Force field (physics)

In physics, a force field is a vector field that describes a non-contact force exerted on a particle depending on its location. Specifically, it is a vector field **F**, where **F**(**r**) gives the force a particle would feel if placed at position **r**. **Examples** Gravity is the attractive force between two objects. A gravitational force field represents the influence a massive body (or any amount of energy) extends into the space around it. In Newtonian gravity, a particle of mass M produces a gravitational field **g** = (-GM / r²) **r̂**, where the radial unit vector **r̂** points away from the particle. Near Earth’s surface, the gravitational force on a small mass m is **F** = m**g**, with g being Earth’s gravity. An electric field **E** exerts a force on a point charge q given by **F** = q**E**. In a magnetic field **B**, a moving point charge experiences a force perpendicular to both its velocity and the field direction, described by **F** = q **v** × **B**. **Work** Work depends on both the displacement and the force acting on an object. As a particle moves through a force field along a path C, the work done by the force is a line integral: W = ∫_C **F** · d**r** This value does not depend on the particle’s velocity or momentum along the path. **Conservative force field** For a conservative force field, the work is also independent of the path itself, relying only on the starting and ending points. Thus, the work for an object traveling a closed path is zero, because start and end are the same: ∮_C **F** · d**r** = 0 If the field is conservative, it can be written as the gradient of a scalar potential function: **F** = –∇φ The work done then equals the difference in this potential between the start and end points.