Energy Density In Electrostatic Field / Work Done To Assemble Charges.

- In case, if we wish to assemble a number of charges in an empty system, work is required to do so.


- Also electrostatic energy is said to be stored in such a collection.


- Let us build up a system in which we position three point charges Q₁, Q₂ and Q₃ at position r₁, r₂ and r₃ respectively in an initially empty system.


- Consider a point charge Q₁ transferred from infinity to position r₁ in the system. It takes no work to bring the first charge from infinity since there is no electric field to fight against (as the system is empty i.e. charge free).

Hence, W₁ = 0 J

- Now bring in another point charge Q₂ from infinity to position r₂ in the system. In this case we have to do work against the electric field generated by the first charge Q₁.

Hence, W₂ = Q₂ V₂₁

where V₂₁ is the electrostatic potential at point r₂ due to Q₁.


- Work done W₂ is also given as:

- Now bring in another point charge Q₃ from infinity to position r₃ in the system. In this case we have to do work against the electric field generated by Q₁ and Q₂.

Hence, W₃ = Q₃ V₃₁ + Q₃ V₃₂ = Q₃ ( V₃₁ + V₃₂ )

where V₃₁ and V₃₂ are electrostatic potential at point r₃ due to Q₁ and Q₂ respectively.


- The work done is simply the sum of the work done against the electric field generated by point charge Q₁ and Q₂ taken in isolation: 

- Thus the total work done in assembling the three charges is given as:

W_E = W₁ + W₂ + W₃

                               = 0 + Q₂ V₂₁ + Q₃ ( V₃₁ + V₃₂ )

- Also total work done ( W_E ) is given as:

- If the charges were positioned in reverse order, then the total work done in assembling them is given as:

W_E = W₃ + W₂+ W₁

                            = 0 + Q₂V₂₃ + Q₃( V₁₂+ V₁₃)

where V₂₃ is the electrostatic potential at point r₂ due to Q₃ and V₁₂ and V₁₃ are electrostatic potential at point r₁ due to Q₂ and Q₃ respectively.


- Adding the above two equations we have,

2W_E = Q₁ ( V₁₂ + V₁₃) + Q₂ ( V₂₁ + V₂₃) + Q₃ ( V₃₁ + V₃₂)

                                          = Q₁ V₁ + Q₂ V₂ + Q₃ V₃

Hence

W_E =1 / 2 [Q₁V₁ + Q₂V₂ + Q₃V₃]

where V₁, V₂ and V₃ are total potentials at position r₁, r₂ and r₃ respectively.


- The result can be generalized for N point charges as:

- The above equation has three interpretation:

          a) This equation represents the potential energy of the system.
  
          b) This is the work done in bringing the static charges from infinity and assembling them in the required system.
 
          c) This is the kinetic energy which would be released if the system gets dissolved i.e. the charges returns back to infinity.


- In place of point charge, if the system has continuous charge distribution ( line, surface or volume charge), then the total work done in assembling them is given as:

- Since ρ_v = ∇ . D and E = - ∇ V,

Substituting the values in the above equation, work done in assembling a volume charge distribution in terms of electric field and flux density is given as:

 

- The above equation tells us that the potential energy of a continuous charge distribution is stored in an electric field.


- The electrostatic energy density w_E is defined as:

ALSO READ: 

- Gauss's Law - Theory.

- Gauss's Law - Application To a Point charge.

- Gauss's Law - Application To An Infinite Line Charge.

- Gauss's Law - Application To An Infinite Sheet Charge.

- Gauss's Law - Application To a Uniformly Charged Sphere.

- Numericals / Solved Examples - Gauss's Law.

- Scalar Electric Potential / Electrostatic Potential (V).

- Relationship Between Electric Field Intensity (E) and Electrostatic Potential (V).

- Electric Potential Due To a Circular Disk.

- Electric Dipole.

- Numericals / Solved Examples - Electric Potential and Electric Dipole.

- Energy Density In Electrostatic Field / Work Done To Assemble Charges.

- Numericals / Solved Examples - Electrostatic Energy and Energy Density.

- Numericals / Solved Examples - Gauss's law...


- Short Notes/FAQ's

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