infinitesimal section of the wire individually. For example, consider the magnetic field of a loop of radius carrying a current For a point a distance along the center line of the loop, the magnetic field vector at that point is:where is the unit vector of along the center-line of the loop (and the loop is taken to be centered at the origin). Loops such as the one described appear in devices like the Helmholtz coil, the solenoid, and the Magsail spacecraft propulsion system. Calculation of the magnetic field at points off the center line requires more complex mathematics involving elliptic integrals that require numerical solution or approximations. There is also a 2D version of the Biot–Savart equation, used when the sources are invariant in one direction. In general, the current need not flow only in a plane normal to the invariant direction and it is given by (current density). The resulting formula is: Electric current density (throughout conductor volume) The formulations given above work well when the current can be approximated as running through an infinitely-narrow wire. If the conductor has some thickness, the proper formulation of the Biot–Savart law (again in SI units) is: where is the vector from dV to the observation point , is the volume element, and is the current density vector in that volume (in SI in units of A/m2). In terms of unit vector Constant uniform current In the special case of a uniform constant current I, the magnetic field is i.e., the current can be taken out of the integral. Point charge at constant velocity In the case of a point charged particle q moving at a constant velocity v, Maxwell's equations give the following expression for the electric field and magnetic field: where is the unit vector pointing from the current (non-retarded) position of the particle to the point at which the field is being measured, and θ is the angle between and . When , the electric field and magnetic field can be approximated as These equations were first derived by Oliver Heaviside in 1888. Some authors call the above equation for the "Biot–Savart law for a point charge" due to its close resemblance to the standard Biot–Savart law. However, this language is misleading as the Biot–Savart law applies only to steady currents and a point charge moving in space does not constitute a steady current. Magnetic responses applications The Biot–Savart