# Video: Component Form of Electric Field from Equipotentials

In a region containing an electric field, equipotential surfaces have the potentials of π = 100 V for π§ = 0.00 m, π = 200 V for π§ = 0.50 m, and π = 300 V for π§ = 1.00 m. What is the electric field in this region?

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### Video Transcript

In a region containing an electric field, equipotential surfaces have the potentials of π equals 100 volts for π§ equals 0.00 meters, π equals 200 volts for π§ equals 0.50 meters, and π equals 300 volts for π§ equals 1.00 meters. What is the electric field in this region?

We can call the electric field that we want to solve for in this region πΈ, where πΈ is a vector having π₯-, π¦-, and π§- components. Weβre told in the problem statement various potentials corresponding to different values of π§. Letβs start by sketching out the π§-values and the corresponding potentials.

Here, is a three-dimensional plot with π₯-, π¦- and π§-dimensions marked out. On the π§-axis, weβre told potential values for three different values of π§: at the origin, at 0.50 meters and at 1.00 meters. At those points, we have equipotential surfaces of 100, 200, and 300 volts, respectively. If we consider this progression as a gradient caused by an electric field in the π§-direction, which we can call πΈ sub π§, we can see from these points that over a distance of one meter, the electric potential increases by 200 volts. So the electric field along this dimension is 200 volts per meter.

Now what about πΈ sub π₯ and πΈ sub π¦? All the data we have indicates that π, the potential, only changes in the π§-direction. It doesnβt change at all, as far as we know, in the π₯ or π¦. So πΈ π₯ and πΈ π¦ are both zero based on the information given.

So what is πΈ, the electric field that includes all π₯-, π¦-, and π§- dimensions? The vector πΈ equals πΈ sub π₯, πΈ sub π¦, πΈ sub π§. Since weβve solved for each of these three components, we can now insert them to write out πΈ.

The electric field in this region πΈ is zero πΈ sub π₯, zero πΈ sub π¦, 200 πΈ sub π§ volts per meter.