Section 14–5 Potentials and fields

(Potential = (−field).(ds) / F_x = −∂U/∂x / Field = grad potential)

In this section, Feynman discusses the integral of (−field).(ds), partial derivative of force, and gradient of potential.

1. Potential = (−field).(ds):

“Since the potential energy, the integral of (−force).(ds) can be written as m times the integral of (−field).(ds), a mere change of scale… (Feynman et al., 1963, section 14–5 Potentials and fields).”

The potential energy function U(x, y, z) is equal to the integral of (−force).(ds) and it can be written as m times the integral of (−field).(ds). (When the mass of an object is equal to 1 kg, the force is numerically equal to the field and the potential energy is numerically equal to the potential.) Strictly speaking, whether the work done is positive or negative is dependent on the convention chosen. One explanation is that an external force on an object within a gravitational field results in positive work and store “more” potential energy. On the other hand, the work done by the gravitational field may release the stored energy and thus, the potential energy is reduced through the negative work. In other words, the work done by an internal (gravitational) force results in a decrease in the potential energy of an object-Earth system.

In the last chapter, Feynman has already explained that there is no work done in moving an object from one place to another where the potential energy is constant because there is no (conservative) force. In this section, he suggests the convenience of giving a scalar function Ψ instead of writing three complicated components of a vector function C. Some may criticize Feynman for the use of the symbol C instead of g, which may refer to gravitational field strength or acceleration vector that is due to gravity. However, one may guess the symbol C is chosen to represent a conservative field. More important, a definition of conservative field could be provided and specifically, it may refer to a gravitational field or electrostatic field.

2. F_x = −∂U/∂x:

“Therefore, we find that the force in the x-direction is minus the partial derivative of U with respect to x: F_x = −∂U/∂x (Feynman et al., 1963, section 14–5 Potentials and fields).”

In general, the derivative of a function U(x, y, z) with respect to x can be written as dU/dx if we ignore other variables such as y and z. Feynman explains that we can write ∂U/∂x, or include a line beside it with a subscript yz at the bottom (∂U/∂x|_yz), which means “partial derivative of U with respect to x, while keeping y and z constant.” In addition, he mentions that the mathematicians have invented a new symbol to remind us to be very careful when we are differentiating such a function, by considering only x varies, whereas y and z do not vary. Historically, in Memoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations, Adrien Marie Legendre (1786) first used the “curly d” in the form of ¶u/¶x.

Partial derivatives differ from ordinary derivatives in five ways (Roundy et al., 2015): (1) Symbolic: we need to specify varying quantities and “fixed” quantities for a ratio of small changes. (2) Graphical: a partial derivative becomes the slope of the tangent plane (or tangent space for higher dimensions) in a given direction at a specific point. (3) Algebraic procedure: the steps to obtain a partial derivative are identical to an ordinary derivative if the variables are independent of each other. (4) Verbal: we must mention the independent variables, e.g., the ratio of change of volume as pressure is changed and temperature is held constant. (5) Experimental: The representation of derivatives in experiments must include measurable (dependent) variable, manipulated (independent) variables and controlled (constant) variables.

3. Field = grad potential:

“The x-component of this “grad” is ∂/∂x the y-component is ∂/∂y, and the z-component is ∂/∂z, and then we have the fun of writing our formulas this way… (Feynman et al., 1963, section 14–5 Potentials and fields).”

According to Feynman, the mathematicians have invented a glorious new symbol, ∇, called “grad,” which is an operator that makes a vector from a scalar function. In short, we can write ∇U instead of ∂U/∂x i + ∂U/∂y j + ∂U/∂z k. The symbol ∇ (pronounced as “del”) is not a specific operator, but rather a convenient mathematical notation that simplifies many equations. (It has a different meaning when it is written as ∇².) Historically, William Rowan Hamilton (1853) invented the ∇ symbol (or “nabla”) in his Lectures on Quaternions. Writing the symbol ∇ together with a scalar function U indicate a directional derivative that always points in the direction of greatest increase of U, and it has a magnitude equal to the maximum rate of increase at the point.

Feynman ends the section by saying it is easy to show that the force on a particle due to magnetic fields is always at right angles to its velocity. He concludes that no work is done by the magnetic field on a moving charge because the motion is at right angles to the force. However, this does not mean that the magnetic field is always a conservative field. In general, if a magnetic field is changing, then it generates an electric field according to Faraday’s law of electromagnetic induction. Notably, the induced electric field can form closed paths that are non-conservative. As a result, some energy can be dissipated as internal energy in resistive materials or radiated as light energy (electromagnetic waves).

Questions for discussion:

1. How would you explain the gravitational potential energy is equal to the integral of (−force).(ds)?

2. Why does the force in the x-direction can be more precisely written as the negative partial derivative of U with respect to x: F_x = −∂U/∂x instead of F_x = −dU/dx?

3. Are magnetic forces always conservative?

The moral of the lesson: the potential energy and the integral of (−force).(ds) can be written as m times the integral of (−field).(ds), or equivalently, force is the (negative) rate of change of potential energy and field is the negative rate of change of potential.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith, Dublin.

3. Legendre, A. M. (1786). Mémoire sur la manière de distinguer les maxima des minima dans le Calcul des Variations. Mémoires de l’Acad. roy. des Sciences, 1788, 7–37.

4. Roundy, D., Weber, E., Dray, T., Bajracharya, R. R., Dorko, A., Smith, E. M., & Manogue, C. A. (2015). Experts’ understanding of partial derivatives using the partial derivative machine. Physical Review Special Topics-Physics Education Research, 11(2), 020126.