Finding E from V

Section 25.3 Finding E from V

Activity 25.3.1. Warm-Up.

Starting with what you learned about Determining the Electric Potential Difference from the Electric Field: suppose you instead knew the electric potential \(V\) and you want to figure out the electric field \(\vec{E}\text{.}\) What mathematical operation could you perform on \(V\) to get \(\vec{E}\text{?}\)

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Video Lesson 25.3.1. Finding E from V.

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Subsubsection Key Ideas

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Now you have seen how we can go back and forth between \(V\) and \(\vec{E}\text{.}\) To find \(\vec{E}\) from \(V\text{,}\) you can take a derivative!

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Definition 25.3.2. Determining the Electric Field from the Electric Potential.

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The electric field can be directly calculated given a known electric potential using the equation

\begin{equation*} \vec{\mathit{E}} = -\vec{\nabla}\mathit{V} \end{equation*}

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Notation 25.3.3. Cartesian Coordinates.

In Cartesian coordinates, the gradient can be represented in terms of partial derivatives as:

\begin{equation*} \vec{\mathit{E}} = -\vec{\nabla}\mathit{V} = - \frac{\partial V}{\partial x}\hat{x} - \frac{\partial V}{\partial y}\hat{y} - \frac{\partial V}{\partial z}\hat{z} \end{equation*}

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Subsubsection Activities

Activity 25.3.2. Electric Field Estimate.

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Consider the Example Equipotential Graph. Use the equation above to estimate a numerical value for the magnitude of the electric field at position \(\vec{r} \approx (0.5 \mathrm{~m}) \hat{x}\text{.}\)

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