Question

Potential functions Potential functions arise frequently in physics and engineering. A potential function has the property that a field of interest (for example, an electric field, a gravitational field, or a velocity field) is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter \(17 .)\) The gravitational potential associated with two objects of mass \(M\) and \(m\) is \(\varphi=-G M m / r,\) where \(G\) is the gravitational constant. If one of the objects is at the origin and the other object is at \(P(x, y, z),\) then \(r^{2}=x^{2}+y^{2}+z^{2}\) is the square of the distance between the objects. The gravitational field at \(P\) is given by \(\mathbf{F}=-\nabla \varphi,\) where \(\nabla \varphi\) is the gradient in three dimensions. Show that the force has a magnitude \(|\mathbf{F}|=G M m / r^{2}\) Explain why this relationship is called an inverse square law.

Short answer

**Answer:** The gravitational field is found by taking the negative gradient of the gravitational potential function. The relationship between the gravitational field and the gravitational potential function is given by \(\mathbf{F} = -\nabla \varphi\). The magnitude of the force \(|\mathbf{F}|\) in the gravitational field is given by \(\frac{GMm}{r^2}\), which shows that the force is inversely proportional to the square of the distance between the objects (\(r^2\)). This relationship is called an inverse square law because the force is inversely related to the square of the distance between the objects.

Step-by-step solution

Calculate gradient of potential function

To find the gradient of the potential function \(\varphi\) in three dimensions, we have to calculate the partial derivatives of \(\varphi\) with respect to \(x\), \(y\) and \(z\). Since \(r = \sqrt{x^2 + y^2 + z^2}\), we can rewrite \(\varphi\) as \(\varphi = -\frac{GMm}{\sqrt{x^2 + y^2 + z^2}}\). Now, we calculate the partial derivatives: $$ \frac{\partial \varphi}{\partial x} = \frac{GMmx}{(x^2+y^2+z^2)^{3/2}}, \\ \frac{\partial \varphi}{\partial y} = \frac{GMmy}{(x^2+y^2+z^2)^{3/2}}, \\ \frac{\partial \varphi}{\partial z} = \frac{GMmz}{(x^2+y^2+z^2)^{3/2}}. \\ $$

Find the gravitational field

Now, we find the gravitational field \(\mathbf{F}\) by taking the negative of the gradient of potential function \(\varphi\). The gravitational field is a vector with components \((-\frac{\partial \varphi}{\partial x}, -\frac{\partial \varphi}{\partial y}, -\frac{\partial \varphi}{\partial z})\). Therefore, $$ \mathbf{F} = \left(-\frac{GMmx}{(x^2+y^2+z^2)^{3/2}}, -\frac{GMmy}{(x^2+y^2+z^2)^{3/2}}, -\frac{GMmz}{(x^2+y^2+z^2)^{3/2}}\right). \\ $$

Find the magnitude of the force

To find the magnitude of the force \(|\mathbf{F}|\), we calculate the square root of the sum of the squares of the components of the gravitational field: $$ |\mathbf{F}| = \sqrt{\left(-\frac{GMmx}{(x^2+y^2+z^2)^{3/2}}\right)^2 + \left(-\frac{GMmy}{(x^2+y^2+z^2)^{3/2}}\right)^2 + \left(-\frac{GMmz}{(x^2+y^2+z^2)^{3/2}}\right)^2} \\ = \frac{GMm}{(x^2+y^2+z^2)}. \\ $$ Since \((x^2+y^2+z^2) = r^2\), we have: $$ |\mathbf{F}| = \frac{GMm}{r^2}. $$

Explain why this relationship is an inverse square law

The magnitude of the force, \(|\mathbf{F}|\), is given by the formula \(\frac{GMm}{r^2}\). This means the force is inversely proportional to the square of the distance between the objects (\(r^2\)). As the distance between the objects increases, the force decreases as the square of the distance, and as the distance decreases, the force increases as the square of the distance. This relationship is called an inverse square law because the force is inversely related to the square of the distance between the objects.

Key concepts

Gradient of Potential Function

In calculus, the concept of a potential function is intimately connected to fields such as gravitational or electric fields. The potential function, often represented by \( \varphi \), possesses the property where the field of interest is the gradient (or sometimes the negative gradient) of this potential. The gradient essentially measures how much the potential function changes at a particular point in space, providing a vector that points in the direction of the greatest rate of increase of the function.

When calculating the gradient of a potential function in three dimensions, we compute the partial derivatives with respect to each spatial variable—\( x \) for width, \( y \) for depth, and \( z \) for height. For example, the gravitational potential \( \varphi = -\frac{GMm}{r} \) results in the gradient components being calculated as follows: \( \frac{\partial \varphi}{\partial x} \) for the \(x\)-component, \( \frac{\partial \varphi}{\partial y} \) for the \(y\)-component, and \( \frac{\partial \varphi}{\partial z} \) for the \(z\)-component, where \( r \) is the distance from the point of interest to the origin, and \( GMm \) represents the constant multiplicative factors involving the gravitational constant \( G \) and the masses \( M \) and \( m \).

The significance of the gradient in this context lies in its ability to quantify the force experienced by an object in the field, which leads us directly into the concept of a gravitational field.

Gravitational Field Calculus

Continuing from the gradient, a gravitational field can be understood in terms of calculus as well. When we deal with gravitational fields using calculus, we are often considering the influence one mass exerts on another. The field at any point \( P(x, y, z) \) is represented mathematically as \( \mathbf{F} = -abla \varphi \), with \( abla \varphi \) being the gradient of the potential function. Negative sign indicates that the force is attractive and points towards the source mass.

By determining the components of the gravitational field using the gradient, we can articulate the force vector \( \mathbf{F} \) at any given point in space. This vector points directly towards the mass creating the field and its magnitude can be calculated by finding the square root of the sum of the squares of its components, as demonstrated in the step-by-step solution. It is crucial for students to not only follow the computational steps but also to grasp the underlying principles that describe how and why a mass creates a gravitational pull in the space around it.

Inverse Square Law

The inverse square law is a fundamental concept in physics that emerges not only in the context of gravity but also in other forces like electromagnetism. It states that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity.

In the context of gravity, it explains how the magnitude of the force between two masses decreases as the distance between them increases, and vice versa. This relationship is expressed as \( |\mathbf{F}| = \frac{GMm}{r^2} \), where \( r \) is the distance between the centers of the two masses, and \( |\mathbf{F}| \) is the magnitude of the gravitational force between them.

Understanding the inverse square law helps explain a range of natural phenomena, from the illumination of light to the orbits of planets. It's a powerful tool in the sciences because it applies universally across several forces of nature. By learning about the inverse square law, students gain insight into why celestial bodies behave the way they do and can better grasp concepts such as orbits and tides, which are governed by this law.