Physics : Electrostatics: Electrostatic Potential and Potential Energy : Electrostatic potential energy for collection of point charges

**Electrostatic potential**** ****energy for collection of
point charges**

The electric potential
at a point at a distance *r* from point charge *q*_{1} is
given by

This potential *V*
is the work done to bring a unit positive charge from infinity to the point.
Now if the charge *q*_{2} is brought from infinity to that point
at a distance *r* from *q*_{1}, the work done is the product
of *q*_{2} and the electric potential at that point. Thus we have

W = q_{2}V

This work done is stored
as the electrostatic potential energy U of a system of charges q_{1}
and q_{2} separated by a distance r. Thus we have

The electrostatic
potential energy depends only on the distance between the two point charges. In
fact, the expression (1.45) is derived by assuming that *q*_{1} is
fixed and *q*_{2}* *is brought from infinity. The equation*
*(1.45) holds true when *q*_{2} is fixed and *q*_{1}
is brought from infinity or both *q*_{1} and *q*_{2}
are simultaneously brought from infinity to a distance *r* between them.

Three charges are
arranged in the following configuration as shown in Figure 1.30.

To calculate the total
electrostatic potential energy, we use the following procedure. We bring all
the charges one by one and arrange them according to the configuration as shown
in Figure 1.30.

(i) Bringing a charge q_{1}
from infinity to the point A requires no work, because there are no other
charges already present in the vicinity of charge q_{1}.

(ii) To bring the second
charge q_{2} to the point B, work must be done against the electric
field created by the charge *q*_{1}. So the work done on the
charge *q*_{2} is *W *=* q*_{2}* V *_{1B.}*
*Here* V*_{1B}* *is the electrostatic* *potential due
to the charge *q*_{1} at point *B*.

Note that the expression
is same when *q*_{2} is brought first and then *q*_{1}
later.

(iii) Similarly to bring
the charge q_{3} to the point C, work has to be done against the total
electric field due to both charges *q*_{1}* *and* q*_{2}.
So the work done to bring* *the charge *q*_{3} is = *q*_{3}
(*V*_{1C} + *V*_{2C}). Here *V*_{1C}* *is
the electrostatic potential due* *to charge *q*_{1} at point
C and *V*_{2C} is the electrostatic potential due to charge *q*_{2}
at point C.

The electrostatic
potential is

(iv) Adding equations
(1.46) and (1.47), the total electrostatic potential energy for the system of
three charges q_{1,} q_{2} and q_{3} is

Note that this stored
potential energy U is equal to the total external work done to assemble the
three charges at the given locations. The expression (1.48) is same if the
charges are brought to their positions in any other order. Since the Coulomb
force is a conservative force, the electrostatic potential energy is
independent of the manner in which the configuration of charges is arrived at.

**EXAMPLE 1.15**

Four charges are arranged at the
corners of the square PQRS of side a as shown in the figure.(a) Find the work
required to assemble these charges in the given configuration. (b) Suppose a
charge qâ€² is brought to the center of the square, by keeping the four charges
fixed at the corners, how much extra work is required for this?

**Solution**

(a) The work done to arrange the
charges in the corners of the square is independent of the way they are
arranged. We can follow any order.

(i) First, the charge +q is brought
to the corner P. This requires no work since no charge is already present, W_{P}
= 0

(ii) Work required to bring the
charge â€“q to the corner Q = (-q) x potential at a point Q due to +q located at
a point P.

W_{Q} = âˆ’q Ã— 1/4Ï€Îµ . q/ = âˆ’
1/4Ï€Îµ . q^{2}/a

(iii) Work required to bring the charge +q to
the corner R= q Ã— potential at the point R due to charges at the point P and Q.

(iv) Work required to bring the
fourth charge â€“q at the position S = q Ã— potential at the point S due the all
the three charges at the point P, Q and R

(b) Work required to bring the
charge qâ€² to the center of the square = qâ€² Ã— potential at the center point O
due to all the four charges in the four corners

The potential created by the two +q
charges are canceled by the potential created by the -q charges which are
located in the opposite corners. Therefore the net electric potential at the
center O due to all the charges in the corners is zero.

Hence no work is required to bring
any charge to the point O. Physically this implies that if any charge qâ€² when
brought close to O, then it moves to the point O without any external force.

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12th Physics : Electrostatics : Electrostatic potential energy for collection of point charges |

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