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| Potential Energy of a System of Charges |
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| Consider two charges q1 and q2 with position vectors r1 and r2 respectively, relative to some origin. To calculate externally the work done in building this configuration, we have to consider the charges q1 and q2 initially at infinity and determine the work done by an external agency to bring the charges to a given point. |
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| Suppose the charge 'q1' is brought from infinity to a point r1, there is no external field against which work needs to be done. Here the work done is zero, but the charge produces a potential in space and is given by |
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| where 'r1p' is the distance of any point 'p' in space from location of q1. Therefore, by definition of potential, work done in bringing q2 from infinity to point r2 is q2 times the potential at r2 due to q1. |
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| where 'r12' is the distance between points 1 and 2. As the electrostatic force is conservative, the work done is stored as potential energy and is given by |
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| The potential 'V' would be same, even if the charge q2 is brought from infinity to a point first and then q1, which in turn indicates potential energy is path - independent of work for a electrostatic force. |
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| If the charges q1, q2 > 0, the potential energy is positive, that is for like charges electrostatic force is repulsive and a positive amount of work has to be done against the repulsive force, to bring the charges from infinity to a finite distance apart. |
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| If the charges q1, q2 < 0, electrostatic force is attractive. In this case, a positive amount of work has to be done against the attractive force to take the charge from a given location to infinity. |
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| Similarly, the potential energy of a system of three charges q1, q2, q3 located at r1, r2, r3 respectively can be calculated. |
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| Work done in bringing q1 from infinity to r1 is zero. |
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| To bring q1 from infinity to r2 is |
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| Similarly for bringing 'q3', we have |
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| From equations (1) and (2), the total work done is given by |
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| This again proves that the potential energy is path independent of work. Hence, potential energy is the characteristic of the present state of configuration and not how the state was achieved. |
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