Identical +1.39 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?
{V_T} &= {V_1} + {V_2} + {V_3}\\
{V_T} &= \dfrac{{k{Q_1}}}{l} + \dfrac{{k{Q_2}}}{{l\sqrt 2 }} + \dfrac{{k{Q_3}}}{l}
0 &= \dfrac{{k{Q_1}}}{l} + \dfrac{{k{Q_2}}}{{l\sqrt 2 }} + \dfrac{{k{Q_3}}}{l}\\
\dfrac{k}{l}\left( {{Q_1} + \dfrac{{{Q_2}}}{{\sqrt 2 }} + {Q_3}} \right) = 0\\
{Q_3} + {Q_1} + \dfrac{{{Q_2}}}{{\sqrt 2 }} &= 0\\
{Q_3}& = - \left( {7.74 \times {{10}^{ - 6}}\;{\rm{C + }}\dfrac{{7.74 \times {{10}^{ - 6}}\;{\rm{C}}}}{{\sqrt 2 }}} \right)\\
{Q_3}& = - 1.32 \times {10^{ - 5}}\;{\rm{C}}
To find the charge that should be fixed to one of the empty corners, we need to consider the concept of electric potential.
Electric potential is a scalar quantity that represents the electric potential energy per unit charge at a specific point in space. It is defined as the amount of work done to bring a unit positive charge from infinity to that point in an electric field.
In this case, we have identical +1.39 µC charges fixed to adjacent corners of a square. Let's label these charges A and B. The distance between these charges can be represented as d.
To find the charge that should be fixed to one of the empty corners, we need to calculate the electric potential at the remaining empty corner. Since the total electric potential at this corner is 0V, the sum of the electric potential due to charges A and B must be equal in magnitude and opposite in sign.
First, let's calculate the electric potential at the empty corner due to charge A. The electric potential due to a point charge can be calculated using the formula:
V = k * (q / r)
Where:
V is the electric potential
k is the electrostatic constant (9 x 10^9 N m^2 / C^2)
q is the charge
r is the distance between the charge and the point
In this case, the electric potential at the empty corner due to charge A will be:
V_A = k * (q / d)
Next, let's calculate the electric potential at the empty corner due to charge B. Similarly, the electric potential at this corner due to charge B will be:
V_B = k * (q / d)
Since the total electric potential at the empty corner is 0 V, we can write the equation:
V_A + V_B = 0
Substituting the respective formulas and values:
k * (q / d) + k * (q / d) = 0
2k * (q / d) = 0
Cancelling out the common factors:
2q / d = 0
Now, we can solve for q, the charge to be fixed at the empty corner:
2q = 0
q = 0 C
Therefore, 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 V is 0 Coulombs. This means that no charge needs to be fixed at the corner for a potential of 0 V.