Force charges +q,-q,+q,-q placed at 4 corners of a square of sides 'a' calculate the resultant force on +Q charge at the intersection of 2 diagonals
If those are at consecutive corners, they cancel each other.
To calculate the resultant force on the +Q charge at the intersection of the diagonals, we need to consider the forces acting on it due to each of the individual charges at the corners of the square.
The force between two charges is given by Coulomb's Law:
\[ F = k \frac{q_1 \cdot q_2}{r^2} \]
Where:
- F is the force between the charges
- k is Coulomb's constant (\(9 \times 10^9 \, \text{Nm}^2/\text{C}^2\))
- \(q_1\) and \(q_2\) are the magnitudes of the charges
- r is the distance separating the charges
Given that the charges at the corners are \(+q\), \(-q\), \(+q\), and \(-q\), we can calculate the forces between these charges and the +Q charge using Coulomb's Law. The forces acting on the +Q charge due to each corner charge will be along the diagonals of the square.
Let's label the corners of the square as A, B, C, and D, in a counterclockwise direction starting from the bottom left corner, with the +Q charge at the intersection of the diagonals.
1. Force between +Q and +q at corner A:
- The magnitude of the charge at corner A is \(+q\).
- The distance between the +Q charge and corner A is \(\sqrt{2}a\) (using Pythagorean theorem).
- Using Coulomb's Law, we can calculate the force between these charges.
2. Force between +Q and -q at corner B:
- The magnitude of the charge at corner B is \(-q\).
- The distance between the +Q charge and corner B is \(\sqrt{2}a\) (using Pythagorean theorem).
- Using Coulomb's Law, we can calculate the force between these charges.
3. Force between +Q and +q at corner C:
- The magnitude of the charge at corner C is \(+q\).
- The distance between the +Q charge and corner C is \(\sqrt{2}a\) (using Pythagorean theorem).
- Using Coulomb's Law, we can calculate the force between these charges.
4. Force between +Q and -q at corner D:
- The magnitude of the charge at corner D is \(-q\).
- The distance between the +Q charge and corner D is \(\sqrt{2}a\) (using Pythagorean theorem).
- Using Coulomb's Law, we can calculate the force between these charges.
To find the resultant force, we need to consider the vector sum of the forces acting on the +Q charge at the intersection of the diagonals. Since the forces are acting along the diagonals, they will have equal magnitudes but opposite directions.
The resultant force is the vector sum of these forces. You can calculate it by adding the forces along the x-axis separately and the forces along the y-axis separately. The x-component will give you the net force in the x-direction, and the y-component will give you the net force in the y-direction.
Finally, you can calculate the magnitude and direction of the resultant force using the Pythagorean theorem and trigonometry with the net forces in the x and y directions.