idential point charges of 1.7e-6 C are fixed diagonally on opposite corners of a square. A third charge is then fixed at the center of the square, such that it causes the potentials at the empty corners to change signs without changing their magnitudes. Find the sign and magnitude of the third charge in the center of the square.
i am absolutely clueless.
Let the charges at the two opposite corners be q, and the added charge at the center be q'. Before q' is added, the potential at the empty corners is
2 k q/a, where a is the length of a side of the square.
The distance of the center of the square from either empty corner is a/sqrt2
The potential at the emptry corners with q' added must be
-2kq/a = 2kq/a + kq' sqrt2/a
The k's and a's cancel and you are left with
-2q = +q' *sqrt2
q' = -sqrt2*q (since 2/sqrt2 = sqrt2)
To solve this problem, we can use the concept of electric potential and Coulomb's law. Here's how you can approach it step by step:
1. Let's denote the charges at the diagonally opposite corners of the square as Q1 and Q2, with each having a magnitude of 1.7e-6 C. These charges are fixed, meaning they do not move.
2. Next, we need to find the potential at the empty corners of the square when there is no third charge at the center.
3. Since the potential is a scalar quantity, we can simply add the potentials due to each individual charge to find the total potential at a given point. The equation for the potential due to a point charge Q at a distance r is given by V = kQ/r, where k is the Coulomb's constant (8.99e9 Nm²/C²).
4. The distance from each corner to the center of the square is the length of the side of the square divided by the square root of 2 (since it's a square with diagonal).
5. So, the potential at the empty corners without the presence of the third charge can be calculated by summing up the potentials due to Q1 and Q2 at those points. Since they are diagonally opposite, their contributions will have the same magnitude and can be added directly.
6. Now, we need to find the magnitude of the potential at the empty corners when it changes sign but not magnitude. This means that the potential at the empty corners after adding the third charge is zero.
7. Let's denote the magnitude of the third charge at the center of the square as Q3. So, we need to find the value of Q3 that would make the potential at the empty corners zero.
8. Using the same approach as before, we can calculate the potential at the empty corners due to Q1, Q2, and Q3 and set it equal to zero.
9. Since the potential at the empty corners is zero, the sum of the potentials due to Q1, Q2, and Q3 must be equal to zero. Hence, the equation becomes V1 + V2 + V3 = 0, where V1, V2, and V3 are the potentials due to Q1, Q2, and Q3, respectively.
10. Substitute the values of Q1, Q2, r, and k into these equations and solve for Q3.
11. Once you find the value of Q3, its sign will indicate whether it is positive or negative. A positive sign indicates that it is the same as Q1 and Q2, while a negative sign indicates the opposite.
12. The magnitude of Q3 will be the absolute value of the charge calculated.
By following these steps, you will be able to find the sign and magnitude of the third charge at the center of the square.