Question

Both the Coulomb and gravitational forces lead to potential energies of the form \(U=\gamma / | \mathbf{r}_{1}-\) \(\mathbf{r}_{2} |,\) where \(\gamma\) denotes \(k q_{1} q_{2}\) in the case of the Coulomb force and \(-G m_{1} m_{2}\) for gravity, and \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) are the positions of the two particles. Show in detail that \(-\nabla_{1} U\) is the force on particle 1 and \(-\nabla_{2} U\) that on particle 2.

Short answer

The negative gradient of potential energy gives the forces on particles: \(-\nabla_1 U\) is force on particle 1 and \(-\nabla_2 U\) is force on particle 2.

Step-by-step solution

Understanding the Gradient

The gradient, denoted as \(abla\), is a vector operator used to find the rate and direction of change in scalar fields. For the potential energy \(U\), \(abla U\) is used to find the force associated with the potential energy for any position \(\mathbf{r}\).

Expressing the Potential Energy

Given the potential energy \(U = \frac{\gamma}{| \mathbf{r}_{1} - \mathbf{r}_{2} |}\), where \(\gamma\) could be \(kq_1q_2\) for Coulomb's force or \(-Gm_1m_2\) for gravitational force. Here, \(| \mathbf{r}_{1} - \mathbf{r}_{2} |\) is the distance between the two particles.

Finding the Gradients \(-\nabla_1 U\) and \(-\nabla_2 U\)

The gradient with respect to \(\mathbf{r}_1\) is given by\[ -abla_1 U = -abla_1 \left( \frac{\gamma}{| \mathbf{r}_1 - \mathbf{r}_2 |} \right) \]where \(abla_1\) represents the gradient with respect to \(\mathbf{r}_1\). Applying the chain rule and noting \(\mathbf{r}_1 - \mathbf{r}_2\) gives a force magnitude, modifying the direction to point from \(\mathbf{r}_1\) to \(\mathbf{r}_2\). Use similar logic to find \(-abla_2 U\).

Calculating \(\nabla_1 U\)

By finding the derivative of \(\frac{\gamma}{| \mathbf{r}_1 - \mathbf{r}_2 |}\) with respect to \(\mathbf{r}_1\), we obtain:\[ abla_1 U = \frac{-\gamma \cdot (\mathbf{r}_1 - \mathbf{r}_2)}{| \mathbf{r}_1 - \mathbf{r}_2 |^3} \]Multiplying by -1 gives the force on particle 1:\[ -abla_1 U = \frac{\gamma \cdot (\mathbf{r}_1 - \mathbf{r}_2)}{| \mathbf{r}_1 - \mathbf{r}_2 |^3} \]

Calculating \(\nabla_2 U\)

Similarly, differentiate \(\frac{\gamma}{| \mathbf{r}_1 - \mathbf{r}_2 |}\) with respect to \(\mathbf{r}_2\):\[ abla_2 U = \frac{\gamma \cdot (\mathbf{r}_2 - \mathbf{r}_1)}{| \mathbf{r}_1 - \mathbf{r}_2 |^3} \]Thus, the force on particle 2 is:\[ -abla_2 U = \frac{-\gamma \cdot (\mathbf{r}_2 - \mathbf{r}_1)}{| \mathbf{r}_1 - \mathbf{r}_2 |^3} \]

Relating Force to Potential

According to classical mechanics, force \(\mathbf{F}\) is related to potential energy \(U\) by the negative gradient, such that \(-abla U = \mathbf{F}\). This implies that the forces obtained from \(-abla_1 U\) and \(-abla_2 U\) correspond to that on particles 1 and 2, respectively.

Key concepts

Coulomb Force

The Coulomb force is one of the fundamental interactions in nature and governs the electric force between charged particles. It is described by Coulomb's law, which is mathematically expressed as:

• \( F = k \frac{q_1 q_2}{r^2} \)

Here, \( F \) is the magnitude of the force, \( q_1 \) and \( q_2 \) are the charges of the interacting particles, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant.
The direction of the Coulomb force depends on the nature of the charges. If both charges are similar (either both positive or both negative), the force is repulsive, pushing the particles apart. Conversely, if one charge is positive and the other negative, the force is attractive, pulling them together.
Potential energy associated with Coulomb's force is defined as:

• \( U = \frac{k q_1 q_2}{|\mathbf{r}_1 - \mathbf{r}_2|} \)

This expression is crucial when calculating the force vector using the gradient, as demonstrated in the original problem. Understanding this helps in predicting how charged particles interact in different conditions.

Gravitational Force

Gravitational force is the attractive force acting between any two masses. According to Newton's law of universal gravitation, the gravitational force \( F \) can be defined as:

• \( F = G \frac{m_1 m_2}{r^2} \)

Where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the separation distance between the centers of the two masses.
Despite being one of the weakest forces when intrinsic to a single pair of subatomic particles, gravity becomes significant on a large scale, especially in the context of planetary and cosmic interactions. This force is always attractive, bringing masses closer together.
The gravitational potential energy between two point masses is given by:

• \( U = -\frac{G m_1 m_2}{|\mathbf{r}_1 - \mathbf{r}_2|} \)

The negative sign indicates that work would need to be done against the gravitational attraction if the masses were to be separated. The calculation of gradient forces connected to this potential energy allows us to relate to the dynamics of orbits and other gravitational phenomena.

Gradient in Vector Calculus

The gradient is a crucial concept in vector calculus, representing the rate and direction of change in a scalar field. When applied to potential energy functions like those from Coulomb or gravitational forces, the gradient becomes a powerful tool understanding field dynamics.

• The notation for the gradient is \( abla \).
• It transforms a scalar function into a vector.
• The vector indicates the direction of the steepest ascent of the function, while the magnitude represents the rate of increase.

In the context of the original exercise, calculating gradients \( -abla_1 U \) and \( -abla_2 U \) points us to the direction and magnitude of the force exerted on particles due to their positions. Calculating these involves understanding how the position vector changes the potential field, allowing forces to be calculated as gradients of potential energy.
Using the gradient, we can identify the field forces acting on particles and systems, essential for problems in classical mechanics, especially those involving fields and potential energies, such as electrostatic and gravitational fields.