the Maxwell stress tensor
Electric and magnetic fields push and pull on charges, but where does that force act, and how is it transmitted through empty space? The Maxwell stress tensor answers this: it describes the flow of momentum in the electromagnetic field, letting you compute the force on any object as a pressure and shear exerted across a surface surrounding it.
The stress tensor is a 3x3 array T_ij = epsilon_0(E_i E_j - (1/2) delta_ij E^2) + (1/mu_0)(B_i B_j - (1/2) delta_ij B^2). Its meaning is mechanical: T_ij is the i-th component of force per unit area transmitted across a surface whose normal points in the j-th direction. The total electromagnetic force on the charges inside a region equals the integral of T over the closed surface bounding it (plus, if the fields vary in time, a term for the rate of change of field momentum inside). Field lines then behave like elastic bands under tension along their length and mutual pressure sideways — which is why parallel currents attract and like charges repel.
The stress tensor turns force calculations into surface integrals, often far easier than summing forces on charges directly, and it is essential wherever momentum balance matters: radiation pressure, the force between magnet poles, the pinch of a plasma. It is the spatial part of a deeper object, the electromagnetic stress-energy tensor of relativity, whose time components are the energy density and the momentum density (the Poynting vector over c^2). Sign conventions vary between textbooks, so always check whether tension lies along or across the field lines in a given definition.
To find the force between the two halves of a uniformly charged sphere, you need not sum Coulomb forces pair by pair. Instead integrate the Maxwell stress tensor over any surface separating the halves: the field's own tension along the field lines delivers the outward force directly, a surface integral replacing a volume of pairwise forces.
T_ij = epsilon_0(E_i E_j - (1/2)delta_ij E^2) + (1/mu_0)(B_i B_j - (1/2)delta_ij B^2): momentum flux through a surface.
T_ij is a momentum flux (force per area transmitted across a surface), NOT a force density — the force on a region is its surface integral, not its value at a point. When fields change in time you must add the rate of change of stored field momentum to get the correct mechanical force.