Etheta

E theta, commonly written as Eθ or E_theta, denotes the theta (azimuthal) component of the electric field in cylindrical coordinates (r, θ, z). If a vector field E is expressed as E = E_r e_r + E_θ e_θ + E_z e_z, then E_θ is the component along the unit vector e_θ, which points in the increasing-θ direction around the axis.

The basis vectors e_r and e_θ depend on θ, with e_r = (cos θ, sin θ, 0) and e_θ = (-sin θ, cos θ, 0) in Cartesian coordinates. As a result, E_θ can vary with position even if the Cartesian components of E are constant.

Contexts and use: E_θ appears in problems with cylindrical symmetry, such as coaxial cables, circular waveguides, and rotating or rotating-plasma configurations. It is one of three components of E in cylindrical coordinates (E_r, E_θ, E_z) and interacts with E_r and E_z in the expressions for curl and divergence in this coordinate system.

In electromagnetism, the tangential component E_θ is relevant for boundary conditions at interfaces; the tangential electric field must satisfy continuity relations at dielectric boundaries and conductor surfaces under appropriate conditions.

Computation note: To obtain E_θ from Cartesian components (E_x, E_y), use E_θ = -E_x sin θ + E_y cos θ, where θ is the azimuthal angle at the point. Equivalently, E_θ is the projection of E onto the e_θ direction.