Case 1 - Potential Energy due to two charges : Consider two charges q1 and q2 with position vector r1 and r2 relative to some origin. Consider the charges q1 and q2 initially at infinity and determine the work done by an external agency to bring the charges to the given locations. Suppose, first the charge q1 is brought from infinity to the point vector r1. There is no external field against which work needs to be done, so work done in bringing q1 from infinity to vector r1 is zero. This charge produces a potential in space given by
where r1P is the distance of a point P in space from the location of q1. From the definition of potential, work done in bringing charge q2 from infinity to the point vector r2 is q2 times the potential at vector r2 due to q1:
Work done on q2
where r12 is the distance between points 1 and 2.
Since electrostatic force is conservative, this work gets stored in the form of potential energy of the system. Thus, the potential energy of a system of two charges q1 and q2 is
Obviously, if q2 was brought first to its present location and q1 brought later, the potential energy U would be the same.
Case 2 : Potential Energy due to three charges :
Let us calculate the potential energy of a system of three charges q1, q2 and q3 located at vector r1, r2, r3, respectively. To bring q1 first from infinity to vector r1, no work is required. Next we bring q2 from infinity to vector r2. As before, work done in this step is
The charges q1 and q2 produce a potential, which at any point P is given by
Work done next in bringing q3 from infinity to the point vector r3 is q3 times V1,2 at vector r3
The total work done in assembling the charges at the given locations is obtained by adding the work done in different steps [Eq. (4) and Eq. (6)],
The potential energy is characteristic of the present state of configuration, and not the way the state is achieved.Read more on Sarthaks.com - https://www.sarthaks.com/424469/derive-an-expression-electric-potential-energy-of-a-system-of-charges
Yogita Ingle 4 years, 9 months ago
Case 1 - Potential Energy due to two charges : Consider two charges q1 and q2 with position vector r1 and r2 relative to some origin. Consider the charges q1 and q2 initially at infinity and determine the work done by an external agency to bring the charges to the given locations. Suppose, first the charge q1 is brought from infinity to the point vector r1. There is no external field against which work needs to be done, so work done in bringing q1 from infinity to vector r1 is zero. This charge produces a potential in space given by
where r1P is the distance of a point P in space from the location of q1. From the definition of potential, work done in bringing charge q2 from infinity to the point vector r2 is q2 times the potential at vector r2 due to q1:
Work done on q2
where r12 is the distance between points 1 and 2.
Since electrostatic force is conservative, this work gets stored in the form of potential energy of the system. Thus, the potential energy of a system of two charges q1 and q2 is
Obviously, if q2 was brought first to its present location and q1 brought later, the potential energy U would be the same.
Case 2 : Potential Energy due to three charges :
Let us calculate the potential energy of a system of three charges q1, q2 and q3 located at vector r1, r2, r3, respectively. To bring q1 first from infinity to vector r1, no work is required. Next we bring q2 from infinity to vector r2. As before, work done in this step is
The charges q1 and q2 produce a potential, which at any point P is given by
Work done next in bringing q3 from infinity to the point vector r3 is q3 times V1,2 at vector r3
The total work done in assembling the charges at the given locations is obtained by adding the work done in different steps [Eq. (4) and Eq. (6)],
The potential energy is characteristic of the present state of configuration, and not the way the state is achieved.Read more on Sarthaks.com - https://www.sarthaks.com/424469/derive-an-expression-electric-potential-energy-of-a-system-of-charges
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