Simple Nature - Light and Matter

perpendicular to the constant-potential curves.

The one-dimensional relationshipE=−dV/dxgeneralizes to

three dimensions as follows:

Ex=−

dV

dx

Ey=−

dV

dy

Ez=−

dV

dz

This can be notated as a gradient (page 219),

E=−∇V,

and if we know the field and want to find the potential, we can use

a line integral,

∆V =−

∫

C

E·dr,

where the quantity inside the integral is a vector dot product.

self-check C

Imagine that figure a represents potential rather than height. (a) Con-

sider the stream the starts near the center of the map. Determine the

positive and negative signs of dV/dxand dV/dy, and relate these to

the direction of the force that is pushing the current forward against the

resistance of friction. (b) If you wanted to find a lot of electric charge on

this map, where would you look? .Answer, p. 1059

Figure c shows some examples of ways to visualize field and

potential patterns.

594 Chapter 10 Fields