Problem 56 - University Physics with Modern Physics

Question

22.56. Can Electric Forces Alone Give Stable Rquilibrium? In Chapter \(21,\) several examples were given of calculating the force exerted on a point charge by other point charges in its sur- roundings. (a) Consider a positive point charge \(+q .\) Give an example of how you would place two other point charges of your choosing so that the net force on charge \(+q\) will be zero. \((b)\) If the net force on charge \(+q\) is zero, then that charge is in equilibrium. The equilibrium will be stable if, when the charge \(+q\) is displaced slightly in any direction from its position of equilibrium, the net force on the charge pushes it back toward the equilibrium position. For this to be the case, what must the direction of the electric field \(\overrightarrow{\boldsymbol{E}}\) be due to the other charges at points surrounding the equilibrium position of \(+q ?(c)\) Imagine that the charge \(+q\) is moved very far away, and imagine a small Gaussian surface centered on the position where \(+q\) was in equilibrium. By applying Gauss's law to this surface, show that it is impossible to satisfy the condition for stability described in part (b). In other words, a charge \(+q\) cannot be held in stable equilibrium by electrostatic forces alone. This result is known as Earnshaw's theorem. (d) Parts (a)-(c) referred to the equilibrium of a positive point charge \(+q\) . Prove that Earnshaw's theorem also applies to a negative point charge \(-q\) .

Step-by-Step Solution

Answer

Charges cannot be in stable equilibrium with electrostatic forces alone: Earnshaw's theorem.

Step 1: Understanding the Force Balance for Part (a)

For a charge +q to experience no net force, the electric forces due to other charges must cancel out. Let's consider placing two identical negative charges -q at equal distances from +q along a straight line. These forces from -q will have equal magnitude but opposite directions, thus canceling each other out, resulting in a net force of zero.

Step 2: Analyzing Equilibrium Stability for Part (b)

For stability, if +q is displaced slightly, the forces should ideally push it back to its original position. This requires that the electric field due to other charges at neighboring points promotes restoration. However, for purely electric forces, the direction of the electric field lines usually diverges away from positive charges and converges towards negative charges, indicating that small displacements will result in forces moving +q away rather than backward.

Step 3: Applying Gauss’s Law for Part (c)

Place a Gaussian surface around the equilibrium position of +q. According to Gauss's law, the electric field through this surface is zero if the net charge inside is zero. However, to have stability—where small displacements result in a restoring force—the electric field must exert an inward force, which contradicts having zero charge within the Gaussian surface at the equilibrium point. This illustrates why purely electrostatic forces can't create a stable equilibrium for +q per Earnshaw's theorem.

Step 4: Extending the Argument to Negative Charge for Part (d)

Since the electrostatic force phenomenon is merely inverted for -q (e.g., attractive versus repulsive), the same principles apply in inverse form. The field due to other charges would diverge when the -q charge is slightly displaced, proving that it, too, cannot be held in stable equilibrium by electrostatic forces alone, adhering to Earnshaw's theorem.

Key Concepts

Earnshaw's Theorem

Earnshaw's theorem is an important concept in electrostatics that establishes a significant limitation: it states that it is impossible for a system of point charges to maintain a stable equilibrium configuration using electrostatic forces alone. This theorem applies regardless of whether the charges involved are positive or negative.

Earnshaw's theorem becomes evident when we consider the forces acting on a charge placed in a region where these forces should ideally balance in a stable manner. However, due to the nature of electric field lines, which diverge around positive charges and converge around negative charges, any small displacement of the charge tends to increase this imbalance. This effectively causes the charge to move further away rather than returning to the equilibrium position.

This theorem is pivotal because it tells us that other forces, perhaps magnetic or mechanical, are necessary if we wish to create entirely stable systems with charges.

Gaussian Surface

A Gaussian surface is an imaginary closed surface used in the application of Gauss’s law, a fundamental principle that relates the electric flux passing through the surface to the charge enclosed within it.

When applied to electrostatic equilibrium and Earnshaw's theorem, we consider a Gaussian surface around the point where a charge "+q" was in equilibrium. Gauss’s law helps determine the net electric field through this closed surface. If the net charge enclosed is zero, then the net electric field flux through this surface is also zero.

Nonetheless, achieving stability in electrostatic equilibrium would require that the electric field inside the Gaussian surface act such that it pushes any displaced charge back to the equilibrium position. But, applying Gauss’s law reveals that the required electric field configurations for this restoring force cannot coexist with a zero electric flux condition, thus reinforcing Earnshaw's theorem.

Electric Field

The electric field represents the force per unit charge at a point in space, providing a way to visualize how charged objects interact at a distance.

In our scenario of electrostatic equilibrium, the direction and nature of electric fields are crucial. Electric fields generated by positive charges point outward, diverging away from the charge, while those from negative charges point inward, converging.

For a charge "+q" to remain in stable equilibrium, one might assume that the surrounding electric field would help restore it to its initial position upon slight displacement. However, the divergent nature of fields around a positive charge, and the convergent nature around a negative charge, do not support a restoring action. Rather, they tend to enhance any displacement, working against stability.

Stable Equilibrium

In mechanics and physics, a system is in stable equilibrium if, after a small displacement, it tends to return to its original place of balance. This intuitive concept unfortunately does not translate well into electrostatics when considering electric forces alone.

Stable equilibrium for a point charge surrounded by other charges occurs if, after any slight displacement, the electric forces act to restore the charge to its original position. However, because of the matched configuration of electric fields, any small perturbation in position tends to increase the misalignment instead.

This is because electric fields around positive and negative charges tend not to produce the required patterns to encourage such a return, according to Earnshaw's theorem, leaving the charge moving further from its supposed equilibrium.

Electric Forces

Electric forces are the interactions between charged particles; they act at a distance and can be either attractive or repulsive depending on the nature of the charges involved. These forces are dictated by Coulomb's Law, which defines the magnitude based on the product of the charges and the inverse square of their separation.

Considering a point charge in equilibrium, the analysis of electric forces requires that the sum of all forces from surrounding charges equals zero. Although this results in equilibrium, it does not necessarily lead to a stable one.

When trying to achieve stability, electric forces alone cannot provide the necessary conditions for a charge to be restored to its equilibrium position after being disturbed because of how they distribute around charges. Thus, despite their capacity to achieve force balance, they fall short of maintaining stability without external assistance such as magnetic forces or mechanical constraints.