Identical +6.15 µC charges are fixed to adjacent corners of a square. What charge (magnitude and algebraic sign) should be fixed to one of the empty corners, so that the total electric potential at the remaining empty corner is 0 V?

To determine the charge that should be fixed to one of the empty corners so that the total electric potential at the remaining empty corner is 0 volts, we can utilize the concept of electric potential and the principle of superposition.

First, let's consider the charges fixed at the adjacent corners of the square as shown:

+6.15 µC +6.15 µC
•----------------•
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| |
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•----------------•

To find the charge and sign needed at one of the empty corners, we need to understand that electric potential is a scalar quantity. The electric potential due to point charges is given by the equation:

V = k * q / r

Where:
V is the electric potential,
k is Coulomb's constant (8.99 x 10^9 N·m^2/C^2),
q is the charge, and
r is the distance.

To find the total electric potential at the empty corner, we can calculate the electric potential due to each of the +6.15 µC charges and then add them together:

V_total = V1 + V2

Since the electric potential at the remaining empty corner is required to be 0 volts, we can set up the equation:

0 = V1 + V2

Next, we can substitute the formula for electric potential and rearrange the equation:

0 = k * q1 / r1 + k * q2 / r2

Since the charges at the adjacent corners are identical, q1 = q2 = +6.15 µC.

Our equation becomes:

0 = k * (6.15 µC) / r1 + k * (6.15 µC) / r2

Now, let's consider the distances. Since all the sides of the square are equal, the distances from the charges to the empty corner will also be equal. Let's represent the distance as 'd'.

Therefore, r1 = r2 = d.

The equation now becomes:

0 = k * (6.15 µC) / d + k * (6.15 µC) / d

Simplifying further:

0 = 2 * k * (6.15 µC) / d

Now, we can rearrange the equation to solve for the charge at the empty corner:

q_empty / d = - (q1 + q2) / d

q_empty = - (q1 + q2)

Substituting the values we know:

q_empty = - (6.15 µC + 6.15 µC)

q_empty = - (12.3 µC)

Therefore, in order to achieve a total electric potential of 0 volts at the remaining empty corner, a charge of -12.3 µC should be fixed to that corner.

Note: The negative sign indicates the algebraic sign of the charge, indicating a negative charge.