A positive point charge and a negative point charge have equal magnitudes. One charge is fixed to one corner of a square, and the other is fixed to another corner. On which corners should the charges be placed, so that the same potential exists at the empty corners? What is the reasoning?
The two charges should be placed at diagonally opposite corners, because in that case the open corners are equidistant from the positive and the negative charges, so there is no effect on the potential there.
To determine the placement of the charges so that the same potential exists at the empty corners of the square, we need to consider the electric field and the concept of equipotential surfaces.
Since the positive and negative charges have equal magnitudes, they will create equal electric fields in magnitude but opposite in direction. The electric field lines will be radially outward from the positive charge and radially inward towards the negative charge.
To have the same potential at the empty corners, the electric field lines should be perpendicular to the sides of the square, which means they should be pointing towards or away from the center of the square.
We can achieve this by placing the positive charge at a corner that corresponds to an empty corner with two adjacent sides perpendicular to the other sides of the square (diagonal corner), while placing the negative charge at the corner that corresponds to an empty corner with two adjacent sides parallel to the other sides of the square (opposite corner).
The reasoning behind this placement is that the electric field lines from the positive charge would be perpendicular to the sides of the square, creating an equipotential surface. Similarly, the electric field lines from the negative charge would also be perpendicular to the sides of the square, resulting in the same potential at the empty corners.
The configuration would look like this:
+ (positive charge)
O (empty corner)
O (empty corner)
(negative charge) -
Therefore, the charges should be placed at the diagonal corner (positive charge) and the opposite corner (negative charge) to create the same potential at the empty corners.
To determine the corners on which the charges should be placed so that the same potential exists at the other empty corners, we need to consider the nature of electrostatic potential.
Electrostatic potential is a scalar quantity that represents the amount of work done in bringing a unit positive charge from infinity to a specific point in an electric field. The potential is proportional to the magnitude of the point charge and inversely proportional to the distance from the charge.
In this scenario, we have two charges, one positive and one negative, with equal magnitudes. Let's assume Q represents the magnitude of the charge.
To have the same potential at the empty corners, the electrostatic potential due to each charge should cancel out when they are combined at the empty corners. This means that the sum of the distances between the charges and the empty corners should be equal for both charges.
To achieve this, we place the positive charge at one corner and the negative charge at the diagonally opposite corner. Let's label these corners A and C, respectively.
When we calculate the distance between the positive charge at corner A and the empty corner B, and then calculate the distance between the negative charge at corner C and the same empty corner B, the sum of these distances will be the same.
Similarly, when we calculate the distance between the positive charge at corner A and the empty corner D, and then calculate the distance between the negative charge at corner C and the same empty corner D, the sum of these distances will also be the same.
Therefore, in order to achieve the same potential at the empty corners B and D, we should place the positive charge in one corner and the negative charge in the diagonally opposite corner.
To summarize:
- Place the positive charge at one corner (let's call it corner A).
- Place the negative charge at the diagonally opposite corner (let's call it corner C).
- The empty corners B and D will have the same potential because the distances from each of the charges to these corners will add up to the same sum.
Remember, the key concept here is that the sum of the distances between the charges and the empty corners should be equal for both charges to cancel out each other's potential.
Electrical potential is the force experienced by a unit test charge at a particular point. In this case that potential on the test charge is generated by the positive and the negative charges placed in the two corners of the square.
There are two possible configurations for the "occupied" corner charges: Adjacent and diagonal. One of these configurations would produce identical effects on the test charge no matter which unoccupied corner it was placed on. Do some drawing and i think you will "see" which arrangement works best.