Electricity and magnetism/Electrostatics/The gradient of a potential

When the work of a force does not depend on the path followed, we say that the force field derives from a potential, because we can calculate the force from the potential by taking its gradient. To understand the concept of gradient, the simplest way is to think about the relief of a mountain. The height h(x,y) of a point on the mountain surface above a point (x,y) in a plane at sea level is a scalar field. We can call it an altitude field. From this scalar field we can define a vector field. For each point (x,y) we define a vector v(x,y) whose direction is the line of greatest slope of the mountain at this point, which is directed upwards and whose length is the slope of the line of greatest slope (if we rise 10 m when we move horizontally 100 m, the slope is equal to 10% = 10/100 = 0.1). The field of the vectors thus defined is the gradient of the altitude field: v = grad h File:Gradient2.svg In these images a scalar field is represented by shades of gray. Its gradient is represented by the arrows. If the scalar field represents the altitude of a relief, the first represents a cone, the second, an inclined plane. Level lines are lines of equal altitude. If h were a potential, we would call them equipotential lines. The lines of greatest slope are the lines that always follow the greatest slope. They are always tangent to the vector field at each point. If h were a potential, we would call them field lines. The lines of greatest slope are always perpendicular to the level lines. Likewise, the field lines are always perpendicular to the equipotential lines. 500px The black lines are field lines, and the brown lines are the equipotential lines of the electric field created by two equal and opposite charges. On a terrain map, adjacent contour lines always indicate the same difference in altitude. They are tighter as the slope is greater. If in the same way we draw adjacent equipotential lines for the same potential difference, the lines are tighter as the field is larger. For a scalar field in a two-dimensional space, we calculate its gradient by taking its two partial derivatives. grad has components and . The gradient grad of a scalar field in three-dimensional space is the field of vectors whose components are , and . By definition of the potential, if we move a standard body by the distance dx, its potential variation is dV = -fx dx, where fx is the component of the force f in the x direction. So fx = . Similarly, fy = and fz = - . The force f on a standard body is the opposite of the gradient of the potential : f = -grad This is the general formula that allows us to calculate a force field from a potential.