EM-09 · Electrostatics toolkitTool equip

The Gradient

The gradient extracts from a scalar field the direction of steepest ascent and its steepness as one vector. Sweep a test direction to see the climb is steepest when aligned with the gradient, and that the field points the opposite way (E = -∇V).

Find the steepest climb

On the tilted potential terrain, rotate the test arrow. In which direction is the climb steepest? And what is the slope if you follow a contour (an equipotential line)?

Test direction αα = 110°

Drag to orbit. The slider rotates the test direction α.

The climb in this direction

∇V = (∂V/∂x, ∂V/∂y, ∂V/∂z)

E = −∇V · the field is the gradient flipped downhill

Rate in this direction (vs maximum) · 22%

Along contour · no climb

The gradient, steepest ascent

A scalar field gives one number at each point (EM-02, EM-08). The gradient ∇V points in the direction that number grows fastest, and the arrow’s length is the steepness in that direction. On a hilly terrain it is the single arrow pointing the steepest way up. One scalar field gives birth to one vector field.

Observe∇V = (∂V/∂x, ∂V/∂y, ∂V/∂z)

The gradient gathers the rate of change along each axis.

The directional derivative, by dot product

The rate at which the potential changes as you move in a direction û is the directional derivative. Its value is the dot product of the gradient with that direction, ∇V·û (the same dot product as in EM-05). When û aligns with ∇V the cosine is 1, so it is maximal; at a right angle it is zero. So following a contour gives zero change, and going straight along the gradient gives the most. The size of that maximum is exactly |∇V|.

ChoosedV/ds = ∇V · ?

The rate in a direction is the gradient dotted with the unit vector.

Fill in(dV/ds)_max = ?

Aligned with the gradient the rate is maximal, of size |∇V|.

Perpendicular to equipotentials

Since the direction of zero change is the equipotential, the gradient is always perpendicular to equipotential surfaces (just as in EM-08). It is like contour lines meeting the steepest climb at a right angle on a terrain. So the gradient packs both a direction (the normal to the equipotential) and a magnitude (the slope) at once.

E = -∇V, from scalar to vector

The field points where the potential drops fastest, the exact opposite of the gradient: E = -∇V. The minus sign flips uphill into downhill. This one line pulls the entire vector field E out of the scalar field V. Knowing the single potential gives all three field components by differentiation; conversely, integrating the field along a path returns the potential.

On your ownE = ?

The field is the opposite of the potential’s gradient.

Back to the first screen

On the first screen, rotating the test arrow gave a different climb rate in each direction: zero along the equipotential, largest when aligned with the gradient. The gradient ∇V is exactly that maximal direction and magnitude packed together, and the directional derivative ∇V·û says so through a cosine. The field is this gradient flipped downhill, E = -∇V, which is why it was always perpendicular to the equipotentials.

The gradient ∇V extracts from a scalar field V the direction of steepest ascent and its steepness as one vector. Its components are the partial derivatives (∂V/∂x, ∂V/∂y, ∂V/∂z). The rate of change in a direction û is the dot product ∇V·û, maximal (= |∇V|) when û aligns with ∇V. The gradient is perpendicular to equipotentials, and the field is its opposite, E = -∇V.

Tool equipOnce you equip this tool

The gradient is the tool for pulling a vector field out of a scalar field. Once equipped, you get the field E = -∇V straight from the potential V (recovering EM-08) and find the field by differentiation in any problem with a potential. Together with the divergence ∇· (EM-07) and the curl ∇× (EM-16), it forms the three uses of the nabla operator ∇ that underpin the differential form of Maxwell’s equations.