Calculating the Field from the Potential — Question 1

Question 1

A polynomial potential produces a field with coupled components

A local potential is $V(x,y,z)=Ax^2y-Bz+C$, with constants A,B,C carrying the required SI units.

1. Find all electric-field components.

2. At $(a,2a,0)$ find the field magnitude and the force on charge q; identify which constant is unobservable through force.

Question 1 - Solution

A polynomial potential produces a field with coupled components

(a) Method and calculation

$$\frac{\partial V}{\partial x}=2Axy,\quad\frac{\partial V}{\partial y}=Ax^2,\quad\frac{\partial V}{\partial z}=-B.$$$$\mathbf E=-2Axy\hat{\mathbf x}-Ax^2\hat{\mathbf y}+B\hat{\mathbf z}.$$

(b) Interpretation and further calculation

$$\mathbf E_P=-4Aa^2\hat{\mathbf x}-Aa^2\hat{\mathbf y}+B\hat{\mathbf z},\quad|\mathbf E_P|=\sqrt{17A^2a^4+B^2}.$$$$\mathbf F_P=q\mathbf E_P,\quad\nabla C=\mathbf0.$$

Physical check

Take each partial derivative while holding the other coordinates fixed; C changes assigned voltage but not any force.