Potential Energy of Multiple Charges

Description:

To understand Potential energy of a system of charges, First we deal with two particle system, where two particles (Q and q ) are separated by a distance of ‘r’ in an electric field. Then, the potential energy (U) of the system is given by −

U = qV

Potential at a point B due to charge Q is Q/4πε₀ 1/r

If a charge ‘q’ is placed at point B, some work is needed to be done. That work will be equal to the potential energy of the system

U = W = Q q/4πε₀ 1/r = qV

For 3 particle system

Let there is a charged particle q₁. Some work is done and an another charge q₂ is brought from infinity to point B. The work done or the potential energy of the system is −

U₁ = W = q₁q₂/4πε₀ 1/r₁₂

Now,a third charged particle q₃ is brought to the system at point C. This particle (q₃) is repelled by the other two particles(q₁ and q₂). Against this repulsion, some work is needed to be done to take charge q₃ from infinity to point C.

Work done on q₃ against q₁ −

U₂ = W = q₁q₃/4πε₀ 1/r₁₃

Work done on q₃ against q₁ −

U₃ = W = q₃q₂/4πε₀ 1/r₂₃

Since potential energy is a scalar quantity so, total potential energy of the system is (U);

U = W = U₁ + U₂ + U₃ = q₃q₂/4πε₀ 1/r₂₃ + q₁q₃/4πε₀ 1/r₁₃ + q₁q₂/4πε₀ 1/r₁₂

= 1/4πε₀[q₃q₂/r₂₃ + q₁q₃/r₁₃ + q₁q₂/r₁₂]

So U is the total energy stored in 3 particle system.

For 4 particle system

If a forth charged particle q₄ is brought to the system at point D. This particle (q₄) will be repelled by the other three particles(q₁ q₂ and q₃). Against this repulsion, some work is needed to be done to take charge q₄ from infinity to point D. If q₄ is brought and blocked in the system. Some energy will be enclosed with each pair of charges.

So, energy enclosed between q₁ q₂ −

U₁ = W = q₁q₂/4πε₀ 1/r₁₂

energy enclosed between q₁ q₃

U₂ = W = q₁q₃/4πε₀ 1/r₁₃

energy enclosed between q₁ q₄

U₃ = W = q₁q₄/4πε₀ 1/r₁₄

energy enclosed between q₂ q₃

U₄ = W = q₃q₂/4πε₀ 1/r₂₃

energy enclosed between q₂ q₄

U₅ = W = q₄q₂/4πε₀ 1/r₂₄

energy enclosed between q₃ q₄

U₆ = W = q₄q₃/4πε₀ 1/r₃₄

Total potential energy of the system is (U);

U = W = U₁ + U₂ + U₃ + U₄ + U₅ + U₆ = q₁q₂/4πε₀ 1/r₁₂ + q₁q₃/4πε₀ 1/r₁₃ + q₁q₄/4πε₀ 1/r₁₄ + q₃q₂/4πε₀ 1/r₂₃ + q₄q₂/4πε₀ 1/r₂₄ + q₄q₃/4πε₀ 1/r₃₄

U = 1/4πε₀[q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₁q₄/r₁₄ + q₃q₂/r₂₃ + q₄q₂/r₂₄ + q₄q₃/r₃₄]

Similarly energy ‘n‘ number of particles system can be calculate.