Question

For it must be confessed that the exhaustion of potential energy from a universal medium is a very unintelligible and mysterious matter. When matter is infinitely widely separated, and the forces are least, the potential energy is at its greatest, and when the potential energy is most exhausted, the forces are most energetic!

But I thought that the force only depends on the derivative of the potential at that position and not its actual value i.e. F=−dU/dx
. So why is Heaviside making this point?

I guess the first sentence is referencing the ether, but I still dont understand why Heaviside is confused. What physical intuition led him to the idea that the relationship between gravitational force and potential energy is somehow 'backwards'?

Answer 1

Heaviside's point relates to the relationship between gravitational force and potential energy. Despite the fact that the force is determined by the derivative of the potential, the forces actually become more energetic as the potential energy decreases.

Step-by-step explanation:

Heaviside's point is related to the relationship between gravitational force and potential energy. In general, the force is determined by the derivative of the potential at a given position, as you mentioned (F = -dU/dx). However, Heaviside was highlighting the fact that when the potential energy is most exhausted (meaning it is at its lowest value), the forces are actually most energetic. This seems counterintuitive because we might expect that as the potential energy decreases, the forces would also decrease. But in reality, the forces become stronger as the potential energy decreases.

To understand why this is the case, consider the example of two masses separated by a distance. When the masses are infinitely far apart, the potential energy is at its greatest and the forces between them are least. However, as the masses get closer, the potential energy decreases and the forces increase. This is because as the masses move closer together, positive work must be done against the force of gravity, causing the potential energy to decrease. The forces become more energetic in order to accelerate the masses towards each other.

So, Heaviside's point was to highlight this seemingly contradictory relationship between potential energy and forces. Despite the fact that the force is determined by the derivative of the potential, the forces actually become more energetic as the potential energy decreases.