Introduction
Gravitational field intensity is a crucial concept in physics, particularly in gravitational studies. It helps in understanding how a mass distribution influences the gravitational force at a given point in space. In this article, we explore the gravitational field intensity due to a uniform solid sphere and its implications in competitive exams such as JEE, NEET, and CBSE Class 11 Physics.
Gravitational Field Intensity due to Uniform Solid Sphere
Consider a uniform solid sphere of radius ‘R’ and mass ‘M’. Let us find out the value of gravitational field intensity in all these 3 regions:
- Inside the solid sphere.
- On the surface of a solid sphere.
- Outside the solid sphere.
Gravitational Field Intensity Outside the Solid Sphere (r > R)
For a point outside the solid sphere, the entire mass of the sphere can be assumed to be concentrated at its center. To find the gravitational field intensity at a point ‘P’, which is at a distance ‘r’ from the centre of outside the solid sphere, consider an imaginary sphere about ‘P’, which encloses the entire mass ‘M’.
∴ E = – GM/r2
⇒ E ∝ -1/r2
Gravitational Field Intensity On the Surface of a Solid Sphere (r = R)
To find the gravitational field intensity at a point ‘P’ situated on the surface of the solid sphere,
Distance to the point on the surface is r = R.
Then,
E = -GM/R2
⇒ E = g = Constant
Gravitational Field Intensity Inside the Solid Sphere (r < R)
For a point inside the sphere at a distance from the center, only the mass enclosed within radius contributes to the gravitational field.
To find the gravitational influence at a point ‘P’ situated inside the uniform solid sphere at a distance ‘r’ from the centre of the sphere. If we draw an imaginary sphere about this point, the mass present within this imaginary sphere is given by ‘m’.
For a volume of (4/3) πR3, the mass present is M; for a volume of (4/3) πr3, the mass present is ‘m’.
As the density of the solid sphere remains constant throughout,
m = M × (r3/R3)
Then, the gravitational field intensity at point ‘P’ inside the solid sphere at a distance ‘r’ from the centre of the sphere is given by,
E = -Gm/r2
Where m is the source mass present within the imaginary sphere drawn about point ‘P’. By substituting the value of m in the above equation, we get
E = -GMr/R3
⇒ E ∝ -r
This shows that inside a uniform solid sphere, the gravitational field intensity varies linearly with the distance from the center.
| The Position of Point ‘P’ | Gravitational Field Intensity |
| Inside the uniform solid sphere (r < R) E = -GMr/R3 | |
| On the surface of the uniform solid sphere (r= R) | E = -GM/R2 |
| Outside the uniform solid sphere (r>R) | E = -GM/r2 |
FAQs (Frequently Asked Questions)
Q1: Why is gravitational field intensity maximum at the surface of a solid sphere?
A: Inside the sphere, the field increases with r , but beyond the surface, it decreases with E ∝ 1/r2. Thus, it attains its maximum at r = R.
Q2: How does the gravitational field intensity behave at the center of a solid sphere?
A: At the center (r = 0), the gravitational field intensity is zero because the mass is symmetrically distributed around it, leading to net cancellation.
Q3: What is the difference between gravitational field intensity and gravitational potential?
A: Gravitational field intensity is a vector quantity representing force per unit mass, whereas gravitational potential is a scalar quantity representing the work done to bring a unit mass from infinity to a point.
Multiple-Choice Questions (MCQs)
Q1: The gravitational field intensity inside a solid sphere is proportional to
A) 1/r
B) r
C) 1/r²
D) Constant
Answer: B) r
Explanation: The gravitational field intensity inside a solid sphere is given by E = -GMr/R3, indicating a direct proportionality to r.
Q2: What is the gravitational field intensity at the center of a uniform solid sphere?
A) Maximum
B) Zero
C) Same as at the surface
D) Infinity
Answer: B) Zero
Explanation: At the center, equal and opposite gravitational forces cancel each other, resulting in zero field intensity.
Q3: Outside a uniform solid sphere, the gravitational field behaves as if the entire mass is concentrated at:
A) The surface
B) The center
C) A point at a distance R from the center
D) It varies randomly
Answer: B) The center
Explanation: According to Newton’s Shell Theorem, a uniform solid sphere acts as a point mass for points outside it.
Test Your Knowledge
Gravitational Field Intensity Due to Solid Sphere Quiz
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