What is the boundary condition for E_tangential across a boundary in electrostatics with no time-varying magnetic fields?

What is the boundary condition for E_tangential across a boundary in electrostatics with no time-varying magnetic fields?

• A. E1_tangent = E2_tangent.
• B. E1_tangent = − E2_tangent.
• C. E1_tangent − E2_tangent = σ/ε0.
• D. E1_tangent = 0.

Answer: (A)

In electrostatics with no time-varying magnetic fields, the electric field is curl-free (curl E = 0). This means E behaves as a conservative field, E = −∇V, so the potential is well behaved and, importantly, the tangential behavior must be continuous across a boundary. Imagine a tiny loop that straddles the boundary, with sides parallel and perpendicular to the interface. Faraday’s law tells us the line integral of E around this loop equals −dΦB/dt, which is zero when the magnetic field isn’t changing. The contributions from the sides perpendicular to the boundary vanish as the loop becomes infinitesimal, leaving the tangential contributions on the two sides to cancel. That requires the tangential components of E on each side to be equal: the tangential component is continuous across the boundary. Note that the normal component can jump in the presence of surface charge, but the tangential part remains continuous when there’s no time variation in B.

In electrostatics with no time-varying magnetic fields, the electric field is curl-free (curl E = 0). This means E behaves as a conservative field, E = −∇V, so the potential is well behaved and, importantly, the tangential behavior must be continuous across a boundary.

Imagine a tiny loop that straddles the boundary, with sides parallel and perpendicular to the interface. Faraday’s law tells us the line integral of E around this loop equals −dΦB/dt, which is zero when the magnetic field isn’t changing. The contributions from the sides perpendicular to the boundary vanish as the loop becomes infinitesimal, leaving the tangential contributions on the two sides to cancel. That requires the tangential components of E on each side to be equal: the tangential component is continuous across the boundary.

Note that the normal component can jump in the presence of surface charge, but the tangential part remains continuous when there’s no time variation in B.