Electric Field As Derivative of Potential Video Lessons

Concept

# Problem: What is the magnitude of the electric field at the point(−1.00î −2.00ĵ +4.00k̂ )mif the electric potential in the region is given byV=2.00xyz2, where V is in volts and coordinates x, y, and z are in meters?

###### FREE Expert Solution

In this problem, we have an electric field and electric potential. Thus, we know that that we are dealing with the relationship between the two. The electric field is equal to the derivative of the potential with respect to a given direction.

For a 3D electric field vector:

$\overline{){\mathbf{E}}{\mathbf{=}}{\mathbf{-}}{\mathbf{\nabla }}{\mathbit{V}}}$

Where,

$\overline{){\mathbf{\nabla }}{\mathbf{=}}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{x}}\stackrel{\mathbf{^}}{\mathbf{i}}{\mathbf{+}}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{y}}\stackrel{\mathbf{^}}{\mathbf{j}}{\mathbf{+}}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{z}}\stackrel{\mathbf{^}}{\mathbf{k}}}$

Power rule of derivation:

$\overline{)\frac{\mathbit{d}}{\mathbit{d}\mathbit{t}}\mathbf{\left(}{\mathbit{x}}^{\mathbit{n}}\mathbf{\right)}{\mathbf{=}}{\mathbit{n}}{{\mathbit{x}}}^{\mathbit{n}\mathbf{-}\mathbf{1}}}$

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###### Problem Details

What is the magnitude of the electric field at the point

(−1.00î −2.00ĵ +4.00k̂ )m

if the electric potential in the region is given by

V=2.00xyz2

, where V is in volts and coordinates x, y, and z are in meters?