Identical +3.36 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 potential at the remaining empty corner is 0 V?
Well, it seems like we've stumbled upon an electrifying problem! To find what charge should be fixed to one of the empty corners, let's do some electrifying calculations.
Since the charges fixed to adjacent corners are identical and have a magnitude of +3.36 μC, we can assume they have the same charge of +3.36 μC each.
Now, to make the total potential at the remaining empty corner 0V, we need to calculate the distance between the charges and use Coulomb's Law. We can assume that the charges form a diagonal cross, and the distance between the two charges is the length of the square's side.
So, let's assign the fixed charge on the empty corner as Q. According to Coulomb's Law, the potential due to a single charge is given by:
V = k * (Q/r)
Where k is the electrostatic constant (approximately 9 × 10^9 N·m^2/C^2) and r is the distance between the charges.
Since we want the total potential at the remaining empty corner to be 0V, it means the potentials due to the two fixed charges should cancel each other out.
So, we have:
V1 + V2 = 0
(k * Q/r) + (k * Q/r) = 0
Now, substituting the values we have, we get:
(9 × 10^9 N·m^2/C^2 * Q/r) + (9 × 10^9 N·m^2/C^2 * Q/r) = 0
Simplifying this equation, we get:
(2 * 9 × 10^9 N·m^2/C^2 * Q/r) = 0
Now, let's rearrange the equation to solve for Q:
Q = -r * 0 / (2 * 9 × 10^9 N·m^2/C^2)
Since anything multiplied by 0 is 0, we find that Q is equal to 0!
So, the magnitude and algebraic sign of the charge that should be fixed to one of the empty corners is 0.
To find the charge needed to make the total potential at the remaining empty corner zero, we can use the principle of superposition. The potential at a point due to multiple charges is the sum of the potentials due to each individual charge.
Let's label the corners of the square as A, B, C, and D. Charges Q1 and Q2 are fixed at corners A and B, respectively, and we need to find the charge needed at corner D to make the total potential at corner C zero.
Since the charges at corners A and B are identical, they have the same magnitude of charge, Q1 = Q2 = +3.36 μC.
The potential at corner C due to charge Q1 is given by V1 = k * Q1 / r1, where k is the electrostatic constant (8.99 x 10^9 N m^2/C^2) and r1 is the distance between charge Q1 and corner C.
Similarly, the potential at corner C due to charge Q2 is given by V2 = k * Q2 / r2, where r2 is the distance between charge Q2 and corner C.
To make the total potential at corner C zero, we need the sum of V1 and V2 to be zero.
V1 + V2 = 0
Substituting the expressions for V1 and V2:
k * Q1 / r1 + k * Q2 / r2 = 0
Plugging in the given values:
(8.99 x 10^9 N m^2/C^2) * (3.36 x 10^-6 C) / r1 + (8.99 x 10^9 N m^2/C^2) * (3.36 x 10^-6 C) / r2 = 0
Simplifying:
(8.99 x 10^9 N m^2/C^2) * (3.36 x 10^-6 C) * (1/r1 + 1/r2) = 0
Now, to make the total potential at corner C zero, the term inside the parentheses must be zero:
1/r1 + 1/r2 = 0
Since the distance between any two corners of a square is the same, we have r1 = r2 = R, where R is the distance between the corners.
So, 1/R + 1/R = 0
2/R = 0
This implies that R must be infinity, which is not possible in this scenario.
Therefore, it is not possible to place a charge at the empty corner D to make the total potential at corner C zero.
To start solving this problem, let's break it down into steps:
Step 1: Understand the scenario
We have a square with two fixed charges on adjacent corners. These charges are identical and have a magnitude of +3.36 μC. We need to determine the charge and its sign required on one of the empty corners so that the total potential at the remaining empty corner is 0 V.
Step 2: Determine the potential due to each charge
The potential due to a point charge can be given by the formula:
V = k * (q / r)
Where:
V is the potential,
k is the Coulomb's constant (k ≈ 9 × 10^9 Nm^2/C^2),
q is the charge, and
r is the distance from the charge.
In this scenario, since we have identical charges on each corner, the distances from the fixed charges to the remaining empty corner are equal.
Step 3: Calculate the potential due to the fixed charges
Let's denote the charge on one of the fixed corners as "Q" and the distance from each fixed charge to the remaining empty corner as "r". Since the charges are identical, the potential at the remaining empty corner due to each fixed charge is given by:
V_fixed = k * (Q / r)
Step 4: Equate the total potential to zero
To find the magnitude and algebraic sign of the charge required on the empty corner, such that the total potential at the remaining empty corner is zero, we equate the sum of the potentials due to the two fixed charges to zero:
V_fixed + V_fixed + V_empty = 0
Since the magnitude of the charges on the fixed corners is the same, we can rewrite this equation as:
2V_fixed + V_empty = 0
2(k * (Q / r)) + k * (q_empty / r) = 0
Step 5: Solve for the charge on the empty corner
To find the charge (magnitude and algebraic sign) on the empty corner, we rearrange the equation:
2(k * (Q / r)) = - k * (q_empty / r)
We simplify:
2Q = -q_empty
Finally, we find the magnitude of the charge:
|q_empty| = 2Q
q_empty = ± 2Q
Since the two fixed charges have a magnitude of +3.36 μC, we can substitute Q = 3.36 μC into the equation:
|q_empty| = 2(3.36 μC)
q_empty = ± 6.72 μC
Therefore, to make the total potential at the remaining empty corner zero, a charge of either +6.72 μC or -6.72 μC should be fixed to one of the empty corners.