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.
Gravitational Field Intensity Due to Solid Sphere Quiz
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{ q: “What is the gravitational field intensity outside a solid sphere?”, options: [“Same as a point mass”, “Zero”, “Inversely proportional to radius”, “Depends on volume”], answer: 0, explanation: “Outside the sphere, the field behaves as if all mass were concentrated at the center, following g = Gm/r².” },
{ q: “At what point inside a uniform solid sphere is the gravitational field intensity maximum?”, options: [“At the center”, “At the surface”, “At half the radius”, “Just below the surface”], answer: 1, explanation: “The field intensity inside a solid sphere increases linearly and reaches its maximum at the surface.” },
{ q: “What happens to the gravitational field intensity if the radius of the solid sphere increases while keeping mass constant?”, options: [“Increases”, “Decreases”, “Remains constant”, “Becomes infinite”], answer: 1, explanation: “Increasing radius spreads the mass over a larger volume, reducing field intensity outside the sphere.” },
{ q: “What is the formula for gravitational field intensity inside a uniform solid sphere?”, options: [“g = Gm/r²”, “g = GMr/R³”, “g = Gm/r”, “g = GM/r³”], answer: 1, explanation: “Inside a uniform solid sphere, field intensity is given by g = GMr/R³.” },
{ q: “How does gravitational field intensity behave at the center of a solid sphere?”, options: [“Maximum”, “Zero”, “Equal to surface value”, “Infinity”], answer: 1, explanation: “At the center of a solid sphere, the field intensity is zero because all forces cancel out symmetrically.” },
{ q: “Which law governs the gravitational field inside a solid sphere?”, options: [“Newton’s First Law”, “Gauss’s Law”, “Inverse Square Law”, “Kepler’s Law”], answer: 1, explanation: “Gauss’s Law helps in determining the gravitational field inside a solid sphere.” },
{ q: “What happens to the gravitational field intensity if the mass of the sphere doubles?”, options: [“Doubles”, “Halves”, “Remains the same”, “Becomes zero”], answer: 0, explanation: “Since g = Gm/r² outside the sphere, doubling mass doubles the field intensity.” },
{ q: “How does gravitational field intensity change as we move from the center to the surface of a solid sphere?”, options: [“Increases linearly”, “Decreases”, “Remains constant”, “Becomes zero”], answer: 0, explanation: “Inside a solid sphere, gravitational field intensity increases linearly with distance from the center.” },
{ q: “What is the gravitational field intensity outside a solid sphere of mass M and radius R at a distance r (r > R)?”, options: [“Gm/R²”, “Gm/r²”, “Gm/r³”, “Zero”], answer: 1, explanation: “Outside the sphere, gravitational field follows the inverse square law: g = Gm/r².” }
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Author and Publisher Details
Proprietor: NIRMAL ANAND Educations
Written by: Neeraj Anand
Published by: Anand Technical Publishers, under Anand Classes
