Question

Four point charges, A, B, C, and D, are placed at the cornersof a square with side length .Charges A, B, and C have charge ,and D has charge .

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If you calculate ,the amount of work it took to assemble this charge configuration ifthe point charges were initially infinitely far apart, you willfind that the contribution for each charge is proportional to $\frac{kq^2}{L}$ . In the space provided, enter the numeric valuethat multiplies the above factor, in .

=      x(kq²/L)                  

Answer 1

Concept and reason

The required concept to solve this question is electric potential and work of charges.

First find the work done to place the negative charge at the bottom left corner. Next find work done to place remain three positive charges. Finally add the all works.

This work done is the amount of work it took to assemble this charge configuration.

Fundamentals

An electric potential is the amount of work needed to move a unit positive charge from a reference point (infinitely far apart) to a specific point inside the field.

The expression for the electrical potential is,

$V = \frac{{kq}}{r}$

Here, k coulombs constant, q is the charge, and r is the distance where to be potential measured.

If a charge of Q is moved through a potential difference of V, then the work done is,

$W = QV$

To achieve this four charge configuration, place each charge one by one at their places.

The expression for the work done is,

$W = \frac{{kQq}}{r}$

Here, Q is the moving charge, q is the charge which provides potential field, and r is the distance between the two charges.

The work done to place the charge D at the corner is given by,

$W = QV$

Substitute –q for Q and 0 for V in $W = QV$ .

$\begin{array}{c}\\{W_D} = \left( { - q} \right)\left( 0 \right)\\\\ = 0\\\end{array}$

The work done to place the charge C at the bottom right corner as follows:

Substitute –q for Q and L for r in the equation $W = \frac{{kQq}}{r}$ .

${W_C} = \frac{{ - kqq}}{L}$

The work done to place the charge B at the top right corner is,

${W_B} = \frac{{ - kqq}}{{\sqrt 2 L}} + \frac{{kqq}}{L}$

The work done to place the charge A at the top left corner is,

${W_A} = \frac{{ - kqq}}{L} + \frac{{kqq}}{{\sqrt 2 L}} + \frac{{kqq}}{L}$

The work done to assemble this configuration is,

$\begin{array}{c}\\W = 0 - \frac{{kqq}}{L} - \frac{{kqq}}{{\sqrt 2 L}} + \frac{{kqq}}{L} - \frac{{kqq}}{L} + \frac{{kqq}}{{\sqrt 2 L}} + \frac{{kqq}}{L}\\\\ = 0\\\end{array}$

Therefore, the amount of work it took to assemble this charge configuration is zero.

Ans:

The amount work is $W = 0 \times \left( {\frac{{k{q^2}}}{L}} \right)$ .