You have two negative balloons hanging close to one another. To increase the amount of potential energy stored in the field between the two balloons, which way would a force need to move the balloons—toward each other or apart?

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Towards each other. The basic quantity is potential in the field . The potential is property of the field showing that energy is stored in the field.

When a body which can interact with the field is put in the field the capacity of doing work comes from the interaction of the body with field.

Potential energy

Potential energy is the energy that is needed to move a charge against the electric field.

The gravitational potential at a point in the gravitational field is the gravitational potential energy of a unit mass placed at that point.

Therefore, it will need to move towards each other.

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