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Electric Potential and Electric Field
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Physics Textbooks Boundless Physics Electric Potential and Electric Field Overview
Physics Textbooks Boundless Physics Electric Potential and Electric Field
Physics Textbooks Boundless Physics
Physics Textbooks
Physics
Concept Version 13
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Potentials and Charged Conductors

Electric potential within a charged conductor is equal to zero, but can be calculated as a nonzero value outside of a charged conductor.

Learning Objective

  • Determine the electric potential within and outside a charged conductor


Key Points

    • Electric potential (∆V) and field (E) are related according to the integral: $\Delta V = - \int_i^f \! \vec E \cdot \mathrm{d} \vec l$ where l is the distance between two points between which the potential difference is being found.
    • Given that the electric field is constantly 0 for any location within the charged conductor, it is impossible for potential difference in that same volume to have any value other than 0.
    • For points outside a conductor, potential is nonzero and can be calculated according to field and distance from the conductor.

Terms

  • electric potential

    The potential energy per unit charge at a point in a static electric field; voltage.

  • work

    A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.

  • electric field

    A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.


Full Text

When a conductor becomes charged, that charge distributes across its surface until electrostatic equilibrium is reached. Its surface is equipotential.

All points within a charged conductor experience an electric field of 0. This is because field lines from charges on the surface of the conductor oppose one another equally . However, having the electric field equal to zero at all points within a conductor, the electric potential within a conductor is not necessarily  equal to zero for all points within that same conductor. This can be proven by relating electric field and potential.

Electrical Charge at a Sharp Point of a Conductor

Repulsive forces towards the more sharply curved surface on the right aim more outward than along the surface of the conductor.

Given that work is the difference in final and initial potential energies (∆U), we can relate this difference to the dot product of force at every infinitesimal distance l along the path between the points within the conductor:

$\Delta U = - \int_i^f \! \vec F \cdot \mathrm{d} \vec l$

This is the equation for work, with ∆U substituted in place of W. Rewriting U as the product of charge (q) and potential difference (V), and force as the product of charge and electric field (E), we can assert:

$\Delta (qV) = - \int_i^f \! (q\vec E) \cdot \mathrm{d} \vec l$

Dividing both sides by the common term of q, we simplify the equation to:

$\Delta V = - \int_i^f \! \vec E \cdot \mathrm{d} \vec l$

Finally we derive the equation :

$dV = - \vec E \cdot \mathrm{d} \vec l = 0$

Thus we can conclude that, given that the electric field is constantly 0 for any location within the charged conductor, the potential difference in that same volume needs to be constant and equal to 0.

On the other hand, for points outside a conductor, potential is nonzero and can be defined by the very same equation, according to field and distance from the conductor.

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