zero curl
The curl of E is given by
| x^ ŷ ẑ |
| |
curl E = | ∂/∂x ∂/∂y ∂/∂z |
| |
| 2xy²+z³ 2x²y 3xz² |
= x^(0−0) + ŷ(3z²−3z²) + ẑ(4xy−4xy) = 0. (12.136)
634 Solutions to the problems
To find the associated potential φ, we could evaluate the (negative) line integral of E from a given reference point to a general point (x,y,z). But an easier method is simply to find a function φ that satisfies E = −$\nabla\varphi$.
Looking at the x component of this relation, we see that we need 2xy² + z³ = −$\partial\varphi/\partialx.Hence\varphi$ must take the form
φ = −x²y² − xz³ + f(y,z). (12.137)
The arbitrary function f(y,z) won't ruin the 2xy² + z³ = −$\partial\varphi$/$\partial$x equality because $\partial$f(y,z)/$\partial$x = 0. Similarly, the y component of E = −$\nabla\varphiyields2x2y=−\partial\varphi$/$\partial$y, so $\varphi$ must take the form
$\varphi$ = −x²y² + f(x,z). (12.138)
And the z component yields 3xz² = −$\partial\varphi/\partialz,so\varphi$ must take the form
φ = −xz³ + f(x,y). (12.139)
You can quickly check that the only function consistent with all three of these forms is φ = −x²y² − xz³ + C, where C is an arbitrary constant.
If the curl of E weren't zero, then there would exist no function consistent with the three required forms.