Understanding Gravitational Field Intensity

Gravitational field intensity describes the strength of the gravitational field at a point in space due to a mass. It quantifies the force that a unit mass would experience when placed at that point. This is a fundamental concept in the study of gravitation and is essential for analyzing the motion and interaction of masses under gravitational influence.

Fundamentals of Gravitation and Gravity

Gravity is a fundamental force that causes objects with mass or energy to attract each other. Its effects, while universally present, become significant on large scales such as planetary or stellar interactions. On microscopic or subatomic scales, gravity is negligible compared to other fundamental forces.

The classical explanation of gravity was provided by Newton’s law of universal gravitation. It states that every pair of masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. For more foundational details, refer to Gravitation Overview.

Definition of Gravitational Field Intensity

Gravitational field intensity at a point is defined as the gravitational force experienced per unit mass placed at that point. It is a vector quantity, always directed towards the source mass.

If $F$ is the gravitational force acting on a test mass $m$ at a point, the gravitational field intensity $E_g$ is given by:

$E_g = \dfrac{F}{m}$

The SI unit of gravitational field intensity is newton per kilogram (N/kg).

Gravitational Field Intensity due to a Point Mass

Consider a point mass $M$ placed at the origin. The field intensity at a point located at distance $r$ from the mass is:

$E_g = -\dfrac{GM}{r^2}\,\hat{r}$

Here, $G$ is the universal gravitational constant, and $\hat{r}$ is a unit vector directed radially outward from the mass. The negative sign indicates that the field is directed towards the mass.

Superposition Principle for Gravitational Field Intensity

The net gravitational field intensity at a point due to several masses is the vector sum of the field intensities due to individual masses. If $E_1, E_2, \ldots, E_n$ are the intensities due to $n$ point masses at a given point, the total intensity $E$ is:

$E = E_1 + E_2 + \ldots + E_n$

This linear superposition allows calculation of net gravitational effects from complex mass distributions.

Gravitational Field Intensity Due to Extended Mass Distributions

For continuous mass distributions, the field intensity at a point is determined by integrating contributions from infinitesimal mass elements over the entire distribution. This approach is required for objects like rings, discs, and spheres.

Gravitational Field Intensity Due to a Solid Sphere

Outside a uniform solid sphere of mass $M$ and radius $R$, the gravitational field intensity at a distance $r$ from the center ($r \geq R$) is identical to that of a point mass $M$ at the center:

$E_g = -\dfrac{GM}{r^2}$

Inside the sphere ($r < R$), the intensity is given by:

$E_g = -\dfrac{GM r}{R^3}$

This shows that within the sphere, field intensity increases linearly with distance from the center.

Gravitational Field Intensity Due to Other Bodies

For a thin ring of total mass $M$ and radius $a$, the gravitational field intensity at a point located on its axis at distance $x$ from the center is:

$E_g = -\dfrac{GM x}{(a^2 + x^2)^{3/2}}$

For a thin disc of mass $M$ and radius $R$, the expression at a point on the axis perpendicular to the center can be derived using integration.

Dimensional Formula and Units

The dimensional formula of gravitational field intensity is $[M^0 L^1 T^{-2}]$, which is identical to that of acceleration.

Its SI unit is newton per kilogram (N/kg). This is equivalent to metre per second squared ($\mathrm{m/s}^2$).

Characteristics of Gravitational Field Intensity

• It is always directed towards the source mass.
• It decreases with the square of the distance.
• It is a vector quantity.
• It can be superposed for multiple sources.

Relation to Gravitational Potential

Gravitational field intensity is related to gravitational potential by the negative gradient. The field at a point is the rate of change of potential with respect to distance.

For more information on potential energy, visit Gravitational Potential Energy Explained.

Applications of Gravitational Field Intensity

Understanding gravitational field intensity is central to satellite motion, planetary orbits, and artificial gravity design in rotating frames. It is also essential in analyzing objects suspended or floating in different media. Study more about such applications at Conditions for Objects to Float.

Precise knowledge of the Earth's gravitational field assists in geophysical studies, satellite trajectory determination, oceanography, and measurements involving GPS.

Examples of Gravitational Field Intensity Calculations

• At the Earth’s surface, $E_g \approx 9.8$ N/kg downward.
• For the Sun–Earth system, field intensity explains orbital motion.
• Inside a uniform sphere, intensity is highest at the surface, zero at centre.

Frequently Asked Questions on Gravitational Field Intensity

The direction of gravitational field intensity is always towards the mass creating the field. The field lines are closer where intensity is stronger. On the Earth's surface, the field can often be considered uniform over small regions.

The field produced by rings, discs, or other bodies can be derived using principles such as superposition and integration. In all cases, the field at a point indicates force per unit mass at that location.

Mass involved in gravitational field intensity refers to gravitational mass, which is experimentally found to be equivalent to inertial mass for all practical scenarios.

For practice questions and advanced understanding, try the Gravitation Practice Paper.

Explorations of advanced theories, such as the connection between gravitational field and spacetime curvature, can be linked to the fundamental contributions of Galileo and Newton. For more, see Galileo and Newton on Gravity.

The field intensity at different positions (such as inside, at the surface, or outside a sphere) provides critical information for various physics and engineering applications. Its measurements and models support domains including satellite navigation, geophysics, and oceanography.

The relationship between acceleration due to gravity and gravitational field intensity is direct, as $g$ at any location is numerically equal to the field intensity. Learn more at Acceleration Due to Gravity.