Physics - electrostatic exercise with 4 charges at the corners of a square

In summary, the electrostatic exercise involving four charges placed at the corners of a square explores the interactions between these charges, calculating the resultant electric field and potential at various points. The arrangement can lead to various scenarios depending on the magnitudes and signs of the charges, and the symmetry of the square simplifies the analysis of forces acting on each charge. Key concepts such as Coulomb's law, superposition principle, and vector addition are applied to determine net forces and energies in the system.
  • #1
Hanz_
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Homework Statement
A charge Q1 is fixed in each of the two opposite vertices of the square,
a Q2 charge is placed in each of the other two opposite vertices.
a) Express Q1 through Q2 if the resulting electrostatic
the force acting on each charge Q1 is zero.
b) There is such a value of Q2 for which the resultant electrostatic force would
acting on each of the four charges was zero? Can you explain it to me please
Relevant Equations
I do not understand the procedure of integrations/derivations from which a formula will be created according to which I will calculate.
tablet-2023-12-25 3175114.png
 
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  • #2
Welcome to PF, Hanz.

Is there a figure that is associated with this exercise? If so, can you scan it and use the "Attach files" link to upload it? I'm not sure I'm understanding the problem statement if the two Q1 have the same value and the two Q2 have the same value...
 
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  • #3
After thinking about it more I think I see a solution for part (a). Start with the Q1 charges negative, and the Q2 charges positive, and draw them at the vertices of the square. Then draw the resultant force vectors due to the electrostatic charges, and think about what the values have to be to balance out the forces on the Q1 charges.
 
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  • #4
In your drawing, ##\vec F_{123}## is the hypotenuse of a right triangle with right sides ##\vec F_{12}## and ##\vec F_{13}##. What should the relation of ##\vec F_{14}## be to these for the net force on the charge in the upper left corner to be zero?
 
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  • #5
In your diagram, you have an expression for ##\vec F_{12}## and one for ##\vec F_{123}##. Can you combine them?
You have not written an expression for ##\vec F_{14}##.

There are no integrations nor differentiations to be done.
 
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FAQ: Physics - electrostatic exercise with 4 charges at the corners of a square

How do you calculate the net force on one charge due to the other three charges?

To calculate the net force on one charge due to the other three charges, you need to use Coulomb's Law to find the force between each pair of charges. Then, use vector addition to sum up the forces. The force between two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by \( F = k_e \frac{q_1 q_2}{r^2} \), where \( k_e \) is Coulomb's constant. Calculate the forces between the charge in question and each of the other three charges, and then add these vectorially to get the net force.

What is the potential energy of the system of four charges?

The potential energy of a system of four charges can be found by summing the potential energy of each pair of charges. The potential energy between two charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by \( U = k_e \frac{q_1 q_2}{r} \). Calculate this for each unique pair of charges and sum them to get the total potential energy of the system.

How does the symmetry of the square affect the net force and potential energy calculations?

The symmetry of the square simplifies the calculations because the distances between the charges are either the side length of the square or the diagonal. This symmetry allows you to use geometric relationships to simplify the vector addition of forces and the calculation of potential energy. For example, if the charges are identical, the forces along the diagonals will cancel out in certain directions due to symmetry.

How do you determine the electric field at the center of the square?

To determine the electric field at the center of the square, calculate the electric field contribution from each charge at the center and then sum these vectorially. The electric field due to a single charge \( q \) at a distance \( r \) is \( E = k_e \frac{q}{r^2} \). For a square, the distance from each corner to the center is \( \frac{a}{\sqrt{2}} \), where \( a \) is the side length of the square. Calculate the electric field due to each charge and sum the vector components to find the net electric field at the center.

What happens to the forces and potential energy if one of the charges is different from the others?

If one of the charges is different from the others, the symmetry is broken, and the calculations become more complex. The net force on each charge will need to be recalculated individually, as

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