Why are Field Lines ALWAYS Perpendicular to Conducting Surfaces?

Let us consider a charged conductor, such as an electrically charged metal body. The charged conductor produces an electric field that is always perpendicular to its surface. Why?

The electric charges, namely the electrons, inside the conductor can move freely. Charges near the surface of the conductor are called surface charges.

• The surface charges can move freely along the conductor surface, that is, parallel to it.
• In contrast, surface charges cannot move perpendicular to the conductor surface. If they could, they would leave the surface. The conductor would no longer be charged and would therefore not generate an electric field at all.

The electrons inside the conductor generate an electric field \(\class{purple}{\boldsymbol{E}}\). The electric field is a vector quantity. It therefore has a magnitude and a direction. The electrons experience an electric force in the direction of the electric field.

Let us assume that the electric field \(\class{purple}{\boldsymbol{E}}\) generated by the electrons at the surface would not emerge vertically from the surface. As shown in illustration 1, it points in any direction that is not vertical.

Mathematics tells us that we can split every vector, including the electric field vector \(\class{purple}{\boldsymbol{E}}\), into two parts:

• Into a component \(\boldsymbol{E}_{\perp}\) perpendicular to the conductor surface.
• And into a component \(\boldsymbol{E}_{||}\) parallel to the conductor surface.

The electric field \(\boldsymbol{E}_{||}\) parallel to the surface leads to an electric force on the neighboring electrons, so that these electrons move parallel to the surface. The electrons continue to move along the surface until an equilibrium is reached. In this equilibrium, the parallel force due to the mutual repulsion of the electrons is cancelled out and thus the electric field pointing parallel to the surface also disappears. No resulting electric force, therefore also no electric field. If the parallel field component would still be present, the electrons would continue to move along the surface.

The electric field \(\boldsymbol{E}_{\perp}\) perpendicular to the surface, however, is still there. This means: If a charge is placed outside the conductor, it will experience an electric force perpendicular to the surface. It will not move parallel to the conductor surface because there is no parallel field component. The component of the electric field perpendicular to the surface does not cancel out because for this to happen, the surface charges would have to leave the surface.

Thus, the electric field of a charged conductor is always perpendicular to its surface. If you draw the electric field lines, they would also exit the surface perpendicularly.