In this article, we are going to derive an expression for potential energy of a system of three points charges. So keep reading till end..
DERIVATION FOR THE POTENTIAL ENERGY OF A SYSTEM OF THREE POINTS CHARGES
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Let’s consider a system of three points charges q1q1, q2q2 and q3q3 having position vector r1r1, r2r2 and r3r3 respectively from the origin as shown in following figure.
If we bring charge q1q1 first from infinity to position r1r1 then there is no work done required to do so, it is because when we bring charge q1q1 from infinity to position r1r1 then at that position there is no any source which can produce electric field. If there is no electric field then there is no any opposing force. Hence work done is zero.
∴W1=0∴W1=0
But when we bring charge q2q2 from infinity to the position r2r2 then in this, we have to do work done because, here a opposing field is present due to charge q1q1. So we have to do work done in against the electric field produced by the first electric charge q1q1.
The work done in bringing charge q2q2 from infinity to the position r2r2 is-
W2=q2V(r2)W2=14πϵ0q1q2r12W2=q2V(r2)W2=14πϵ0q1q2r12
Where r12r12 is the position vector between charge q1q1 and q2q2 and V is the electric potential due to charge q1q1 at position vector r12r12.
Now, charges q1q1 and q2q2 will produce a electric potential at any point say P. Think that point P denotes the position of charge q3q3. The position vector between charge q1q1 and q3q3 will be r13r13 and in between charge q2q2 and q3q3 will be r23r23.
Now the electric potential due to the charge q1q1 and q2q2 at point P is given as-
V1,2=14πϵ0(q1r13+q2r23)V1,2=14πϵ0(q1r13+q2r23)
So the work done in bringing charge q3q3 from infinity to the position r3r3 is-
W3=q3V1,2(r3)=14πϵ0(q1q3r13+q2q3r23)W3=q3V1,2(r3)=14πϵ0(q1q3r13+q2q3r23)
The total work done in assembling the charges at the given location is equal the total potential energy of the system and According to the superposition principle, this total potential energy can be obtained by adding the work done of individual charges.
U=W1+W2+W3=0+14πϵ0q1q2r12+14πϵ0(q1q3r13+q2q3r23)U=W1+W2+W3=0+14πϵ0q1q2r12+14πϵ0(q1q3r13+q2q3r23)
U=14πϵ0(q1q2r12+q1q3r13+q2q3r23)U=14πϵ0(q1q2r12+q1q3r13+q2q3r23)
This result can also be expressed in the form of summation as follows-
U=[14πϵ03∑i=13∑i=1j≠iqiqjrij]U=⎡⎣14πϵ03∑i=13∑i=1j≠iqiqjrij⎤⎦
If we want to obtain the value of electric potential energy of a system of N point charges then we also obtained it. The value of electric potential energy due to a system of N point charges is equal to the total amount of work done in assembling all the charges to the given position from infinity.
U=[14πϵ0N∑i=1N∑i=1j≠1qiVj]whereVj=N∑i=1j≠i14πϵ0qirijU=⎡⎣14πϵ0N∑i=1N∑i=1j≠1qiVj⎤⎦whereVj=N∑i=1j≠i14πϵ0qirij
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We know that electrostatic force is conservative in nature, so the value of U is independent of the manner in which the configuration of charge is assembled.
The SI unit of electric potential energy is joule (J) and it’s another unit is electron volt (eV)
1eV=1.6×10−19×1V=1.6×10−19J1eV=1.6×10−19×1V=1.6×10−19J