Gavin Howe
Gauss's law, also known as Gauss's flux theorem, is a fundamental principle in physics, particularly in the study of electromagnetism and electrostatics. This law relates the distribution of electric charge to the resulting electric field. Interestingly, the question arises as to whether this law can be applied to gravitational fields. The answer is yes; Gauss's law can be applied to gravity, and this is known as Gauss's law for gravity or Gauss's flux theorem for gravity. This law is mathematically similar to Gauss's law for electrostatics and is often more convenient to use than Newton's law of universal gravitation. By using Gauss's law for gravity, we can easily derive the gravitational field in certain scenarios where applying Newton's law directly would be more challenging.
| Characteristics | Values |
|---|---|
| Name | Gauss's Law for Gravity |
| Other Names | Gauss's Flux Theorem for Gravity, Gauss's Theorem |
| Named After | Carl Friedrich Gauss |
| Law Equivalent To | Newton's Law of Universal Gravitation |
| Law Similar To | Gauss's Law for Electrostatics |
| Use Case | More convenient to work from than Newton's Law |
| Formula | \(\displaystyle \oint _{\partial V}\mathbf {g} \cdot d\mathbf {A} =\int _{V}\nabla \cdot \mathbf {g} \,dV\) |
| Formula (Divergence Theorem) | \(\displaystyle \int _{V}\nabla \cdot \mathbf {g} \ dV=-4\pi G\int _{V}\rho \ dV\) |
| Formula (Rewritten) | \(\displaystyle \int _{V}(\nabla \cdot \mathbf {g} )\ dV=\int _{V}(-4\pi G\rho )\ dV\) |
| Formula (Integral Form) | \(\displaystyle \oint_{S} \vec g \cdot \vec n dA = -4 \pi r^2 \frac{GM}{r^2} = -4 \pi G \int_V \rho dV\) |
| Formula (Differential Form) | \(\displaystyle \nabla \cdot \mathbf{g} \,=\,4\pi G\rho\) |
| Formula (Cylindrical Symmetry) | \(\displaystyle \oint_{\partial V}\mathbf {E} \cdot d\mathbf {A} =\int _{V}\nabla \cdot \mathbf {E} \,dV\) |
| Formula (Planar Symmetry) | \(\displaystyle \oint_{\partial V}\mathbf {E} \cdot d\mathbf {A} =\int _{V}\nabla \cdot \mathbf {E} \,dV\) |
| Formula (Spherical Symmetry) | \(\displaystyle \oint_{\partial V}\mathbf {E} \cdot d\mathbf {A} =\int _{V}\nabla \cdot \mathbf {E} \,dV\) |
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What You'll Learn
- Gauss's Law for Gravity is a law of physics
- It is equivalent to Newton's law of universal gravitation
- Gauss's Law can be applied to any function where the quantity is inversely proportional to distance squared
- The gravitational field inside a hollow sphere is the same as if the sphere were not there
- The law can be used to derive the gravitational field in cases where Newton's law would be difficult to apply
Gauss's Law for Gravity is a law of physics
Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is indeed a law of physics. Named after Carl Friedrich Gauss, it is equivalent to Newton's law of universal gravitation. Gauss's law states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed.
Gauss's law for gravity is often more convenient to work with than Newton's law. This is because it is mathematically similar to Gauss's law for electrostatics, one of Maxwell's equations, which relates the distribution of electric charge to the resulting electric field. This means that Gauss's law for gravity has the same mathematical relation to Newton's law as Gauss's law for electrostatics has to Coulomb's law. Both Newton's law and Coulomb's law describe inverse-square interaction in a 3-dimensional space.
The integral form of Gauss's law for gravity states that dA is a vector whose magnitude is the area of an infinitesimal piece of the surface ∂V. The direction of dA is the outward-pointing surface normal. M is the total mass enclosed within the surface ∂V. According to the law, the flux of the gravitational field is always negative (or zero) and never positive. This is because mass can only be positive, while charge can be either positive or negative.
Gauss's law for gravity can be applied to any function where the quantity (force) is inversely proportional to distance squared. For example, a hollow sphere does not produce any net gravity inside. The gravitational field inside is the same as if the hollow sphere were not there. This can be derived easily using Gauss's law for gravity, but it would take several pages of calculus to derive directly using Newton's law of gravity.
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It is equivalent to Newton's law of universal gravitation
Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss.
Gauss's law for gravity states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. The gravitational field g (also called gravitational acceleration) is a vector field – a vector at each point of space (and time). Gauss's law for gravity is often more convenient to work with than Newton's law.
Newton's law of universal gravitation describes gravity as a force stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres of mass. In other words, any two bodies are attracted by a force proportional to their mass and inversely proportional to their separation squared. Newton's law of universal gravitation can be expressed as: F (the gravitational force acting between two objects), m1 and m2 (the masses of the objects), r (the distance between the centres of their masses), and G (the gravitational constant).
Gauss's law for gravity can be derived from Newton's law of universal gravitation. Newton's law states that the gravitational field due to a point mass is: M is the mass of the particle, which is assumed to be a point mass located at the origin. Gauss's law can be used to easily derive the gravitational field in certain cases where a direct application of Newton's law would be more difficult but not impossible. For example, for an infinite, flat plate (Bouguer plate) of any finite thickness, the gravitational field outside the plate is perpendicular to the plate, towards it, with magnitude 2πG times the mass per unit area, independent of the distance to the plate.
