But in general you can say that these equations

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But, in general, you can say that: These equations demonstrate that if you move with the electric field, your electric potential will decrease. Electric Potential
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Electric Potential As it turns out you can calculate all components of an electric field this way. Suppose you have a charge moving in an electric field, if you can measure the potential of the charge as it moves then you can find E with: This is how you can calculate the electric field (vector) from voltage (scalar). E x = ! " V " x E y = ! " V " y E z = ! " V " z ! E = E x ˆ i + E y ˆ j + E z ˆ k
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Electric Potential Conversely, you can also calculate voltage from the electric field by integrating over a distance (two points: initial and final): This gives us the following relationships: Many times you will be asked to find one variable given the expression of one of the others, just recall how they are related. V f ! V i = ! ! E " d ! s i f #
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Electric Potential For example, in Chapter 22 we found that the electric field on the z-axis from a charged disk was: If we wanted to find the electric potential we can turn to: Evaluating the right side we get: E disk = ! 2 " o 1 ! z z 2 + R 2 " # $ $ % & ' ' V f ! V i = ! ! E " d ! s i f # ! E ! d ! s i f " = ! 2 " o 1 # z z 2 + R 2 $ % & & ' ( ) ) * z " dz
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Electric Potential The first term is easy, but the second term needs a u-substitution, let u = (z 2 +R 2 ) Turning back to our equation for potential: We usually define a place (like V i to be zero at infinite distance), yielding: V f ! V i = ! ! 2 " o z ! z 2 + R 2 ( ) V = ! 2 " o z 2 + R 2 ! z ( ) ! E ! d ! s i f " = ! 2 " o z # z 2 + R 2 ( ) ! E ! d ! s i f " = ! 2 " o z # z 2 + R 2 ( ) $ z
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Clicker Question 24C-1 An electron is released from rest at point B (as shown to the right), where the potential is 0V.
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