which is given by: \[g = \frac{Gm}{r^2}\] Here, \(g\) is the intensity of the gravitational field, \(G\) is the gravitational constant, \(m\) is the mass of the object, and \(r\) is the distance between the center of the mass and the point where we want to find the gravitational field intensity.

Step 2: Apply the Principle of Superposition

Due to the uniform mass distribution and symmetry of the spherical shell, we can determine the gravitational field at its center by calculating the gravitational field created by each mass element of the shell and summing them up. By symmetry, we know that the gravitational field created by opposite mass elements will cancel each other out. This means that at the center of the spherical shell, the net gravitational field intensity will be zero.

Step 3: Conclusion

By applying the principle of superposition and considering the symmetry of the problem, we found out that the intensity of the gravitational field at the center of a spherical shell is zero. Therefore, the correct answer is (C) zero.

Key Concepts

Spherical Shell

In physics, a spherical shell is a hollow object with a spherical shape, often used in theoretical problems to simplify complex calculations. Imagine a soccer ball with no interior, just its outer surface. This hollow nature is crucial in understanding gravitational fields, because it allows the mass to be distributed evenly over the surface. A spherical shell is used in many physics problems because its symmetry simplifies calculations. Plus, it often helps to idealize real-world objects by approximating them as spherical shells, making the mathematics more manageable. In the context of gravitational fields, this concept plays a positive role since the forces can be considered to act from the center of mass of the shell, making it easier to calculate their impact.

Superposition Principle

The superposition principle is a key concept in physics, applying to various fields like electrodynamics and acoustics, but also incredibly useful in gravitational field calculations. This principle states that the net effect of multiple forces acting at a point is the vector sum of all individual forces. In terms of gravitational fields, the superposition principle allows us to calculate the total gravitational force as the sum of the fields generated by individual mass elements. This is particularly useful for complex objects like a spherical shell, as it enables us to break down the calculation into smaller, more manageable parts. Each small part of the spherical shell exerts a gravitational force, but when we add up all these tiny forces at the center, they cancel each other out due to symmetry, leading to a net gravitational field intensity of zero.

Gravitational Field Intensity

Gravitational field intensity is a measure of the gravitational force exerted per unit mass at a specific point in space. It's represented by the symbol \( g \), and its formula \( g = \frac{Gm}{r^2} \) encapsulates several important parameters:

• \( G \) is the universal gravitational constant, expressing the force of attraction between two bodies.
• \( m \) represents the mass exerting the gravitational force.
• \( r \) is the distance between the mass's center and the point where the force is being calculated.

Gravitational field intensity tells us how strong the gravitational field is at a particular location. In the case of a spherical shell, while there is mass present, the distribution of mass causes inner points like the center to have a zero gravitational field intensity. This is largely due to the symmetric nature of the spherical shell.

Symmetry

Symmetry is a fascinating concept in physics and plays a crucial role in simplifying complex problems, such as those involving gravitational fields. It involves identifying patterns that repeat themselves in different orientations, allowing for simplifications due to predictable outcomes. For a spherical shell, symmetry implies that the mass distribution is uniform everywhere on the shell's surface. This uniformity is incredibly beneficial when calculating gravitational fields because it leads to forces from opposite sides of the shell canceling each other out. Thus, at the center of a perfectly symmetrical spherical shell, the net gravitational field is zero. This is a direct consequence of how balanced and uniformly distributed the mass is all around the spherical shape.