The two forms of Gauss's law for gravity are mathematically equivalent. The divergence theorem states: {\displaystyle \oint _{\partial V}\mathbf {g} \cdot d\mathbf {A} =\int _{V}\nabla \cdot \mathbf {g} \,dV} where V is a closed region bounded by a simple closed oriented surface ∂V and dV is an infinitesimal piece of the volume V. The gravitational field g must be a continuously differentiable vector field defined on a neighbourhood of V.
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Gauss's Law can be applied to any function where the quantity is inversely proportional to distance squared
Gauss's law, also known as Gauss's flux theorem, is a fundamental principle in physics, specifically in the study of electromagnetism. It relates the distribution of electric charge to the resulting electric field and is one of Maxwell's equations. The law states that the flux (surface integral) of the electric field passing through a closed surface is directly proportional to the electric charge enclosed by that surface, regardless of the distribution of the charge. This law is particularly useful for determining the electric charge within a closed surface, as it provides a mathematical relationship between the electric field and the charge.
Gauss's law is not limited to electromagnetism and can be applied to other areas of physics, including gravity. Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is mathematically similar to its application in electrostatics. It states that the flux of the gravitational field over any closed surface is proportional to the mass enclosed. This law is equivalent to Newton's law of universal gravitation and is often more convenient to work with.
The inverse-square law, a scientific principle, states that the intensity of a physical quantity is inversely proportional to the square of the distance from the source of that quantity. In other words, as the distance from the source increases, the intensity of the quantity decreases proportionally to the square of the distance. This law is applicable in various contexts, including acoustics and optics, and is fundamental in understanding the behaviour of energy and intensity over distances.
Gauss's law can be applied to any function where the quantity follows an inverse-square relationship with distance. This includes physical quantities that radiate outward from a point source in three-dimensional space. For example, the intensity of sound or light follows an inverse-square law, where the intensity decreases as the distance from the source increases. Similarly, in the context of gravity, the gravitational force between two masses is inversely proportional to the square of the distance between them.
In summary, Gauss's law is a versatile mathematical tool that can be applied to any function where the quantity exhibits an inverse-square relationship with distance. This includes situations in electromagnetism, gravity, and other physical phenomena, such as sound and light intensity. By utilising Gauss's law, scientists can more easily derive complex fields and understand the behaviour of physical quantities over distances.
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The gravitational field inside a hollow sphere is the same as if the sphere were not there
Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. Gauss's law for gravity is more convenient to work with than Newton's law in certain cases.
The integral form of Gauss's law for gravity is expressed as:
> dA is a vector, whose magnitude is the area of an infinitesimal piece of the surface ∂V, and whose direction is the outward-pointing surface normal, M is the total mass enclosed within the surface ∂V. The left-hand side of this equation is called the flux of the gravitational field. Note that according to the law, it is always negative (or zero), and never positive.
Gauss's law can be applied to a hollow sphere to determine the gravitational field inside it. The law states that a hollow sphere does not produce any net gravity inside. This means that the gravitational field inside a hollow sphere is the same as if the sphere were not there. In other words, the resultant field includes the masses inside and outside the sphere, excluding the sphere itself.
This principle is also known as the shell theorem, which states that a spherically symmetric shell (a hollow ball) exerts no net gravitational force on any object inside, regardless of the object's location within the shell. This theorem was proved by Isaac Newton and has particular applications in astronomy.
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The law can be used to derive the gravitational field in cases where Newton's law would be difficult to apply
Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. Gauss's law states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. This law is often more convenient to work with than Newton's law, especially in certain cases where applying Newton's law directly would be more challenging.
Gauss's law can be used to derive the gravitational field in cases where Newton's law would be difficult to apply directly. One such example is when dealing with an infinite, flat plate (Bouguer plate) of any finite thickness. By using a "Gaussian pillbox", Gauss's law can determine that the gravitational field outside the plate is perpendicular to the plate, towards it, with a magnitude of 2πG times the mass per unit area, regardless of the distance to the plate. This is in contrast to Newton's cumbersome calculus, which requires several pages of calculations to derive the same result.
Another application of Gauss's law is in the case of a mass distribution with a density that depends on only one Cartesian coordinate, z. In this scenario, Gauss's law simplifies the calculation by stating that gravity at any z-value is 2πG times the difference in mass per unit area on either side of that z value. This approach is more straightforward than applying Newton's law, which would involve complex calculations for each bit of mass in the system.
Additionally, Gauss's law can be used to determine the gravitational field between two parallel, infinite plates with equal mass per unit area. In this case, the law concludes that there is no gravitational field produced between the plates. This conclusion can be reached more easily using Gauss's law than through a direct application of Newton's law, which would require intricate calculations for each mass element.
Overall, Gauss's law for gravity provides a valuable alternative approach to deriving the gravitational field in situations where Newton's law becomes cumbersome or challenging to apply directly. By considering the flux of the gravitational field and its relationship to the enclosed mass, Gauss's law offers a more convenient and efficient method for solving complex gravitational problems.
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Frequently asked questions
Gauss's Law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed.
Gauss's Law can be applied to gravity in cases where a direct application of Newton's law would be more difficult. It can be used to derive the gravitational field, especially in cases where there is symmetry. Gauss's Law for gravity is mathematically similar to Gauss's Law for electrostatics, one of Maxwell's equations.
The integral form of Gauss's Law for gravity states that the flux of the gravitational field over a closed surface is proportional to the mass enclosed within that surface. The law states that the flux is always negative (or zero) because mass can only be positive.
